The Prime Pair (1049,1051)

The Prime Pair (1049,1051)

Let's write twin primes in binary.

For example, the twin prime pair (1049,1051)=(10000011001,10000011011).

Now concatenate the binary bit streams to get 1000001100110000011011=2149403 which is a prime!

Primes generating primes! Quite amazing.

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

Google Math Trick - For Math Teachers & Math Students

Google Math Trick - For Math Teachers & Math Students

Need to plot a function?

Just punch your request into Google to get an instant plot.

Example: to plot the function f(x)=x^2 just enter plot x^2 into Google. Of course, it can also handle much more complex functions, for example try plot x^3+1/x or try one of my favorites, just enter plot 1/(1+x^2)

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

Your chance of winning the Lottery

Your chance of winning the Lottery

We all know the probability is low. But how low?

Let's take the Powerball, an incredibly popular lottery in the USA. Last week someone won $571 million in the Powerball.

So what are your chances of winning?

If your 5 numbers plus the Powerball match the winning six numbers drawn, then you win or share the Grand Prize. If the jackpot is not won in any drawing, the First Prize Pool Money is carried forward and is added to the next Powerball Jackpot.

Your chance of getting 5 + Powerball - Grand Prize is 1 in 292,201,338

Powerball offers 9 ways to win, and here are the probabilities..

So does this mean you should not buy a lottery ticket? If you don't buy your chance of winning is absolutely zero!

Good luck.

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

Twin Primes - A Surprising Result

Twin Primes - A Surprising Result

Let's write twin primes in binary. For example the prime pair (281,283)=(100011001,100011011).

Now concatenate the two binary string to get 100011001100011011=144155 which is not a prime. But just reverse the order of concatenation to get 100011011100011001=145177 and this is a prime!

Here's an even more impressive example.

The prime pair (1049,1051)=(10000011001,10000011011) then concatenate the binary bit streams to get 1000001100110000011011=2149403 which is a prime!

Primes generating primes! Quite amazing. Please contact me with your results.

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

Quantum Mechanics Explained Fast

Quantum Mechanics Explained Fast

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 just correspond to the electron spinning clockwise or counter clockwise). So 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

Transcendental Numbers - A Simple Definition

Transcendental Numbers - A Simple Definition

Take any number and ask if it can be converted to a rational number by raising it to an positive integer power. If it can we say it's coupled to the integers. If not we say it's decoupled from the integers.

All rational numbers are coupled by definition. But so are many irrational numbers, for example sqrt(2) is irrational, but (sqrt(2))^2=2. Even many complex numbers are coupled, for example i^2=-1

So it seems most numbers are coupled to the integers, making integers fundamental. Not true. In fact Cantor showed the opposite is true, most numbers are decoupled from the integers. These are the (still mysterious) transcendental numbers. And the name is good, transcendental numbers "transcend" the integers.

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

What is a Prime Number?

What is a Prime Number?

Here's a nice simple definition of a prime number..

Suppose you have a set of objects and I ask you to divide them into subsets of equal size. The key thing here is "equal size".

This can often be done. But there are some sets where it cannot be done.

When it cannot be done we call the number of objects in the set a prime number.

Example: A set of 12 objects can be divided into subsets of equal size (actually, in several different ways), so 12 is not a prime number. A set of 13 objects can never be divided into subsets of equal size so 13 is a prime number.

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

The Mathematics of Drinking

The Mathematics of Drinking

First we define a fundamental unit - the "standard drink". It's a drink that needs one hour for its alcohol content to be metabolized. In other words it takes one hour to get out of your system.

Then using this fundamental unit we can measure drinks..

Liquor (80 Proof - 40% Alcohol by volume)

“Bottle” 750ml = 17 Standard Drinks
“Martini” 200ml = 4.5 Standard Drinks
“Miniature” 50ml = 1 Standard Drink

Wine (13% Alcohol by volume)

“Bottle” 750ml = 5.5 Standard Drinks
“Big Glass” (10 fluid ounces) = 2 Standard Drinks
“Standard Glass” (5 fluid ounces) = 1 Standard Drink

Beer (5% Alcohol by volume)

“Pint” (16 fluid ounces) = 1.2 Standard Drinks
“Bottle” (12 fluid ounces) = 1 Standard Drink

I think the biggest shock here is the Martini. Wow!

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

Information and Dimension - Are They Related?

Information and Dimension - Are They Related?

This is an extreme example, but it shows there may be a fundamental relationship between the availability of information and the concept of dimension.

Sitting on your desk is a sphere. As in mathematics, let's assume an axiom - your sphere is an information void. What's that? It's a closed surface which allows absolutely no knowledge of its interior. The interior of the object is off limits to any form of investigation. Think of its surface as an information barrier that stops any information about the interior from getting out.

So what properties does your sphere have?

You can't break it open to look at the interior, that would violate the axiom, so the object must be infinitely strong.

You can measure its diameter - just use a ruler. You can measure its surface area - just take a small unit square and see how many times you can paste it on the surface.

What about volume? Be careful, knowing volume means you have information about the interior and that's impossible. Of course you can use the formula volume=(4/3)*pi*r^3 where r is the radius. But this is not a measurement of the interior, so the best you can do is call this volume the "external volume". You have no knowledge of the geometry or topology of the interior, so the "internal volume" could be very different!

Now, if you can never have any knowledge of the interior you come to a simple conclusion, the object is totally defined by whatever is on its surface. The object certainly looks 3D but can be fully described as if it was 2D. Interesting object!

Your sphere has no reality inside - just like bubbles in a liquid have no liquid inside. If you own one please handle it carefully!

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

Prime Cores - The Core of a Prime Number

Prime Cores - The Core of a Prime Number

What's a Prime Core? Take a prime number in binary, then strip off the first and last digits (which, for all primes except 2 are always 1's) then interpret the binary string you have left as an integer, and that's the prime core.

Example, the prime 79 in binary is 1001111 so its core is 00111 which is 7. So using C to represent the prime core operation, we have C(79)=7.

Then here's an interesting question: "when is the core of a prime also a prime?"

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

The Problem with Blockchain

The Problem with Blockchain

Don't get me wrong. It's clever. The mathematics behind it is very clever. It would make a great PhD thesis. But it's a solution looking for a problem.

Blockchain is many things to many people. But let's take Bitcoin, the most famous blockchain app.

A payment method? Really? VISA can process up to 47,000 transactions/second. Bitcoin maxes out at 7 transactions/second. Blockchain is the bottleneck.

A peer-to-peer system? Really? Nobody cares about peer-to-peer. Most people like a central authority.

So what is blockchain? It's an example of technology in love with itself. And with no concept of consumer marketing or psychology.

Watch Bitcoin Trading in Realtime

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

Quantum Computing Gets Closer

Quantum Computing Gets Closer

Studying computer programming in hopes of getting a great job?

An experienced programmer looking to boost your career?

Perhaps the really hot jobs in the future will be in quantum computing.

It's worth keeping up to speed in this fast developing field, so take some time to learn quantum programming!

Here's a nice quantum computing tutorial. It teaches you step-by-step how to write a program for these amazing machines.

How to Program a Quantum Computer

Also, IBM is very active in quantum computing. Their main quantum computer research Lab is the Watson Research Center in Yorktown Heights. They have a quantum computer up and running and provide developer tools to let you write and run programs. It's called the "IBM Q experience".

IBM Q experience

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

Hydrogen Cars Explained in 5 Minutes

Hydrogen Cars Explained in 5 Minutes

Hydrogen cars are very different from all other cars (gasoline, battery, hybrid). Here's how they work..

A hydrogen car is a 100% electric car, but instead of a battery it uses hydrogen fuel cells. These are simple, feed them hydrogen gas and they generate electricity. They are like a battery that never need recharging. There is no combustion, nothing burns, hydrogen gas is simply fed to the fuel cells to produce electricity.

So where do we get the hydrogen? The cars carry high pressure tanks of hydrogen gas to supply their fuel cells. The tanks get refilled at a hydrogen gas station, and the filling process is very similar to regular gasoline filling. The hydrogen gas station produces its hydrogen on the spot. How? By electrolysis of water.

Yes, the raw material to generate hydrogen gas is water!

So, the hydrogen gas station makes hydrogen from water. Cars refill their hydrogen tanks. The hydrogen goes through the car's fuel cells and generates electricity to drive the car. And the exhaust? The car's exhaust is water vapor. There is zero pollution.

The whole thing is a water to water cycle!

The range of a hydrogen car is about the same as a gasoline car, and the refueling time is about the same.

Also, hydrogen is the most abundant element in the universe. So I doubt we'll run out!

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

Simple Definition of a Prime Number

Simple Definition of a Prime Number

Suppose you have a set of objects and I ask you to divide them into subsets of equal size. The key thing here is "equal size".

This can often be done. But there are some sets where it cannot be done.

When it cannot be done we call the number of objects in the set a prime number.

Example: A set of 12 objects can be divided into subsets of equal size (actually, in several different ways), so 12 is not a prime number. A set of 13 objects can never be divided into subsets of equal size so 13 is a prime number.

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

Lineland - Life in 1 Dimension

Lineland - Life in 1 Dimension

You're a point mass and you live on the x axis. That's your entire world. It's "Lineland". What's your life like?

First, as regards moving, you only have two directions, forward and backwards. And if you meet another point mass you cannot pass. So you can only know two other masses. You have just two friends maximum!

You have no reason to count objects beyond two, so you might be slow in developing the concept of integers. Or perhaps you never develop the concept at all. You simply have no need for it.

What about Physics in Lineland? You're a point mass, so you have mass, let's say m. Another point mass could have a different mass, say M. So at least gravity exists, right? It does, but it has a strange form. Newton's formula for the gravitational force F between two masses m and M 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. Putting n=1 for Lineland 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 objects are. So physics in Lineland is very different.

This is Lineland on the x axis. What if Lineland is the circumference of a circle? That's even more interesting. Would you be aware that Lineland had a "curvature"? What does gravity do now that Lineland is a closed loop? What happens if Lineland is a closed loop that intersects itself at several points? What happens at these intersection points and how do they contribute to gravity? How do things change as the number of point masses in Lineland changes? It turns out that even 1 dimension can be very complex!

Just think, there's probably a 4 dimensional world somewhere with math teachers looking for a nasty problem to set on an exam. Finally they come up with one, "explain how math would have developed if our world was constrained to just 3 dimensions".

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

Bitcoin Trading Explained Fast

Bitcoin Trading Explained Fast

Bitcoin trades 24/7 around the world on many exchanges. So there's no one Bitcoin price and arbitrage opportunities exist.

If you plan to trade Bitcoins the choice of exchange is very important and should be researched carefully before you open an account.

Watch Bitcoin Trading in Realtime

Here's one of many Bitcoin Exchanges. This one trades Bitcoin in US Dollars, which is called BTCUSD.

Bitcoin Trading Platform

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

Can Reality Vanish?

Can Reality Vanish?

Quantum Mechanics deals with incredibly small objects and this makes it difficult to visualize what's happening. But let's bring a quantum object up to desktop size and see how it might behave.

Sitting on your desk is a sphere. As in mathematics, let's assume an axiom - your sphere is an information void. What's that? It's a closed surface which allows absolutely no knowledge of its interior. The interior of the object is off limits to any form of investigation. Think of its surface as an information barrier that stops any information about the interior from getting out.

So what properties does your sphere have?

You can't break it open to look at the interior, that would violate the axiom, so the object must be very strong.

You can measure its diameter - just use a ruler. You can measure its surface area - just take a small unit square and see how many times you can paste it on the surface.

What about volume? Be careful, knowing volume means you have information about the interior and that's impossible. Of course you can use the formula volume=(4/3)*pi*r^3 where r is the radius. But this is not a measurement of the interior, so the best you can do is call this volume the "external volume". You have no knowledge of the geometry or topology of the interior, so the "internal volume" could be very different!

Now, if you can never have any knowledge of the interior you come to a simple conclusion, the object is totally defined by whatever is on its surface. The object certainly looks 3D but can be fully described as if it was 2D. Interesting object!

Your sphere has no reality inside - just like bubbles in a liquid have no liquid inside. If you own one please handle it carefully!

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

Shannon Entropy Explained in 5 Minutes

Shannon Entropy Explained in 5 Minutes

Shannon Entropy (also called Information Entropy) is a concept used in physics and information theory. Here's the scoop..

Suppose you have a system with n states i.e whenever you make an observation of the system you find it's in one of the n possible states.

Now make a large number of observations of the system, then use them to get the probability pi that if you make an observation the system is in state i. So for every state of the system you have a probability pi.

Now construct this crazy sum = p1*log(p1) + p2*log(p2) +... + pn*log(pn) where the sum is over all the states of the system.

If the log is base 2 then (-1)*sum is called the "information entropy" of the system.

Note that "information entropy" applies to a complete system, not individual states of a system.

Here's a simple example..

My system is a penny and a table.
I define the system to have 2 states.. penny lying stationary on the table with heads up or with tails up.

My experiment is to throw the penny and then observe which state results.

I throw the penny many times and make notes. It lands heads up 1% of the time and tails up 99% of the time (it's biased).

The crazy sum is 0.01*log(0.01) + 0.99*log(0.99) = 0.01*(-6.643856) + 0.99*(-0.0145) = -0.08079356

So the information entropy of the system is (-1)*(-0.08079356) = 0.08079356

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

Fractions Explained in 5 Minutes

Fractions Explained 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 a few 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
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Elementary Particle Physics - The Spin Inversion Operator

Elementary Particle Physics - The Spin Inversion Operator

I just had an idea for a "Spin Inversion Operator". It takes an elementary particle and inverts its spin. Example: the Graviton with spin 2 would map into a spin 1/2 particle. Likewise a spin 1/2 particle would map into a spin 2 particle.

The Photon is interesting, it has spin 1, so the Spin Inversion Operator does not change the spin. It's invariant.

Coming back to the Graviton again, assuming charge is conserved, the Spin Inversion Operator maps it into a neutral spin 1/2 particle. The only one known is the Neutrino. So is there some deep connection between the Graviton and the Neutrino?

The Spin Inversion Operator works for the Graviton and all elementary particles in the Standard Model - except for the Higgs Boson.

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

Simple Harmonic Oscillator - A Different Approach

Simple Harmonic Oscillator - A Different Approach

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 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. 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.

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!

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. 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 different 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.

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

DNA Explained in 5 Minutes

DNA Explained in 5 Minutes

The DNA sequencing industry is developing at lightning speed. It's poised to bring massive change.

But what's DNA anyway?

It's a molecule.

But not just any molecule.. the human DNA molecule is about 1.5 meters long and incredibly thin.

The shape of the molecule is very clever. Think of a ladder with 4 different color rungs. The sequence of colors is the information!

Now imagine a ladder with about 3 billion rungs and with an amazing twist. Literally. Nature twists the molecule into a corkscrew (helix) shape. This simply gives it extra strength, because a break in the DNA molecule would have disastrous consequences.

That's the amazing human DNA molecule!

Nature takes very good care of the huge DNA molecule - it winds it up and packs it into separate containers called Chromosomes. This is just an efficient method to make sure the molecule fits into a tiny space and is protected.

The whole packaging system, with the DNA molecule wound and packed into Chromosomes, is referred to as a Genome. But it's just a molecule!

The DNA molecule encodes data that acts as a program to run each cell in the body. Just like computer data can be reduced to strings of 0s and 1s (2 units), DNA uses 4 units.

The order (sequence) of these units is the program.

And Nature is massively parallel. No central processor here.. every cell in the body contains a copy of the DNA molecule.

Think of a cell as a factory. It manufactures all sorts of substances, and the instruction on how to do this is provided by the DNA. Many of these substances are proteins, so DNA has special instruction sequences (called Genes) that tell exactly how to make different proteins. The sections of DNA between the Genes are a bit of a mystery. It's not clear exactly what these instructions do, if anything.

So the cell simply reads the DNA instructions and makes the appropriate proteins. It's a wonder of molecular manufacturing!

DNA can be analyzed at many levels, for example..

- Just look at the chromosomes for abnormal shape.

- Sequence (read the order of the units) a gene in one of the chromosomes.

- Sequence all genes in one of the chromosomes.

- Sequence all genes in all the chromosomes.

- Sequence all the DNA (even the instructions between the genes) in all the chromosomes. This is called "full sequencing".

A major producer of DNA sequencing machines is a company called Illumina. They reduced the cost to sequence a human DNA molecule from $100 million in 2001 to about $1,000 in 2014. The rate of progress is staggering!

Do we all have the same exact DNA?
No. We are all 99.9% the same, but that 0.1% means about three million differences between your DNA and anyone else's. It's these differences that are used in DNA testing.

Oh, and our DNA is about 99% the same as our closest relative, the chimpanzee.

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

Quantum Computing Gets Closer

Quantum Computing Gets Closer

Studying computer programming in hopes of getting a great job?

An experienced programmer looking to boost your career?

Perhaps the really hot jobs in the future will be in quantum computing.

It's worth keeping up to speed in this fast developing field, so take some time to learn quantum programming!

Here's a nice quantum computing tutorial. It teaches you step-by-step how to write a program for these amazing machines.

How to Program a Quantum Computer

Also, IBM is very active in quantum computing. Their main quantum computer research Lab is the Watson Research Center in Yorktown Heights. They have a quantum computer up and running and provide developer tools to let you write and run programs. It's called the "IBM Q experience".

IBM Q experience

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

Mobius Strip - Yet Another Strange Surprise

Mobius Strip - Yet Another Strange Surprise

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

Now give the strip of paper 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?

But things get strange fast..

This time give the strip of paper 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 it flips into a double thickness Mobius Strip. One surface has gone, and so have 3 half twists!

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

Physics and the Number 3

Physics and the Number 3

It seems that Physics likes the number 3 at all levels, from macroscopic down to the quantum level. Consider this..

1. There are 3 spacial dimensions.
But you can think of this as 3 families each containing 2 members..

Up/Down
Left/Right
Backward/Forwards

2. Quarks consist of 3 families each containing 2 members..

Up/Down
Charm/Strange
Top/Bottom

3. Quarks are bound together to form composite particles (such as protons) by an incredibly strong force known as the color force. Gluons are the particles that mediate this force. For this model to work quarks must have a color charge - and it comes in 3 families each containing 2 members..

Red/Anti-Red
Green/Anti-Green
Blue/Anti-Blue

4. Leptons consist of 3 families each containing 2 members..

Electron/Electron Neutrino
Muon/Muon Neutrino
Tau/Tau Neutrino

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

LHC Dashboard - Watch the LHC in Action

LHC Dashboard - Watch the LHC in Action

Check beam energy, ramps, beam status, machine tests, detector status and more. Plus, see messages from machine operators.

Watch LHC operations

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

Transcendental Numbers - A Simple Definition

Transcendental Numbers - A Simple Definition

Take any number and ask if it can be converted to a rational number by raising it to an positive integer power. If it can we say it's coupled to the integers. If not we say it's decoupled from the integers.

All rational numbers are coupled by definition. But so are many irrational numbers, for example sqrt(2) is irrational, but (sqrt(2))^2=2. Even many complex numbers are coupled, for example i^2=-1

So it seems most numbers are coupled to the integers, making integers fundamental. Not true. In fact Cantor showed the opposite is true, most numbers are decoupled from the integers. These are the (still mysterious) transcendental numbers. And the name is good, transcendental numbers "transcend" the integers.

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

Driverless Cars - Why They Will Never Happen

Driverless Cars - Why They Will Never Happen

Oh sure, you'll see driverless cars on the road. A few here and a few there. But they will never happen in a major commercial way for a very simple reason: people love to drive.

I do. It's a skill that took me a while to learn. I'm proud of my skill and I enjoy using it. A driverless car takes that pleasure away. Can you imagine being in a car that obeys all traffic laws including speed limits. It would be incredibly infuriating and amazingly boring.

Plus, on todays aggressive roads it would be outright dangerous. Go ahead, plod along at the 30 mph speed limit while 18 wheeler trucks swerve past you at 55 mph. No thanks.

But there's one good thing about driverless cars - as more come on the road they will put my driving skills to the test in order to avoid them!

Driverless cars are putting me to sleep. Wake me up when we have driverless trains and driverless subways.

You gotta love the driverless car industry - a brilliant solution to a problem that doesn't exist.

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

At What Age do Kids Understand Numbers?

At What Age do Kids Understand Numbers?

My grand daughter Evy is three. So we went to see her in Gym Class.

They hand out "shaker" noise makers at the beginning of each class. Evy said she would get Shakers for all of us. She came back with exactly 5. There were 5 of us including her. Coincidence? I wonder.

Or perhaps this is simply an example of Georg Cantor's "1-to-1 correspondence".
Content written and posted by Ken Abbott abbottsystems@gmail.com

Atomic Structure Explained in 5 Minutes

Atomic Structure Explained in 5 Minutes

After a century of 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 discover the internal structure of the proton.

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

Elementary Particle Physics - Spin Inversion Operator

Elementary Particle Physics - Spin Inversion Operator

I just had an idea for a "Spin Inversion Operator". It takes an elementary particle and inverts its spin. Example: the Graviton with spin 2 would map into a spin 1/2 particle. Likewise a spin 1/2 particle would map into a spin 2 particle.

The Photon is interesting, it has spin 1, so the Spin Inversion Operator does not change the spin. It's invariant.

Coming back to the Graviton again, assuming charge is conserved, the Spin Inversion Operator maps it into a neutral spin 1/2 particle. The only one known is the Neutrino. So is there some deep connection between the Graviton and the Neutrino?

The Spin Inversion Operator works for the Graviton and all elementary particles in the Standard Model - except for the Higgs Boson.

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

Ancient Egyptian Mathematics - The Great Pyramid of Giza

Ancient Egyptian Mathematics - The Great Pyramid of Giza

The Egyptians built The Great Pyramid at Giza as an amazing burial monument for Pharaoh Khufu. But they did more. They also used it as a showcase for their mathematical and engineering skills.

Pi is probably the most famous number in mathematics. Draw any circle, then measure the length of the circumference and the length of the diameter. Divide the two numbers and you get pi. But a circle was not used in the design of The Great Pyramid of Giza, right? Wrong! Not only was a circle used it was totally fundamental to the design. Here's how..

The Khufu Pyramid (The Great Pyramid of Giza) had a design height of 280 royal cubits and a base length of 440 royal cubits. The Pyramid we see today is slightly different due to erosion and theft of stone. So let's stick with the original design size.

So the distance around the base of the pyramid is simply 4*440=1760

Let's divide this by twice the height, which is 560, so we get..

1760/560=22/7

The number 22/7 is the most famous approximation of pi, and it's a pretty good one, in fact..

22/7-pi=0.001

So the Khufu Pyramid was built on circular geometry. Which means the Egyptians knew pi by the time they built the Great Pyramid (2560 BC). For all we know they may have known it much earlier.

But it gets stranger, not only did they know pi, they used it to define the dimensions of their most sacred monument. What does this mean?

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

Mobius Strip - Yet Another Surprise

Mobius Strip - Yet Another Surprise

The Mobius strip is a simple object yet full of surprises. Here's one you may not know about..

Take a strip of paper, 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?

But here's the surprise..

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!

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

Prime Numbers in Base 2

Prime Numbers in Base 2

We're used to seeing numbers represented in base 10 "decimal" notation, and almost all prime number lists use base 10. 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 2 symbols 0,1

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

In binary the positions represent 1,2,4,8,16,32,.. so the general representation of a positive integer n is..

n=sum{ai*(2^i)} where the coefficients {ai} are all 0 or 1 and the sum is over i from 0 onward.

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

Writing numbers in binary can help spot patterns we might not notice in other bases. For example..

In binary all prime numbers except 2 begin and end with 1.

The first 2 digits of the prime 71 is the prime 3 and the last 5 digits is the prime 17. So we could define a "+" operation and say that 3+17=71. Notice that the + operation depends on order, so 17+3=113 is different, but it's still a prime!

The prime 13 is just the prime 11 written backwards. The same is true for 23 and 29 and lots more. Many primes are just an earlier prime written backwards!

Some primes have all digits set to 1, so these primes are of the form (2^n)-1 where n is just the number of binary digits. Primes of this form are called Mersenne primes, named after Marin Mersenne, a French monk who studied them in the 17th century.

I wonder what we might discover if we used sophisticated computer pattern recognition on prime numbers in binary format?

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

Teaching Fractions - The Fractional Bakery

Teaching Fractions - The Fractional Bakery

Imagine you own a bakery and the only thing you sell are loaves of bread. Not only that, but all your loaves are identical, which means they are all the same size and everything else is the same.

So when customers come into your shop they just tell you how many loaves they want..

{1,2,3,4,...}

One day a customer comes in and explains that they love your bread but your loaves are too big. They ask if you make smaller loaves. You don't. But then you have a clever idea. You take a knife and cut a loaf into 2 equal sized pieces. You sell one piece to the customer and they are happy.

But what did you just sell? It was not a loaf. It was something less. You chopped a loaf into 2 equal pieces and sold one of the pieces. You sold "one out of two", so you could write that as 1/2.

This idea is popular with your customers. Soon you are chopping your loaves into 5 equal pieces and selling customers 1, 2, 3 or 4 of the pieces. That's 1/5, 2/5, 3/5 or 4/5.

Of course, if you sold a customer 5 out of 5 then that's the same as the whole loaf, so 5/5=1.

One day a really fussy customer comes into your bakery and asks for 5/8. You know exactly what to do. You take a loaf, chop it into exactly 8 equal size pieces and then sell the customer 5 of the pieces.

These funny looking things like 1/2 and 5/8 are called fractions. Mathematicians call them rational numbers. That's the fancy mathematical name for them.



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

Physics - Who was JJ Thompson?

Physics - Who was JJ Thompson?

The electron was discovered over 100 years ago, in 1897 by JJ Thompson at the University of Cambridge in England. It was the first elementary particle to be discovered and it's still as elementary as ever. Elementary means no internal structure has been found - so far.

Moreover, it now rules our world. Without it there would be no electricity. That means no lights, no TV, no batteries, no computers, no Internet, no iPhones.. to name just a few. Who says elementary particles are abstract objects!

Who was JJ Thompson? He was born in Manchester. His mom came from a local textile family. His dad ran a bookstore. He changed the world forever. His list of students reads like a who's who of physics. He won a Nobel Prize and so did many of his students.

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

IXL Math - For Kids, Parents and Teachers

IXL Math - For Kids, Parents and Teachers

IXL Math is a online math education resource for ages Pre-K through 12th grade. It covers everything from counting to Calculus.

IXL Math comes in two major versions - one for parents and one for teachers.

Parents use IXL Math at home to help their kids, and teachers use it in the classroom.

IXL Math is offered on a subscription basis and has about 6 million subscribers. It's used in over 190 countries. Their website offers guest access where you can try over 6,000 interactive math skills for FREE. Try before you buy.

IXL Math is a product of IXL Learning, 777 Mariners Island Blvd., Suite 600, San Mateo, CA 94404 (USA)

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

Why I Write a Math & Physics Blog

Why I Write a Math & Physics Blog

There's three reasons I write this blog..

First, I want to introduce people to math and physics concepts in a simple and casual way. It's easy to use textbooks without really understanding the basic ideas. I want to avoid that. I'm interested in explaining fundamental concepts and ideas.

Second, I want to get people interested in math and physics. Good teachers don't just teach, they create a lifelong desire to learn.

Last but not least, I want to learn math and physics. There's no better way to learn a topic than to explain it clearly and simply to others. I spend a lot of time trying to improve my posts.

So far so good. My blog has worldwide readership and I get a lot of feedback. It helps. I'm always going back and tweaking posts to try and make them better.

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

Differential Calculus Explained in 5 Minutes

Differential Calculus Explained 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 different 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.

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

Braille Explained in 5 Minutes

Braille Explained in 5 Minutes

Braille was invented by Louis Braille in 1837 and was the first binary form of writing developed in the modern era.

Braille was based on a tactile military code called night writing, developed by Charles Barbier in response to Napoleon's demand for a means for soldiers to communicate silently at night without a light.

Today, computer professionals will instantly recognize this as 6-bit encoding. Perhaps the first byte was 6 bits!



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

Prime Number Distribution and Shannon Entropy

Prime Number Distribution and Shannon Entropy

The Prime Number Theorem is one of the most famous theorems in mathematics. It tells us something about the distribution of the prime numbers.

How many of the first n integers 1,2,3,4,....,n are prime? The Prime Number Theorem says the number of primes is approximately n/log(n)

This is not an exact count, n/log(n) is only an approximation, but as n gets bigger the approximation gets better and better.

The Prime Number Theorem is also a statement about the Shannon Entropy of the primes! Here's how..

Suppose you have a machine with a big red button. Each time you punch the button the machine responds by displaying an integer in the range 1,2,3,....,n. After much experimentation you discover that the probability of getting integer j is pj. Then physics defines the Shannon Entropy of this machine as..

Shannon Entropy=(-1)*sum (pj*log(pj)) for j=1,2,3,...,n

In the special case where all numbers occur with equal probability pj=1/n for all j and we get the famous result for the Shannon Entropy of the machine..

Shannon Entropy=(-1)*n*(1/n)*log(1/n)=log(n)

Now imagine this is "distributed" equally across all numbers, so on average an individual integer has log(n)/n entropy.

If the integers 1,2,3,....,n contain m primes then the Shannon Entropy of the primes is simply m*log(n)/n

But the prime number theorem says that m=n/log(n) approximately. So the approximate Shannon Entropy becomes..

Shannon Entropy=m*log(n)/n=1

and as n approaches infinity this approximation becomes exact. So we can say that..

"The Shannon Entropy of the primes is 1".

This statement is equivalent to the prime number theorem. How strange!

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

Do Black Holes Have Cores?

Do Black Holes Have Cores?

To make a deep prediction about black holes and 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.

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 flip 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's 2 dramatic predictions..

1. Black holes contain a neutral spin 1/2 core
You could imagine that graviton oscillation requires extremely high graviton pressure - meaning it only occurs in super intense gravitational fields - such as the center of a black hole. This means the center of a black hole would not be a singularity as predicted by General Relativity, it would be core consisting of neutral spin 1/2 particles. You can think of this core as a "graviton condensate".

2. 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.

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 even 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!

So 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 new objects that were discovered. The basic principles were the same, but the mathematics got even 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 became legends. 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!

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

Primes Within Primes - Prime Numbers that contain Prime Numbers

Primes Within Primes - Prime Numbers that contain Prime Numbers

If you write a prime number in binary you can sometimes split it into 2 segments that are also primes.

For example, the prime number 7591 is 1110110100111 in binary (I use the left to right convention). 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!

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

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. 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

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. 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!

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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

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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

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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.

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

Prime Numbers Explained a Different Way

Prime Numbers Explained 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.

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.

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

A Strange Black Hole Prediction

A Strange Black Hole 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.

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.

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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 just correspond to the electron spinning clockwise or counter clockwise). So 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?

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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.

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