Search This Blog

Loading...

Quantum Gravity - 4 Simple Predictions

Quantum Gravity - 4 Simple Predictions

To make some deep predictions about quantum gravity we first need to play with paper strips!

Take a strip of paper, join the ends, so you have a band. Let's use this as a model for a spin 0 particle.

Now give the paper strip 1 half twist before joining the ends. This is our model for a spin 1/2 particle.

Now give the paper strip 2 half twists before joining the ends. This is our model for a spin 1 particle.

But things get strange..

Now give the paper strip 4 half twists before joining the ends. This is a spin 2 particle. The only one known is the hypothetical graviton, carrier of the gravitational force. But if you play around with this thing for a while it will suddenly fold itself into a double thickness band with 1 half twist!

This implies a graviton (spin 2) can transform into a spin 1/2 particle. Assuming charge is conserved this spin 1/2 particle must be neutral and that means a neutrino or some as yet unknown neutral particle. So a Graviton can oscillate into a spin 1/2 neutral particle.

This is just a simple model, but if graviton oscillation exists the implications are deep. Graviton oscillation would change physics as we know it. Here are a 4 predictions..

1. Black holes evaporate
You could imagine that graviton oscillation requires high graviton pressure - meaning it only occurs in very intense gravitational fields such as black holes. This means black holes evaporate into spin 1/2 neutral particles. If the neutral spin 1/2 particle is a neutrino then black holes are neutrino factories!

2. The Universe is expanding
An asymmetry in the oscillation (meaning graviton to spin 1/2 particle occurs more easily than spin 1/2 particle to graviton) would lead to weakened gravity and this would cause the universe to expand. And the rate of expansion would be a measure of the rate of graviton oscillation.

3. Intense gravitational fields are the source of dark matter
The neutral spin 1/2 particle could account for dark matter i.e. dark matter is produced by the decay of black holes.

4. The Law of Conservation of Angular Momentum is violated
A graviton oscillating into a spin 1/2 particle is a Boson to Fermion transition. Most physicists will hate this because it means the law of conservation of angular momentum is violated. But such a violation could be a neat way to detect graviton oscillation.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Does Graviton Oscillation Exist?

Does Graviton Oscillation Exist?

To make some deep predictions about quantum gravity we first need to play with paper strips!

Take a strip of paper, join the ends, so you have a band. Let's use this as a model for a spin 0 particle.

Now give the paper strip 1 half twist before joining the ends. This is our model for a spin 1/2 particle.

Now give the paper strip 2 half twists before joining the ends. This is our model for a spin 1 particle.

But things get strange..

Now give the paper strip 4 half twists before joining the ends. This is a spin 2 particle. The only one known is the hypothetical graviton, carrier of the gravitational force. But if you play around with this thing for a while it will suddenly fold itself into a double thickness band with 1 half twist!

This implies a graviton (spin 2) can transform into a spin 1/2 particle. Assuming charge is conserved this spin 1/2 particle must be neutral and that means a neutrino or some as yet unknown neutral particle. So a Graviton can oscillate into a spin 1/2 neutral particle.

This is just a simple model, but if graviton oscillation exists the implications are deep. Graviton oscillation would change physics as we know it. Here are a 4 predictions..

1. Black holes evaporate
You could imagine that graviton oscillation requires high graviton pressure - meaning it only occurs in very intense gravitational fields such as black holes. This means black holes evaporate into spin 1/2 neutral particles. If the neutral spin 1/2 particle is a neutrino then black holes are neutrino factories!

2. The Universe is expanding
An asymmetry in the oscillation (meaning graviton to spin 1/2 particle occurs more easily than spin 1/2 particle to graviton) would lead to weakened gravity and this would cause the universe to expand. And the rate of expansion would be a measure of the rate of graviton oscillation.

3. Intense gravitational fields are the source of dark matter
The neutral spin 1/2 particle could account for dark matter i.e. dark matter is produced by the decay of black holes.

4. The Law of Conservation of Angular Momentum is violated
A graviton oscillating into a spin 1/2 particle is a Boson to Fermion transition. Most physicists will hate this because it means the law of conservation of angular momentum is violated. But such a violation could be a neat way to detect graviton oscillation.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Physics Explained in 5 Minutes

Physics Explained in 5 Minutes

In 1665 a recent Cambridge University graduate decided to sit and think about the motion of objects. Isaac Newton had an ambitious goal, he wanted to describe the motion of all objects, from a ball to a planet. He realized that many object move because of gravity and so his thinking included gravity. Within two years he had produced a few simple laws that described the motion of all known objects with great accuracy.

It was a remarkable achievement. Newton's Laws of Motion remained the cornerstone of physics for centuries.

Then in 1905 another recent graduate published a theory that showed something amazing. It showed Newton's equations would fail badly if used to describe objects traveling at very high speed, and it gave more accurate equations. The new equations held a big surprise, they predicted that objects could never go faster than the speed of light. Nature imposes a speed limit!

Einstein's new theory was not just an improvement on Newton's theory, it was a total replacement that gave a deep insight into physics and the world it describes. And all this from a physics student who was average in university and could not find a job when he graduated! Albert Einstein's theory became known as "Relativity". It took just a few years of thinking. It changed physics forever.

So now all was good, Physicists could predict the motion of objects with remarkable accuracy. Some even claimed there was nothing else to do and physics was over!

But nature had other ideas. By the early 1900s experimental physicists were discovering new objects. These new objects were confusing, but they all had one thing in common - they were extremely small. At first many scientists refused to believed they existed, but soon atoms, electrons and photons became accepted.

Of course, describing the motion of these new objects was easy - just use Newton's theory, or if you wanted more accuracy use Einstein's theory. Right? Wrong. When this was tried the results were terrible.

New laws were needed. But this time they were not produced by a recent graduate. It took a generation of physicists, each contributing critical parts to the puzzle, and it was not finished until the 1950s. It was a long hard slog. The laws, designed specifically for small objects, became known as Quantum Mechanics. The mathematics was complex but the accuracy was there. The theory was incredibly accurate!

If you had to summarize Quantum Mechanics in one sentence try this.. "small objects behave very differently than large objects". Who knew!

What happened to Einstein?
He never made the transition to the new world of Quantum Mechanics. He understood the mathematics and even made some critical contributions in the early days, but he rejected its underlying philosophy. However, he had one more giant trick up his sleeve. In 1915, ten years after producing his Relativity theory be produced a much broader theory that included gravity - the thing that got Newton started 250 years earlier. His new theory was called "General Relativity". It's still in use today over 100 years later, because nobody has found a better description of gravity.

What happened to Quantum Mechanics?
It got more complex. The laws that describe atoms, electrons and photons had to be revised to describe the structure of new objects like protons. The basic principles were the same, but the mathematics was way more complex. It turned out that Quantum Mechanics was not a simple theory.

So physics is built on three massive achievements: Newton's Laws, Einstein's General Relativity and Quantum Mechanics. The first two were produced by single individuals who's names became legend. The third required a huge team - a whole generation of physicists.

Where does physics stand today?
Using giant accelerators such as the LHC at CERN physicists are finding even smaller objects and there are hints that Quantum Mechanics may have problems describing them. Not only that, but physicists want to unify General Relativity and Quantum Mechanics into one theory.

Perhaps we need a new physics graduate - preferably one who can't find a job!

Content written and posted by Ken Abbott abbottsystems@gmail.com

Primes Within Primes

Primes Within Primes

If you write a prime number in binary you can sometimes split it into 2 segments each beginning and ending with 1 that are also primes.

For example, the prime number 7591 is 1110110100111 in binary. Now snip out the first segment 11101 which is 29 and prime. Then snip out the second 10100111 which is 167 and prime.

So, denoting a binary string concatenation operator by "+" we can say 7591=29+167

Notice that our concatenation operator depends on order, so n+m is not the same as m+n. Mathematicians call this kind of operator "non-commutative".

Let's do a simple example with our new operator..

In binary 3 is 11 and 5 is 101 so 3+5=11101 which is 29.

But 5+3=10111 which is 23.

By the way, notice that all these numbers 3, 5, 23, 29 are prime!

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Collatz Conjecture as a Computer Program

Collatz Conjecture as a Computer Program

One of the most famous unsolved mathematical conjectures totally lends itself to computer investigation.

It's the Collatz Conjecture, named after Lothar Collatz, who first proposed it in 1937. The great mathematician Stanisław Ulam not only failed to prove it but said, "perhaps mathematics is not ready for such problems".

Here it is as a computer program..

Pick any positive integer n
INFINITE LOOP
If n is even replace it by n/2
If n is odd replace it by 3*n+1
If n=1 bail out of loop
LOOP

The Collatz Conjecture says that no matter what number you start with you'll always bail out of the loop. In other words, no matter what number you start with you'll always reach 1.

The number of cycles needed to reach 1 is called the stopping time of n and denoted s(n). It turns out that the stopping time of a number is an interesting property and by no means simple.. for example s(27)=111

Mathematical statements phrased in terms of iteration seem to be especially nasty to prove. Perhaps Ulam was correct.. meaning mathematics was never designed for such problems!

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Learn Atomic Physics in 5 Minutes

Learn Atomic Physics in 5 Minutes

After a century of hard work physicists have established some impressive facts about atoms.

If you could make yourself amazingly small and walk up to an atom the first thing you hit would be a cloud of electrons whizzing at great speed, so they look like a blur. The electron was the first elementary particle to be discovered and it's still as elementary as ever. Elementary means no internal structure has been detected - so far.

You plow your way through the cloud of electrons and finally come to a peaceful empty space. You travel this for ages and then on the horizon you see a small object. This is the nucleus of the atom. As you get closer you see that it has structure. It consists of two type of particles - protons and neutrons - bound together very tightly.

The number of protons in the nucleus is exactly equal to the number of electrons you passed in the outer cloud, but the number of neutrons in the nucleus can vary.

For a while physicists thought protons and neutrons were elementary, but not so. It turns out they are complex objects with internal structure. Plus, the neutron is very similar to the proton. So physicists built particle colliders to learn more about the structure of the proton. The most famous is the LHC (Large Hadron Collider) at CERN. The technique is simple, smash two protons together an incredible speed and see what comes out. From this physicists can deduce what's inside the proton.

It seems a lot is going on inside the proton. It's built from more elementary particles called quarks, bound together by an incredible strong force field generated by particles called gluons.

Given than the proton is amazingly small why would nature decide to give it such complex internal structure?

That's a philosophical question physicists can never answer. But they are working hard to document the internal structure of the proton.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Learn Differential Calculus in 5 Minutes

Learn Differential Calculus in 5 Minutes

Differential calculus is one of the two branches of calculus, the other is integral calculus. Most mathematicians refer to both branches together as simply calculus.

Calculus is all about functions, so there's no point in studying calculus until you understand the idea of a function.

Let's take a simple function, say f(x)=x^2

What's the value of this function at a specific point, say x=a? That's easy, it's f(a)=a^2. But now we ask an interesting question, can we possibly know anything else about the function at point a? At first glance this seems impossible, the value of the function at a is f(a), so surely that's all we can know, right? Wrong. It turns out there's a process called "differentiation" that can tell us more. Here's how it works..

Take a very small number, say q, and ask what the function is doing at a+q, in other words at a point very close to a..

f(a+q)=(a+q)^2=a^2+2*a*q+q^2

But we can make q as small as we please, which means q^2 is much smaller, so to a good approximation we can ignore it, and we get..

f(a+q)=(a+q)^2=a^2+2*a*q

Notice the first term, a^2, is just the value of the function at a, f(a), so now we have..

f(a+q)=f(a)+2*a*q

Which means..

f(a+q)-f(a)=2*a*q

And so..

(f(a+q)-f(a))/q=2*a

What is the meaning of the expression on the left? If you draw a diagram you'll see that the term on the left is simply the slope of the curve f(x) close to x=a. So this gives us some valuable information about what's going on near a. Now all we need to do is keep making q smaller so we get closer and closer to a. In fact, we can use the concept of a limit to say..

Limit(f(a+q)-f(a))/q as q goes to zero is 2*a

Of course we could do this for any point a, so in general..

Limit(f(x+q)-f(x))/q as q goes to zero is 2*x

This is called the "derivative" of f(x) and is often written as df/dx, or sometimes as f'. So, to summarize..

The derivative of the function f(x)=x^2 is 2*x and is written df/dx=2*x and it's the slope of the f(x) curve at x. Of course, a slope is simply a rate of change, so we can also say that df/dx=2*x is the rate of change of the function f(x).

Congratulations, you just did some calculus! You differentiated the function f(x)=x^2 and got the result 2*x

To generalize this example, the derivative of the function f(x)=x^n where n is any integer is..

df/dx=n*x^(n-1)

So for example, if f(x)=x^10 then the derivative is df/dx=10*x^9

So, the essence of differential calculus is this.. in addition to knowing the value of a function f(x) at x=a we also know the rate of change (slope) of the function at a. Differential calculus gives us an extra piece of information!

Much of differential calculus is simply finding ways to differentiate functions. This can get boring, so why bother? Because the derivative of a function is a really useful thing for solving all sorts of problems. It's especially useful in physics and many laws of physics are written as differential equations.

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Simple Harmonic Oscillator - Using Difference Calculus

Simple Harmonic Oscillator - Using Difference Calculus

The Simple Harmonic Oscillator is a famous system in physics. Its equation of motion is written as a second order differential equation which is then solved to give the characteristic "wave" solution. But there's an alternate method which does not need differential equations!

Consider a function f(n) of an integer variable n defined by this..

f(n)=k*f(n-1)-f(n-2)

where n=2,3,4,5,.. and k is a constant.

If k, f(0) and f(1) are given then f(n) can be calculated for any n.

This equation is an example of a difference equation. An area of mathematics called the Theory of Finite Differences or Difference Calculus tells how to solve these equations. The solution is a nice surprise..

f(n)=sin(n*a)

It's the famous sine function, where a is a constant and k=2*cos(a). So the solution is a wave? Yes, if you plot f(n) for n=2,3,4,5,... you get the beautiful sine wave, and the three numbers k, f(0) and f(1) determine the amplitude, wavelength and phase of the wave.

What about n? It plays the role of time, because at time=n the function f(n) is the displacement from the origin for a Simple Harmonic Oscillator.

So the difference equation f(n)=k*f(n-1)-f(n-2) replaces the second order differential equation used to describe the Simple Harmonic Oscillator. It's a nice example of Difference Calculus in action.

Of course, you may have noticed that things are not exactly the same.. time is no longer a continuous variable!

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

AMS - American Mathematical Society

AMS - American Mathematical Society

The AMS (American Mathematical Society) is an professional society who's goal is to advance mathematical research, scholarship and education worldwide. Their headquarters are in Providence, Rhode Island.

AMS Website

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

MAA - Mathematical Association of America

MAA - Mathematical Association of America

The MAA (Mathematical Association of America) is a professional society who's goal is to advance the mathematical sciences, especially at the university level.

It has a broad range of members, from high school students to professional mathematicians. Anyone is invited to join!

MAA Website

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Algebra Explained in 5 Minutes

Algebra Explained in 5 Minutes

Consider this problem, "what number, when added to 5, gives the result 21".

Instead of a sentence, this problem can be written much shorter and clearer as an equation, like this..

5+x=21

where x denotes the number we are trying to find.

Of course, we could also write it as x+5=21 and this is exactly the same equation. Or we could write 21=x+5 which is of course the same thing.

If we manage to find x we say that we've "solved" the equation. Can we solve this equation? Well, we could guess a few numbers for x and try them out. Does x=9 work? Let's see, 5+9=14, so x=9 is not a solution. After a few tries we get the solution, which is x=16.

Guessing a solution is perfectly fine, but it's very time consuming, especially for more complex equations. Of course, we could program a high speed computer to guess solutions and try them out ultra fast until we finally hit on the right solution. And for some very tough equations this is indeed the method used. But this method has a huge flaw.. if it fails to find a solution it does not mean the equation has no solution. That's because even the fastest computer can only make a limited number of tries.. and the actual solution may be something we never get around to trying.

So, coming back to our equation 5+x=21 we should ask if there is a foolproof method that's guaranteed to find the solution. The answer is yes, and it's all about the = sign. Once you truly understand this simple sign solving the equation is easy. The = sign is the secret to learning Algebra!

So what does this sign really mean? It means the "object" on the left of the sign is the same exact object as that on the right. They are the same thing.. exactly the same thing. They are the same exact mathematical object but just written in different ways. So there's really only one object!

OK, so our equation says that 5+x is exactly the same object as 21. So, if I do something to 5+x and then I do the same thing to 21 the results will still be equal. Cool. So lets subtract 5 from 5+x to get the result x. Now do the same exact thing to the other side, I'll subtract 5 from 21 to get the result 16. But these two results must be the same, so I can write them as equal to each other, that is x=16.

Bingo, we've solved the equation without any guessing!

Also, I'm not sure if you noticed this, but we just did some basic Algebra. Don't let Algebra intimidate you, it's just the art of manipulating equations until you get what you want!

Let's look at a slightly more complicated example..

3*x+2=17

To solve it we want to isolate x on one side and get all the other stuff over to the other side. Here's a method I use. It's exactly the same technique as above, but it's faster and easier to handle. Or at least I think so, and I've used it over the years to do massive amounts of algebra!

First move the 2 over to the other side. It was adding, so when it moves over it subtracts, like this..

3*x=17-2=15

Now move the 3 over. It was multiplying, so when it moves over it divides, like this..

x=15/3=5

This technique is quite general and can be used for any equation. But notice that the order in which you do things is important. For example, you need to get the 2 over to the other side before you can handle the 3.

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Teaching Prime Numbers - A Different Way

Teaching Prime Numbers - A Different Way

Let's consider the positive integers greater than 1, that is 2,3,4,5,..

Suppose we are given the first integer and asked to make all other integers using only the multiply operation.

We soon run into problems because 2*2=4 and we have no way to make 3.

OK, we just add 3 to our set of given numbers g, so now g={2,3}

Can we make 4? Yes, 2*2=4

Can we make 5? No, all our tries fail, so we add 5 to our set of given number g={2,3,5}

Can we make 6? Yes, 2*3=6

Can we make 7? No, all our tries fail, so we add 7 to our given numbers g={2,3,5,7}

Can we make 8? Yes, 2*2*2=8

Can we make 9? Yes, 3*3=9

Can we make 10? Yes, 2*5=10

Can we make 11? No, so we add it to the set g={2,3,5,7,11}

Can we make 12? Yes, 2*2*3=12

Can we make 13? No, so add it to the set g={2,3,5,7,11,13}

Can we make 14? Yes, 2*7=14

Can we make 15? Yes, 3*5=15

Can we make 16? Yes, 2*2*2*2=16

What is the set g that we are generating by this process? It's the set of prime numbers! This is simply another way to explain prime numbers.

It's a nice demonstration because it shows how prime numbers generate all numbers using only the multiply operation. You can also see that as g gets bigger we can obviously make more numbers from it, so prime numbers become less and less frequent.

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Numbers Bigger Than Infinity

Numbers Bigger Than Infinity

The prefix "trans" means "beyond". So a transfinite number is one that's beyond the finite. There's only one and that's infinity, right? Wrong. It turns out there are many transfinite numbers. The concept of infinity is just a general concept, and the real mathematics is the study of transfinite numbers.

This work is due to Georg Cantor, who showed that there are many types of infinity, and some are bigger than others! He even developed an arithmetic for working with transfinite numbers. He denoted them by the Hebrew letter "aleph".

His work stands as one of the most elegant pieces of mathematics ever.

So what did Cantor do?

He formalized counting. He started with the integers {1,2,3,...} and asked what other sets could be placed in 1-to-1 correspondence with the integers. Instead of just saying there are an infinite amount of integers he denoted the number of integers by aleph0 and developed an arithmetic that in many ways treated aleph0 as a regular number. But he went further..

He showed that the rational numbers (fractions) could be placed in 1-to-1 correspondence with the integers. So counterintuitively, there are only as many rational numbers as there are integers. Not more!

But when it comes to irrational numbers, there are many more. He called this number aleph1 and he showed that it was different and bigger than aleph0. He proved that the number of subsets of the set of integers {1,2,3,...} is also aleph1 and he produced this amazing result..

aleph1=2^aleph0

He even asked if there was an aleph number between aleph0 and aleph1.

During his lifetime Cantor was ridiculed, not by the general public, but by his fellow mathematicians. Today his work is regarded as brilliant and is taught as part of the standard university mathematics curriculum.

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Black Holes - A Strange Prediction

Black Holes - A Strange Prediction

To make this prediction we first need to play with paper strips!

Take a strip of paper, join the ends, so you have a band. This is a model for a spin 0 particle.

Now give the paper strip 1 half twist before joining the ends. This is a model for a spin 1/2 particle.

Now give the paper strip 2 half twists before joining the ends. This is a spin 1 particle.

But things get strange..

Now give the paper 4 half twists before joining the ends. This is a spin 2 particle. The only one known is the hypothetical graviton, carrier of the gravitational force. But if you play around with this thing for a while it will suddenly flip into a double thickness band with 1 half twist!

This implies a graviton (spin 2) can transform into a spin 1/2 particle. Assuming charge is conserved this spin 1/2 particle must be neutral and that means a neutrino or some as yet unknown particle. So a Graviton can oscillate into a neutrino.

This is just a simple model, but if graviton oscillation exists the implications are deep. Graviton oscillation would change physics as we know it. Here are a few predictions..

Black holes evaporate
You could imagine that graviton oscillation requires high graviton pressure - meaning it only occurs in very intense gravitational fields such as black holes. This means black holes evaporate into spin 1/2 neutral particles.

Black Holes are an intense source of neutrinos
Assuming the neutral spin 1/2 particle is a neutrino then areas of intense gravity (such as black holes) will emit neutrinos. Black Holes are neutrino factories.

The Universe is expanding
An asymmetry in the oscillation (meaning graviton to spin 1/2 particle occurs more frequently than spin 1/2 particle to graviton) would lead to weakened gravity and this would cause inflation. Of course, the rate of inflation need not be constant.

Intense gravitational fields are the source of dark matter
Could the neutral spin 1/2 fermion particle account for dark matter? i.e. dark matter is produced by the decay of black holes.

The Law of Conservation of Angular Momentum is violated
A graviton oscillating into a spin 1/2 particle is a Boson to Fermion transition. Most physicists will hate this because it means the law of conservation of angular momentum is violated. But such a violation may be a way to detect graviton oscillation.

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Sets Explained in 5 Minutes

Sets Explained in 5 Minutes

In mathematics a set is just a collection of distinct objects. What type of objects? Any type. Of course you need a clear way to specify how an object belongs to a set.

Let's consider a simple example, the set containing the first 3 letters of the alphabet..

S={a,b,c}

Can we say anything mathematically interesting about this set?

Well, it's a finite set and contains 3 members. There's also a very clear rule to decide if an object belongs to the set. But we can do more, we can apply a mathematical technique to generate more structure. This technique is very simple, but turns out to be incredibly powerful. It's this..

Once you've defined something ask if contains things like itself. In this case we've defined a set S, so we ask if it contains any subsets. A subset of S is just another set made from the same objects. It's called a subset because you can think of it as contained inside S.

{a} is a subset, so is {a,c}, so is {b,c}

How many subsets does S have in total? In the count we'll include the null set { } which contains nothing and we'll also count the set itself {a,b,c} which contains everything. So here are all the subsets of S..

{ }
{a}
{b}
{c}
{a,b}
{a,c}
{b,c}
{a,b,c}

There are 8 in total, which just happens to be 2^3 where 3 is the number of objects in S. This is no coincidence. If our set contained n objects the number of subsets would be 2^n. This number gets big fast. For example..

A set with 26 members (such as the 26 letters of the alphabet) has 2^26=67108864 subsets. So from just 26 objects we can easily generate 67108864 new objects!

If you don't like the idea of counting { } and {a,b,c} as subsets then just say the number of subsets is (2^n)-2 and this makes almost no difference in the count. In the above example the number of subsets would be 67108862. Mathematicians call these subsets the proper subsets.

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Quantum Mechanics Explained in 5 Minutes

Quantum Mechanics Explained in 5 Minutes

If I was forced to summarize Quantum Mechanics in one sentence I would say this.. "small objects behave very differently than big objects".

It sounds like an innocent statement. But it's not. It's a profound discovery of how nature works.

An example will help..

Spin is something we're all familiar with. We can make any object spin. And not just a top, spin is used to great effect in many sports such as tennis, baseball and cricket. And of course you could never throw a frisbee without spin.

We can spin an object at any speed we please, and as soon as it starts spinning it defines an axis about which the spin occurs.

But that's the spin of a big object, meaning an object we can handle. What about the spin of a really small object such as an electron?

It turns out the electron spins just like a tiny top - but with two big surprises..

- What about the speed of spin?
The electron spins at a fixed rate that can never be changed. There is no known process that can change the spin rate of the electron. This makes electron spin a fundamental quantity.

- What about the axis of spin?
You can measure the spin along any axis you want and you'll always get the same result, +1/2 or -1/2. These two values correspond to the electron spinning clockwise or counter clockwise. The electron behaves as if it's spinning about every axis at the same time!

So electron spin is totally different than the spin of a big object such as a top.

Physicists call the electron a "spin 1/2 particle". And it's not just the electron, all elementary particles have spin except for the recently discovered Higgs boson.

If you plan to study Quantum Mechanics pay attention to spin. It's a wonderful example of how "small objects behave very differently than big objects".

Content written and posted by Ken Abbott abbottsystems@gmail.com

Goldbach Conjecture Explained in 5 Minutes

Goldbach Conjecture Explained in 5 Minutes

In 1742, the German mathematician Christian Goldbach, in a discussion with the mathematician Leonhard Euler, made a simple statement..

Every even integer greater than 2 can be written as the sum of two prime numbers.

Mathematicians have tried to prove this ever since. None have. It's a great example of how a simple statement in mathematics can be amazingly difficult to prove. Computers have checked billions of numbers and shown it to be true for every number tested, but that's not the same as a proof. A proof would show it to be true for all even numbers, period.

It's easy to prove that every integer can be written as a product of primes. This is called the prime decomposition of an integer. So for any integer m we have.. m=(p1^s1)*(p2^s2)*......*(pn^sn)

where p1,p2,...,pn are prime numbers and the s1,s2,s3,...,sn are just integer powers and represent the simple fact that primes may be repeated. By collecting like primes together and raising them to a power we make sure that p1,p2,p3,...,pn are distinct primes with no duplication.

The condition that m be even simply means that one of these primes must be 2. Since the order of multiplication does not matter we can make p1=2. So m now looks like this..

m=(2^s1)*(p2^s2)*...*(pn^sn)

Goldbach's Conjecture says that when m is even there exists two prime numbers, let's call them g1 and g2, such that..

m=(2^s1)*(p2^s2)*...*(pn^sn)=g1+g2

Are we on our way to a proof? No, but it's still fun to try!

Perhaps there's some deep clue in the fact that Goldbach's Conjecture only works if p1=2. In other words, if 2 does not appear in the prime decomposition of an integer then Goldbach's Conjecture does not work. What's so special about the number 2?

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Calculus - Explained in 5 Minutes

Calculus - Explained in 5 Minutes

Differential calculus is one of the two branches of calculus, the other is integral calculus. It's best to learn differential calculus first. So here's the scoop..

Calculus is all about functions, so there's no point in studying calculus until you understand the idea of a function.

Let's take a simple function, say f(x)=x^2

What's the value of this function at a specific point, say x=a? That's easy, it's f(a)=a^2. But now we ask an interesting question, can we possibly know anything else about the function at point a? At first glance this seems impossible, the value of the function at a is f(a), so surely that's all we can know, right? Wrong. It turns out there's a process called "differentiation" that can tell us more. Here's how it works..

Take a very small number, say q, and ask what the function is doing at a+q, in other words at a point very close to a..

f(a+q)=(a+q)^2=a^2+2*a*q+q^2

But we can make q as small as we please, which means q^2 is much smaller, so to a good approximation we can ignore it, and we get..

f(a+q)=(a+q)^2=a^2+2*a*q

Notice the first term, a^2, is just the value of the function at a, f(a), so now we have..

f(a+q)=f(a)+2*a*q

Which means..

f(a+q)-f(a)=2*a*q

And so..

(f(a+q)-f(a))/q=2*a

What is the meaning of the expression on the left? If you draw a diagram you'll see that the term on the left is simply the slope of the curve f(x) close to x=a. So this gives us some valuable information about what's going on near a. Now all we need to do is keep making q smaller so we get closer and closer to a. In fact, we can use the concept of a limit to say..

Limit(f(a+q)-f(a))/q as q goes to zero is 2*a

Of course we could do this for any point a, so in general..

Limit(f(x+q)-f(x))/q as q goes to zero is 2*x

This is called the "derivative" of f(x) and is often written as df/dx, or sometimes as f'. So, to summarize..

The derivative of the function f(x)=x^2 is 2*x and is written df/dx=2*x and it's the slope of the f(x) curve at x. Of course, a slope is simply a rate of change, so we can also say that df/dx=2*x is the rate of change of the function f(x).

Congratulations, you just did some calculus! You differentiated the function f(x)=x^2 and got the result 2*x

To generalize this example, the derivative of the function f(x)=x^n where n is any integer is..

df/dx=n*x^(n-1)

So for example, if f(x)=x^10 then the derivative is df/dx=10*x^9

So, the essence of differential calculus is this.. in addition to knowing the value of a function f(x) at x=a we also know the rate of change (slope) of the function at a. Differential calculus gives us an extra piece of information!

Much of differential calculus is simply finding ways to differentiate functions. This can get boring, so why bother? Because the derivative of a function is a really useful thing for solving all sorts of problems. It's especially useful in physics and many laws of physics are written as differential equations.

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Prime Numbers Explained in 5 Minutes

Prime Numbers Explained in 5 Minutes

A prime number is a positive integer that has no divisors except for 1 and itself. If an integer is not prime it's called composite. Mathematicians do not regard 1 as a prime number, so the first prime number is 2.

Here are two equivalent statements..

If n is prime then a set of n objects cannot be split into subsets of equal size.

If n is prime then it can never be written as the product of two integers except for the trivial case 1*n.

Beginners sometimes get confused by the term "divisor". When mathematicians say that one integer divides another integer they mean with no remainder.

Prime numbers are fundamental numbers due to this fact..

Any integer greater than 2 can be written as a product of prime numbers. So you can think of prime numbers as the "building blocks" from which all numbers are made.

Here are the first 100 prime numbers. There is no known formula that produces the prime number sequence. Primes are indeed mysterious numbers!

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Algebra Explained in 5 Minutes

Algebra Explained in 5 Minutes

Consider this problem, "what number, when added to 5, gives the result 21".

Instead of a sentence, this problem can be written much shorter and clearer as an equation, like this..

5+x=21

where x denotes the number we are trying to find.

Of course, we could also write it as x+5=21 and this is exactly the same equation. Or we could write 21=x+5 which is of course the same thing.

If we manage to find x we say that we've "solved" the equation. Can we solve this equation? Well, we could guess a few numbers for x and try them out. Does x=9 work? Let's see, 5+9=14, so x=9 is not a solution. After a few tries we get the solution, which is x=16.

Guessing a solution is perfectly fine, but it's very time consuming, especially for more complex equations. Of course, we could program a high speed computer to guess solutions and try them out ultra fast until we finally hit on the right solution. And for some very tough equations this is indeed the method used. But this method has a huge flaw.. if it fails to find a solution it does not mean the equation has no solution. That's because even the fastest computer can only make a limited number of tries.. and the actual solution may be something we never get around to trying.

So, coming back to our equation 5+x=21 we should ask if there is a foolproof method that's guaranteed to find the solution. The answer is yes, and it's all about the = sign. Once you truly understand this simple sign solving the equation is easy.

So what does this sign really mean? It means the "object" on the left of the sign is the same exact object as that on the right. They are the same thing.. exactly the same thing. They are the same exact mathematical object but just written in different ways. So there's really only one object!

OK, so our equation says that 5+x is exactly the same object as 21. So, if I do something to 5+x and then I do the same thing to 21 the results will still be equal. Cool. So lets subtract 5 from 5+x to get the result x. Now do the same exact thing to the other side, I'll subtract 5 from 21 to get the result 16. But these two results must be the same, so I can write them as equal to each other, that is x=16.

Bingo, we've solved the equation without any guessing!

Also, I'm not sure if you noticed this, but we just did some basic Algebra. Don't let Algebra intimidate you, it's just the art of manipulating equations until you get what you want!

Let's look at a slightly more complicated example..

3*x+2=17

To solve it we want to isolate x on one side and get all the other stuff over to the other side. Here's a method I use. It's exactly the same technique as above, but it's faster and easier to handle. Or at least I think so, and I've used it over the years to do massive amounts of algebra!

First move the 2 over to the other side. It was adding, so when it moves over it subtracts, like this..

3*x=17-2=15

Now move the 3 over. It was multiplying, so when it moves over it divides, like this..

x=15/3=5

This technique is quite general and can be used for any equation. But notice that the order in which you do things is important. For example, you need to get the 2 over to the other side before you can handle the 3.

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Mobius Strip - A Big Surprise

Mobius Strip - A Big Surprise

Take a strip of paper, join the ends so you have a band. This is a very simple object with 2 surfaces.

Take a strip of paper, but give it 1 half twist before joining the ends. This is the famous Mobius Strip. Despite how it looks it only has 1 surface. Does a half twist clockwise give the same object as a half twist counter clockwise, or are these two different objects?

Take a strip of paper, but give it 2 half twists before joining the ends. This has 2 surfaces. Same question, does 2 half twists clockwise give the same object as 2 half twists counter clockwise, or are these two different objects?

But things get strange fast..

Take a strip of paper, but give it 4 half twists before joining the ends. This has 2 surfaces. Now play around with it for a while. At some point it will suddenly "flip" into a double thickness band with 1 half twist. In other words if flips into a double thickness Mobius Strip. One surface has gone, and so have 3 half twists!

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Gravity Makes News

Gravity Makes News

First the LHC (Large Hadron Collider) folks announced that both the CMS and ATLAS detectors saw a faint blip around 1750 Gev. Speculation abounds, but graviton production is one of the theories. A graviton is the hypothetical particle that carries the gravitational force. More data is required before the blip reaches the 5-sigma level of probability needed to declare a new particle. But gravity!

Then LIGO (Laser Interferometer Gravitational-Wave Observatory) announced the detection of Gravitational Waves. This not only confirms Einstein's prediction from 1915 but opens a whole new view into the Universe.

Both machines have taken decades to build, but the hard work is now starting to payoff big time.

A new era in Physics is about to begin!

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Mirror Primes

Mirror Primes

A formula that generates the prime number sequence is the "holy grail" of mathematics. It's never been found.

But here's a bit of fun..

Write a prime number in binary using the "left to right" convention. The first 20 primes are listed below.

For example 11=1101

Now reflect 11 in a mirror on the x-axis thus..

1101 | 1011

where | donates the mirror.

11 is prime, but it's mirror reflection 1011 (13) is also prime. So 11 and 13 are mirror primes. They are also a prime pair of course.

Some primes are their own mirror, for example 5 is its own mirror, and so is 7.

Of the first 18 primes 14 are mirror primes!

But a word of caution. Be very careful about thinking you've found a property that applies to all primes just because it works for some examples you test.

2 01
3 11
5 101
7 111
11 1101
13 1011
17 10001
19 11001
23 11101
29 10111
31 11111
37 101001
41 100101
43 110101
47 111101
53 101011
59 110111
61 101111
67 1100001
71 1110001

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Solving the Simple Harmonic Oscillator without Calculus

Solving the Simple Harmonic Oscillator without Calculus

The Simple Harmonic Oscillator is a famous example in physics. Its equation of motion is written as a second order differential equation which is then solved to give the characteristic "wave" solution. But there's an alternate method which does not need differential calculus!

Consider a function of an integer variable defined by this..

f(n)=k*f(n-1)-f(n-2)

where n=2,3,4,5,.. and k is a constant.

If k, f(0) and f(1) are given then f(n) can be calculated for any n.

This equation is an example of a difference equation. An area of mathematics called the Theory of Finite Differences or Difference Calculus tells how to solve these equations. The solution is a nice surprise..

f(n)=sin(n*a)

It's the famous sine function, where a is a constant and k=2*cos(a). So the solution is a wave? Yes, if you plot f(n) for n=2,3,4,5,... you get the beautiful sine wave, and the three numbers k, f(0) and f(1) determine the amplitude, wavelength and phase of the wave.

What about n? It plays the role of time, because at time=n the function f(n) is the displacement from the origin for a Simple Harmonic Oscillator.

So the difference equation f(n)=k*f(n-1)-f(n-2) replaces the second order differential equation used to describe the Simple Harmonic Oscillator. It's a nice example of Difference Calculus in action.

Of course, you may have noticed that things are not exactly the same.. time is no longer a continuous variable!

Like this post? Please click G+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com

Newton's Law of Gravitation - Turned Inside Out

Newton's Law of Gravitation - Turned Inside Out

Newton's famous formula for the gravitational force f between two masses m and q at a distance r apart is f=G*m*q/(r^2).

The really interesting thing about this formula is not G,m,q or r but the number 2. Why 2?

If we write f=G*m*q/(r^x) and solve for x we get x=log(G*m*q/f)/log(r). But this is an experimentally measurable quantity! Why would we even expect it to be an integer let alone the integer 2?

Like this post? Please click G+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com

Prime Power Representation of Integers

Prime Power Representation of Integers

There are lots of ways to represent numbers, base 10 and base 2 (binary) being examples. This one is a bit more sophisticated..

Every integer can be written as a product of primes. In fact, if p1,p2,p3,... is the prime number sequence (i.e. 2,3,5,7,..) then the general expression for an integer m is..

m=(p1^s1)*(p2^s2)*......*(pn^sn)

where s1,s2,s3,...,sn are integer powers.

Of course, if a prime pj is not involved in the prime decomposition we still include it, but we set sj=0 so pj^(sj)=1

So, any integer can be uniquely represented by its sequence or powers {sj}. Mathematicians call this representation the canonical representation. Canonical is just a fancy term for "standard".

An example will help..

567=(2)^0*(3)^4*(5)^0*(7)^1 so the prime power representation of 567 is the sequence {0,4,0,1}

Like this post? Please click G+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com

Do Black Holes Evaporate?

Do Black Holes Evaporate?

To draw conclusions about black holes we first need to play with paper strips!

Take a strip of paper, join the ends, so you have a band. This is a model for a spin 0 particle.

Now give the paper 1 half twist before joining the ends. This is a model for a spin 1/2 particle.

Now give the paper 2 half twists before joining the ends. This is a spin 1 particle.

But things get strange..

Now give the paper 4 half twists before joining the ends. This is a spin 2 particle. The only one is the hypothetical graviton, carrier of the gravitational force. But if you play around with this thing for a while it will suddenly flip into a double thickness band with 1 half twist!

This implies a graviton (spin 2) can transform into a spin 1/2 particle. Assuming charge is conserved this spin 1/2 particle must be neutral and that means a neutrino or some as yet unknown particle. So a Graviton can oscillate into a neutrino. This also means Boson <> Fermion transitions are possible.

This is just a simple model, but if graviton oscillations exist the implications are deep. Graviton oscillations would change physics as we know it. Here are a few possible implications..

Black holes evaporate.
You could imagine that graviton oscillation requires high graviton pressure - meaning it only occurs in very intense gravitational fields such as black holes. This means black holes evaporate into spin 1/2 neutral particles.

If the spin 1/2 neutral particle is a neutrino then Black Holes are an intense source of neutrinos.

The Universe is expanding.
An asymmetry in the oscillation (meaning Boson to Fermion is easier than Fermion to Boson) would lead to weakened gravity and this would cause inflation. Of course, the rate of inflation need not be constant.

Intense gravitational fields are the source of dark matter.
Could the neutral spin 1/2 fermion particle account for dark matter? i.e. dark matter is produced by the decay of black holes.

The Law of Conservation of Angular Momentum is violated.
Boson <--> Fermion oscillations imply the law of conservation of angular momentum is violated. Most physicists will not like this, to say the least.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Numbers in Binary

Numbers in Binary

We're used to seeing numbers represented in base 10 "decimal" notation, but we can represent numbers in any base we please. In base 10 we use 10 symbols 0,1,2,3,...,9 and in base n we use n symbols 0,1,2,3,...,(n-1)

The simplest base is 2, because in that base we have only two symbols 0,1

Base 2 is also call "binary" and writing numbers in base 2 makes them look like computer data.

In binary the positions represent 1,2,4,8,16,32,.. so for example, the number 11 in binary is 1101, and this simply means..

11=(1)*1+(1)*2+(0)*4+(1)*8

Here's the first 20 prime numbers in binary..

2 01
3 11
5 101
7 111
11 1101
13 1011
17 10001
19 11001
23 11101
29 10111
31 11111
37 101001
41 100101
43 110101
47 111101
53 101011
59 110111
61 101111
67 1100001
71 1110001

Here's pi in binary..

pi=11.00100100 00111111 01101010 10001000 10000101 10100011 00001000 11010011 00010011 00011001 10001010 00101110 00000011 01110000 01110011 01000100 10100100 00001001 00111000 00100010 00101001 10011111 00110001 11010000 00001000 00101110 11111010 10011000 11101100 01001110 01101100 10001001 01000101 00101000 00100001 11100110 00111000 11010000 00010011 01110111 10111110 01010100 01100110 11001111 00110100 11101001 00001100 01101100 11000000 10101100 00101001 10110111 11001001 01111100 01010000 11011101 00111111 10000100 11010101 10110101 10110101 01000111 00001001 00010111 10010010 00010110 11010101 11011001 10001001 01111001 11111011 00011011 11010001 00110001 00001011 10100110 10011000 11011111 10110101 10101100 00101111 11111101 01110010 11011011 11010000 00011010 11011111 10110111 10111000 11100001 10101111 11101101 01101010 00100110 01111110 10010110..

What is this data stream encoding? Nobody knows.

Like this post? Please click G+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com

Pi in Binary

Pi in Binary

Take any circle, measure the length of its circumference then measure the length of its diameter and divide the two numbers. You get pi, probably the most famous number in all of mathematics and known for thousands of years.

pi=circumference/diameter

pi=3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132..

The decimal places go on forever and appear to be random.

But this is just one representation pi, it's pi represented in base 10.

We can represent numbers in any base we please. In base 10 we have 10 symbols 0,1,2,3,...9 and in base n we have n symbols 0,1,2,3,...,(n-1)

The simplest base is base 2, because in that base we have only two symbols 0,1

Base 2 is also call "binary" and writing pi in base 2 makes it look like computer data and even more mysterious. Here's pi in binary..

pi=11.00100100 00111111 01101010 10001000 10000101 10100011 00001000 11010011 00010011 00011001 10001010 00101110 00000011 01110000 01110011 01000100 10100100 00001001 00111000 00100010 00101001 10011111 00110001 11010000 00001000 00101110 11111010 10011000 11101100 01001110 01101100 10001001 01000101 00101000 00100001 11100110 00111000 11010000 00010011 01110111 10111110 01010100 01100110 11001111 00110100 11101001 00001100 01101100 11000000 10101100 00101001 10110111 11001001 01111100 01010000 11011101 00111111 10000100 11010101 10110101 10110101 01000111 00001001 00010111 10010010 00010110 11010101 11011001 10001001 01111001 11111011 00011011 11010001 00110001 00001011 10100110 10011000 11011111 10110101 10101100 00101111 11111101 01110010 11011011 11010000 00011010 11011111 10110111 10111000 11100001 10101111 11101101 01101010 00100110 01111110 10010110..

What is this data stream encoding? Nobody knows.

Like this post? Please click G+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com

Polynomials Explained in 5 Minutes

Polynomials Explained in 5 Minutes

One of the simplest things you can do with a number is multiply it by itself. If x is the number, we have..

x*x

Mathematicians use a shorthand for this. They write the number x with a small superscript 2 in the upper right corner like this..

x^2=x*x

They describe this in several ways. Sometimes they say the number has been "squared", sometimes they say the number has been "raised to the power of 2".

This notation immediately suggests x^3=x*x*x and x^4=x*x*x*x and so on. So we can raise a number to any power, or at least any positive integer power. For example, x raised to the power of 7 means this: take 7 x's and multiply them all together x^7=x*x*x*x*x*x*x and of course x^1=x

Notation in mathematics is important, it can help manipulate objects, it can save time, and it can help introduce new ideas.

For example, with the above definition we can raise any number to a power that's a positive integer {1,2,3,4...} but can we extend this?

A clue comes from this simple fact..

(x^a)*(x^b)= x^(a+b)

Where a and b are positive integers. So what about this...

x^0

x^0=1 for any x, and this is easy to prove..

(x^0)*(x^1)=x^(1+0)=x^1=x which can only be true if x^0=1

What about negative powers, for example, what's x^(-1)

Well, x^(-1)*x^1=x^(-1+1)=x^0=1

So x^(-1)=1/(x^1)=1/x

Powers also allow us to write a whole new set of equations. For example..

x^2-9=0 which has the solutions x=3 and x=-3
x^5=32 which has the solution x=2
x^4=1 which has the solutions x=1 and x=-1
x^4+x^3+x^2+x-4=0 which has the solution x=1

Equations like this, made from powers, are called polynomial equations.

Like this post? Please click G+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com

Sequences Explained in 5 Minutes

Sequences Explained in 5 Minutes

In mathematics a sequence is simply a list of objects. What kind of objects? Any kind, but they are often numbers. For example, {0,1,1,0} is a sequence of 4 numbers. It turns out that things get more interesting if you have an infinite sequence. For example {1,2,3,...} is the infinite sequence of integers.

Notice that the objects in a sequence are ordered, and objects can repeat. These two properties make sequences very different than sets.

Sequences can "store" a lot of information. For example, even using just two objects [0,1] it's possible to make an infinite number of different sequences!

Another important property of infinite sequences is "convergence". Consider this sequence of fractions..

1/1,1/2,1/3,1/4,1/5,...

The general element of this sequence is (1/n) where n is an integer that goes from 1 to infinity. Notice that as n gets bigger the elements of the sequence get smaller and approach 0. They never reach 0, but they get closer and closer. When this happens mathematicians say that the sequence converges to the limit of 0. Convergence is an important property of infinite sequences.

A sequence whose terms become arbitrarily close together as n gets bigger is called a Cauchy sequence, named for the French mathematician Baron Augustin-Louis Cauchy. The above sequence is an example of a Cauchy sequence. Cauchy sequences are important in several areas of mathematics, probably because of the fact that a sequence is convergent if and only if it is Cauchy.

Sometimes the terms of a sequence are specified by a rule that depends on previous terms. These rules are called "recurrence relations". Here's an example..

Suppose I give you the first two terms of a sequence 0,1 and the rule "the next term of the sequence is the sum of the two prior terms". By repetition of this simple rule you can generate the entire sequence..

0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,...

This is the famous Fibonacci sequence. It was first mentioned by the Indian mathematician Pingala in about 250 BC and has been studied by mathematicians ever since!

Another important class of infinite sequences are the "oscillating" sequences. These don't converge, but they don't diverge either! Here's a couple of examples..

0,1,0,1,0,1,0,1,0...
0,1,2,3,2,1,0,-1,-2,-3,-2,-1,0,1,2,3,2,1,0,-1,-2,-3,-2,-1,0..

The second example clearly shows the "wave like" nature of oscillating sequences. They have a "wavelength" and "amplitude".

Like this post? Please click g+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com
Internet Marketing Consultant

Teaching Prime Numbers a Different Way

Teaching Prime Numbers a Different Way

Let's consider the positive integers greater than 1, that is 2,3,4,5,..

Suppose we are given the first integer 2 and asked to make all other integers using only the multiply operation.

We soon run into problems because 2*2=4 and we have no way to make 3.

OK, we just add 3 to our set of given numbers g, so g={2,3}

Can we make 4? Yes, 2*2=4

Can we make 5? No, all our tries fail, so we add 5 to our set of given number g={2,3,5}

Can we make 6? Yes, 2*3=6

Can we make 7? No, all our tries fail, so we add 7 to our given numbers g={2,3,5,7}

Can we make 8? Yes, 2*2*2=8

Can we make 9? Yes, 3*3=9

Can we make 10? Yes, 2*5=10

Can we make 11? No, so we add it to the set g={2,3,5,7,11}

Can we make 12? Yes, 2*2*3=12

Can we make 13? No, so add it to the set g={2,3,5,7,11,13}

Can we make 14? Yes, 2*7=14

Can we make 15? Yes, 3*5=15

Can we make 16? Yes, 2*2*2*2=16

What is the set g that we are generating by this process? It's the set of prime numbers! This is simply another way to explain prime numbers.

It's a nice demonstration because it shows how prime numbers generate all numbers using only the multiply operation. You can also see that as g gets bigger we can obviously generate more numbers from it, so prime numbers become less and less frequent.

Like this post? Please click g+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com
Internet Marketing Consultant

Atomic Physics Explained in 5 Minutes

Atomic Physics Explained in 5 Minutes

After a century of hard work physicists have established some impressive facts about atoms.

If you could make yourself amazingly small and walk up to an atom the first thing you hit would be a cloud of electrons whizzing at great speed, so they look like a blur. The electron was the first elementary particle to be discovered and it's still as elementary as ever. Elementary means no internal structure has been detected - so far.

You plow your way through the cloud of electrons and finally come to a peaceful empty space. You travel this for ages and then on the horizon you see a small object. As you get closer it reveals its structure. This is the nucleus of the atom. It consists of two type of particles - protons and neutrons - bound together very tightly.

The number of protons in the nucleus is exactly equal to the number of electrons that you passed in the outer cloud, but the number of neutrons in the nucleus can vary.

For a while physicists thought protons and neutrons were elementary, but not so. It turns out they are complex objects with internal structure. Plus, the neutron is very similar to the proton. So physicists built particle colliders to learn more about the structure of the proton. The most famous is the LHC (Large Hadron Collider) at CERN. The technique is simple, smash two protons together an incredible speed and see what comes out. From this physicists can deduce what's inside the proton.

It seems a lot is going on inside the proton. It's built from more elementary particles called quarks, bound together by an incredible strong force field generated by particles called gluons.

Given than the proton is amazingly small why would nature decide to give it such complex internal structure?

That's a philosophical question physicists can never answer. But they are working hard to document the internal structure of the proton.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Imaginary Numbers Explained in 5 Minutes

Imaginary Numbers Explained in 5 Minutes

Imaginary numbers appeared when mathematicians studied polynomial equations. These equations are simple, they're just equations that contain powers, for example..

x^2-9=0 or x^3+x^2+x=9

It turned out that although these equations were easy to write down they were not so easy to solve.

Using integers or whole numbers {..,-3,-2,-1,0,1,2,3,..} it was possible to solve some of these equations, but many were still unsolvable.

Using rational numbers or fractions, a/b where a and b are integers, it was possible to solve many more but still not all.

A major breakthrough occurred when mathematicians introduced a third type of number called irrational numbers which are not integers and not rational numbers. This allowed many more polynomial equations to be solved - but still not all!

In fact, the situation was a bit embarrassing, because one of the simplest polynomial equations could not be solved even with three types on numbers available. It was this..

x^2+1=0

This incredibly simple equation cannot be solved by an integer, it cannot be solved by a rational number, and it cannot be solved by an irrational number!

So what did mathematicians do? No problem, they just invented a new number, and they called it i..

i=square root of -1

It's easy to see that this number solves the original equation, that is..

i^2+1=0

Of course i can be used to define an infinite amount of new numbers, just take any number and multiply it by i. Mathematicians called the new class of numbers complex or imaginary numbers. Not a great name.

Finally, it turned out that with 4 types of numbers available (integer, rational number, irrational numbers and complex numbers) it was possible to solve all polynomial equations.

Of course the name imaginary number sometimes causes non-mathematicians to ask questions like "do imaginary numbers really exist?" This of course is a useless question. In mathematics if you can define something it exists!

There was a big bonus. In addition to helping solve all polynomial equations, the number i turned out to be amazingly useful in other areas of mathematics and it's also used in many areas of physics.

So the number i, defined as "that number which when multiplied by itself gives -1", turned out to be an amazing invention!

But the name? Not so much.

Like this post? Please click g+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com
Internet Marketing Consultant

Elementary Particles - A Paper Band Model

Elementary Particles - A Paper Band Model

Assuming you can't afford to build your own Large Hadron Collider to study elementary particles here's a low cost alternative.

Take a strip of paper, join the ends so you have a band. This is a model for a spin 0 particle.

Give it 1 half twist before joining the ends. This is a spin 1/2 particle. Does a half twist clockwise give the same exact band as a half twist counter clockwise, or are these two bands different objects? If they are different objects then what's the physics interpretation of the clockwise v. counter clockwise twist?

Give it 2 half twists before joining the ends. This is a spin 1 particle. Same question, does 2 half twists clockwise give the same exact band as 2 half twists counter clockwise, or are these two bands different objects? If they are different objects then what's the physics interpretation of the clockwise v. counter clockwise twist?

But things get strange fast..

Give it 4 half twists before joining the ends. This is a spin 2 particle. The only one is the hypothetical graviton. Now, if you play around with this thing for a while it will suddenly "flip" into a double thickness band with 1 half twist! Does this imply that a graviton (spin 2) can transform into a spin 1/2 particle. Assuming charge is conserved, this spin 1/2 particle must be neutral and that means a neutrino or some other as yet unknown particle. So Gravitons can "flip" into neutrinos?

Not only that, but this also means Boson <--> Fermion transitions are possible, and that violates the conservation of angular momentum. Most physicists will hate that.

What about a collision between two bands?

At low energy they might just "bounce" off each other.
At higher energy they might "stick" along their surfaces or edges.
At even higher energy they might "break open" and then reconnect to form something totally new.

----> Read more posts here.

Content written and posted by Ken Abbott abbottsystems@gmail.com

Spin Explained Simply

Spin Explained Simply

Spin is something we're all familiar with. We can make any object spin. And not just a top, spin is used to great effect in many sports such as tennis, baseball and cricket. And of course you could never throw a frisbee without spin.

We can spin an object at any speed we please, and as soon as it starts spinning it defines an axis about which the spin occurs.

But this is the spin of a big object, meaning an object we can handle. What about the spin of a really small object such as an electron?

It turns out the electron spins just like a tiny top - but with a big surprise.

It spins at a fixed rate that can never be changed. There is no known process that can change the spin rate of the electron. This makes electron spin a fundamental quantity.

What about the axis of spin? You can measure the spin along any axis you want and you'll always get the same result, +1/2 or -1/2. These two values correspond to the electron spinning clockwise or counter clockwise. Again, this makes electron spin totally different than the spin of a big object such as a top.

Physicists call the electron a "spin 1/2 particle". And it's not just the electron, quarks have spin 1/2 and so do neutrinos. In fact all elementary particles have spin except for the recently discovered Higgs boson.

If you plan to study Quantum Mechanics pay attention to spin. It's not what you expect. And don't forget my instant summary of Quantum Mechanics, "small objects behave very differently than big objects". Spin is a beautiful example of this.

Like this post? Please click G+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com
Internet Marketing Consultant

Learn Differential Calculus in 5 Minutes

Learn Differential Calculus in 5 Minutes

Differential calculus is one of the two branches of calculus, the other is integral calculus. Most mathematicians refer to both branches together as simply calculus.

Calculus is all about functions, so there's no point in studying calculus until you understand the idea of a function.

Let's take a simple function, say f(x)=x^2

What's the value of this function at a specific point, say x=a? That's easy, it's f(a)=a^2. But now we ask an interesting question, can we possibly know anything else about the function at point a? At first glance this seems impossible, the value of the function at a is f(a), so surely that's all we can know, right? Wrong. It turns out there's a process called "differentiation" that can tell us more. Here's how it works..

Take a very small number, say q, and ask what the function is doing at a+q, in other words at a point very close to a..

f(a+q)=(a+q)^2=a^2+2*a*q+q^2

But we can make q as small as we please, which means q^2 is much smaller, so to a good approximation we can ignore it, and we get..

f(a+q)=(a+q)^2=a^2+2*a*q

Notice the first term, a^2, is just the value of the function at a, f(a), so now we have..

f(a+q)=f(a)+2*a*q

Which means..

f(a+q)-f(a)=2*a*q

And so..

(f(a+q)-f(a))/q=2*a

What is the meaning of the expression on the left? If you draw a diagram you'll see that the term on the left is simply the slope of the curve f(x) close to x=a. So this gives us some valuable information about what's going on near a. Now all we need to do is keep making q smaller so we get closer and closer to a. In fact, we can use the concept of a limit to say..

Limit(f(a+q)-f(a))/q as q goes to zero is 2*a

Of course we could do this for any point a, so in general..

Limit(f(x+q)-f(x))/q as q goes to zero is 2*x

This is called the "derivative" of f(x) and is often written as df/dx, or sometimes as f'. So, to summarize..

The derivative of the function f(x)=x^2 is 2*x and is written df/dx=2*x and it's the slope of the f(x) curve at x. Of course, a slope is simply a rate of change, so we can also say that df/dx=2*x is the rate of change of the function f(x).

Congratulations, you just did some calculus! You differentiated the function f(x)=x^2 and got the result 2*x

To generalize this example, the derivative of the function f(x)=x^n where n is any integer is..

df/dx=n*x^(n-1)

So for example, if f(x)=x^10 then the derivative is df/dx=10*x^9

So, the essence of differential calculus is this.. in addition to knowing the value of a function f(x) at x=a we also know the rate of change (slope) of the function at a. Differential calculus gives us an extra piece of information!

Much of differential calculus is simply finding ways to differentiate functions. This can get boring, so why bother? Because the derivative of a function is a really useful thing for solving all sorts of problems. It's especially useful in physics and many laws of physics are written as differential equations.

Like this post? Please click g+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com
Internet Marketing Consultant

Physics - An Unjustified Assumption?

Physics - An Unjustified Assumption?

When you describe a system using differential equations (or partial differential equations) with respect to space and time you're making the assumption that space-time is continuous (i.e. infinitely divisible). That's a massive assumption, and if it's wrong you could be missing a lot of physics.

Many theories make this assumption, for example Einstein's General Relativity.

Like this post? Please click g+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com
Internet Marketing Consultant

Newton's Law of Gravitation - Derived From Scratch

Newton's Law of Gravitation - Derived From Scratch

A simple way to think about the gravitational field of an object is to imagine a fixed number of "lines of force" that radiate from the object evenly into space.

Let's suppose the number of lines of force produced by an object is directly proportional to its mass, so..

n=k*m

where n is the number of lines of force produced by the mass m and k is a constant.

Now assume the density of the lines at any given point in space represents the strength of the gravitational field at that point. So at a distance r from the object the density of the lines of force is..

n/(surface area of the sphere of radius r)

which is n/(4*pi*r^2)=k*m/(4*pi*r^2)=G*m/(r^2)

where G=k/4*pi is a constant.

This is Newton's famous law of gravitation. G is called the "universal gravitational constant".

Like this post? Please click g+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com
Internet Marketing Consultant

Indian Mathematician Ramanujan

Indian Mathematician Ramanujan

On 16 January 1913, Srinivasa Ramanujan wrote to G. H. Hardy.

Ramanujan was a self taught mathematician from a small village in India. He had almost no formal training in mathematics. Hardy was professor of mathematics at Cambridge University and one of the leading mathematicians in the world.

The letter sent by Ramanujan contained a sampling of theorems he had discovered. Hardy later said, "the theorems defeated me completely; I had never seen anything in the least like them before" and added that "they must be true, because, if they were not true, no one would have the imagination to invent them."

What happened next would change both their lives. Hardy would later write about Ramanujan in his book "A Mathematician's Apology" and say that working with Ramanujan was the most significant event of his life.

Ramanujan produced some amazing infinite series, including several for π that converge extraordinarily fast and form the basis of today's computer algorithms used to calculate π.

His other results involved continued fractions. Ramanujan had a special love of continued fractions and used them to extraordinary effect.

Hardy worked hard to try and discover how Ramanujan produced his remarkable results. He never found out.

Ramanujan died on 26 April 1920. He was 32 years old. Hardy died many years later, on 1 December 1947 at the age of 70.

The Ramanujan story is now part of mathematics legend and his notebooks are still being studied today.

The movie "The Man Who Knew Infinity" tells the story of Ramanujan with Dev Patel in the lead role.

Like this post? Please click g+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com
Internet Marketing Consultant

Mathematician Georg Cantor

Mathematician Georg Cantor

Georg Cantor was a mathematician who proved something quite amazing - there are many different types of infinity! He called these numbers "transfinite numbers" and he even developed an arithmetic for working with them. He denoted them by the Hebrew letter "aleph". In his lifetime Cantor was ridiculed, not by the general public, but by his fellow mathematicians!

Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I. The public celebration of his 70th birthday was canceled because of the war. He died on January 6, 1918 in the sanatorium where he had spent the final year of his life.

Today Cantor's work is part of any university mathematics curriculum and is regarded as one of the most beautiful pieces of mathematics ever created. It stands apart from most advanced math because you don't need to know much mathematics to understand it. In fact, all you need to know is how to count!

Like this post? Please click g+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com
Internet Marketing Consultant

Learn Fractions in 5 Minutes

Learn Fractions in 5 Minutes

The hardest thing about learning fractions is adding them. Multiplication is easy, but addition is usually taught as a complex multi-step process. It's a real pain.

But it doesn't have to be that way. Take the two fractions to be added..

a c
-+-
b d

Take the diagonal numbers (top left and bottom right) and multiply them, a*d

Do the other diagonal, c*b

Add the two results (a*d)+(c*b)

That's the "top" (numerator) of the answer.

The "bottom" (denominator) of the answer is much easier, it's just b*d

You're done!

Did you notice the pattern? It's two diagonals and a base, like this..

X
_

A lot of mathematics involves patterns.

To multiply fractions just multiply the tops and multiply the bottoms. So..

a c
-*-
b d

is (a*c)/(b*d)

That's it. Fractions done in 5 minutes. With the time saved why not study more interesting math topics!

Like this post? Please click g+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com
Internet Marketing Consultant

Could CERN's LHC Produce Gravitons?

Could CERN's LHC Produce Gravitons?

One of the greatest physics insights ever was by Einstein.. that (locally) a gravitational field cannot be distinguished from acceleration.

If gravity and acceleration cannot be distinguished by any experiment then we must conclude that (locally) they are the same thing.

Now consider the protons in the CERN LHC (Large Hadron Collider).

When they collide head-on the acceleration must be staggering. Translation.. the local gravitational field must be ultra intense.

What does an ultra intense gravitational field do? Does it produce gravitons? So the LHC might discover the hypothetical graviton. That would be amazing.

Perhaps this is a step towards Quantum Gravity.

Like this post? Please click G+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com
Internet Marketing Consultant

Gravity - Can it exist in 1 Dimension?

Gravity - Can it exist in 1 Dimension?

Consider a closed loop containing two point masses m and M. The loop is a 1D space.

What's the gravitational force between the two masses?

Newton's formula for the gravitational force F between two masses m and M in 3D space is..

F=G*M*m/(r^2)

where G is a constant and r is the distance between the two masses.

The r^2 term is good in a 3D space, but in general it's r^(n-1) where n is the dimension of the space. So putting n=1 for 1D space we get..

r^(1-1)=r^0=1 so F=G*M*m

Which means F is independent of distance. Gravity has the same strength no matter how far apart the two objects are!

So each object feels the exact same gravitational force to the right and left and the net force is zero. Gravity does not exist in 1D space!

Of course, these calculations have been done using Newton's theory of gravity. I wonder if we would get the same result with Einstein's General Theory of Relativity, which is now 100 years old but still "state-of-the-art" in gravitational theories.

Like this post? Please click g+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com
Internet Marketing Consultant

Mathematician Walter Lederman

Mathematician Walter Lederman

Not all mathematicians get written up in the history books.

I never thought about it, but I suppose that Walter Lederman was the first professional mathematician I knew. I met him during my first year at the University of Sussex in the UK.

It was in 1967. He taught number theory and group theory. Students loved him. Not sure why. Perhaps it was because he had an enthusiasm for his subject. In group theory he recounted the exploits of Evariste Galois like he know him personally. He was real. He gripped his students. He was also very low key. Thanks Walter. I did not forget.

Like this post? Please click g+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com
Internet Marketing Consultant

The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic

Mathematicians often use "number" to refer to the integers {1,2,3,...} so we'll do the same.

Let's pick a number n at random and ask a very simple question. Is it possible to find two other numbers which when multiplied together give n? In other words, can we find two numbers, a and b such that n=a*b

At first glance it seems like this should be true for any n. In fact, some numbers can be broken down like this in many ways, for example..

100=10*10=2*50=25*4=20*5

But it's not true for any n. There are some numbers that can never be broken down into a product of two other numbers. There are some numbers for which n=a*b is never true. These numbers are called prime numbers.

Prime numbers play an important role in mathematics, probably because of this fundamental fact..

Any number bigger than 2 can be written as a product of prime numbers. This is called The Fundamental Theorem of Arithmetic or the Prime Factorization Theorem.

So we can think of prime numbers as the "building blocks" from which all numbers are made. This is an important result in mathematics, but you don't need to be a mathematician to prove it. The proof is easy. Let's do it..

Pick any number n and ask if it can be broken down into the product of two other numbers a and b. If it cannot then n is prime and we're done. If it can then break it down..

n=a*b

Now just repeat. Ask the same question for a and then ask the same question for b. And keep going. At some point this process must stop, because if it never stopped we would wind up with..

n=1*1*...*1=1 which is ridiculous.

So it must stop when n is written as a product of several numbers which cannot be broken down any more. But this is just the definition of a prime number. So it stops when n is written as a product of primes. That was easy!

Here are the first 10 prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

How many prime numbers are there in total? They never stop, there's an infinite amount. Is there a formula that predicts the next prime in the sequence? No, at least not one that any mathematician has discovered so far. Primes are mysterious numbers that seem to elude any type of prediction. They have driven mathematicians to distraction for centuries!

So any number can be written as a product of primes. Let's try one..

20511149=29*29*29*29*29=29^5

Which just illustrates that in the product of primes a prime may be repeated.



Like this post? Please click G+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com
Internet Marketing Consultant

Integral Calculus Explained in 5 Minutes

Integral Calculus Explained in 5 Minutes

Integral calculus is one of the two branches of calculus, the other is differential calculus. Most mathematicians refer to both branches together as simply calculus.

Calculus is all about functions, so there's no point in studying calculus until you understand the idea of a function.

Also, you should study differential calculus before integral calculus because the two have an elegant relationship with each other and this is best explained if you understand differential calculus first.

There are several ways to introduce integral calculus. The method below is a bit abstract, but it's very fast, and it also highlights the elegant relationship between integral and differential calculus. Here it is..

Suppose you have two functions g(x) and f(x) such that when you differentiate g(x) you get f(x), like this..

Dg(x)=f(x) where D is the differentiation operation "take the derivative of".

Then g(x) is called the integral of f(x) and is written as g(x)=Sf(x) where S is the integration operation "take the integral of".

Let's try an example..

Suppose I give you the function f(x)=x^2 and ask you to find a function g(x) such that when you differentiate g(x) you get f(x). The answer is..

g(x)=(x^3/3)+c where c is any constant

Try it. If you differentiate g(x) you get f(x), so the integral of x^2 is (x^3/3)+c

The elegant relationship is that differentiation and integration are "complementary" or "opposite" operations. In other words..

SDf(x)=f(x)+c where c is any constant, and..

DSf(x)=f(x)

SD means doing the two operations in succession.. take the derivative (the D operation) and then integrate the result (the S operation).

If we subtract the two results we get..

SDf(x)-DSf(x)=[SD-DS]f(x)=c

So the operation [SD-DS] reduces any function to a constant! This is the "complimentary" relationship between integration and differentiation. The special operator [SD-DS], is called the "commutator" of S and D.

By the way, the concept of a commutator is not specific to calculus, it's a general concept that applies to many mathematical operations.

The derivative of f(x) at point x=a is just the slope of the curve at that point. Does integration have a simple interpretation? Yes it does..

If we have two points x=a and x=b and we calculate the integral of f(x) at each point, then the difference of these two values is just the area under the f(x) curve between a and b. So differentiation is slope and integration is area.

Much of integral calculus is simply finding ways to integrate functions. This can get a bit boring, so why bother? Because the integral of a function is a really useful thing for solving all sorts of problems.

Like this post? Please click g+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com
Internet Marketing Consultant

Prime Number Patterns

Prime Number Patterns

Let's write the prime numbers in base 2 "binary" format. I like to use the convention of least significant digit to the left, so for example the number 11 in binary is 1101, and this simply means..

11=1*(1)+1*(2)+0*(4)+1*(8)

Here are the first 20 prime numbers in binary..

2 01
3 11
5 101
7 111
11 1101
13 1011
17 10001
19 11001
23 11101
29 10111
31 11111
37 101001
41 100101
43 110101
47 111101
53 101011
59 110111
61 101111
67 1100001
71 1110001

Let's try a pattern algorithm. We'll define a replace operator r(n) as "whenever the pattern 01 occurs in the binary representation of n replace it with 101". Here's an interesting thing..

r(2)=5
r(5)=11
r(11)=23
r(23)=47

But the pattern does not continue, because r(47) is not prime.

Of course, this is just a simple example of pattern recognition. If you have access to serious computing power there are many ideas you can test!

Like this post? Please click g+1 below to share it.
Content written and posted by Ken Abbott abbottsystems@gmail.com