tag:blogger.com,1999:blog-89207959897604283892017-04-26T06:32:27.802-07:00Math MathLearn Math with an eclectic list of short articles. Not your usual Math.Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comBlogger137125tag:blogger.com,1999:blog-8920795989760428389.post-79154286375537406152017-04-03T10:36:00.001-07:002017-04-26T06:32:27.869-07:00Egyptian Mathematics - The Great Pyramid of Giza<font size="4"> <b>Egyptian Mathematics - The Great Pyramid of Giza</b><br><br> 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.<br><br> 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..<br><br> 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.<br><br> So the distance around the base of the pyramid is simply 4*440=1760<br><br> Let's divide this by twice the height, which is 560, so we get..<br><br> 1760/560=22/7<br><br> The number 22/7 is the most famous approximation of pi, and it's a pretty good one, in fact..<br><br> 22/7-pi=0.001<br><br> 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. <br><br> 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?<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-41133329145761032152017-02-16T07:59:00.002-08:002017-02-19T12:00:24.441-08:00Mobius Strip - A New Surprise <font size="4"> <b>Mobius Strip - A New Surprise</b><br><br> The Mobius strip is such a simple object, yet full of surprises. Here's one you may not know about..<br><br> 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? <br><br> <b>But here's the surprise..</b> <br><br> 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!<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-16275628815798117542017-02-15T12:16:00.001-08:002017-02-15T12:16:38.612-08:00Prime Numbers in Base 2<font size="4"> <b>Prime Numbers in Base 2</b><br><br> 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)<br><br> The simplest base is 2, because in that base we have only 2 symbols 0,1<br><br> Base 2 is also call "binary" and writing numbers in binary makes them look like computer data. <br><br> In binary the positions represent 1,2,4,8,16,32,.. so the general representation of a positive integer n is..<br><br> n=sum{ai*2^i} where the coefficients {ai} are all 0 or 1 and the sum is over i from 0 onward.<br><br> For example, the number 11 in binary is 1101, and this simply means..<br><br> 11=1*(1)+1*(2)+0*(4)+1*(8)<br><br> Here's the first 20 prime numbers in binary..<br><br> 2 01<br>3 11<br>5 101<br>7 111<br>11 1101<br>13 1011<br>17 10001<br>19 11001<br>23 11101<br>29 10111<br>31 11111<br>37 101001<br>41 100101<br>43 110101<br>47 111101<br>53 101011<br>59 110111<br>61 101111<br>67 1100001<br>71 1110001<br><br> Writing numbers in binary can help spot patterns we might not notice in other bases. For example..<br><br> In binary all prime numbers except 2 begin and end with 1. <br><br> 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!<br><br> 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!<br><br> 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.<br><br> I wonder what we might discover if we used sophisticated computer pattern recognition on prime numbers in binary format?<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-80644356104368694182017-02-14T12:35:00.002-08:002017-02-15T12:13:45.810-08:00The Atomic Walker <font size="4"> <b>The Atomic Walker</b><br><br> Imagine 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 scientists have not discovered any internal - so far.<br><br> 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 it's made of two type of particles - protons and neutrons - bound together very tightly.<br><br> 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.<br><br> For a while scientists thought protons and neutrons were elementary, but not so. It turns out they are complicated objects. So scientists built particle colliders to learn more about the structure of the proton. The most famous is the LHC (Large Hadron Collider) in Switzerland. The method is simple - crash two protons together an incredible speed to see what comes out. From this scientists learn what's inside the proton.<br><br> It turns to that 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.<br><br> Since the proton is amazingly small why would nature decide to give it such complex internal structure?<br><br> That's a philosophical question scientists can never answer. But they are working hard to learn more about the proton. <br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-88453191606038942392017-02-14T06:29:00.000-08:002017-02-14T06:29:05.826-08:00The Fractional Bakery<font size="4"> <b>The Fractional Bakery</b><br><br> 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. <br /><br /> So when customers come into your shop they just say how many loaves they want..<br /><br /> {1,2,3,4,...} <br /><br /> 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.<br /><br /> 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.<br /><br /> 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.<br /><br /> Of course, if you sold a customer 5 out of 5 then that's the same as the whole loaf, so 5/5=1.<br /><br /> 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.<br /><br /> 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. <br /><br /> <br /><br /> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-36049227518743038582017-02-10T05:39:00.003-08:002017-02-10T05:39:27.113-08:00Who was JJ Thompson?<font size="4"> <b>Who was JJ Thompson?</b><br><br> 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.<br><br> 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! <br><br> 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.<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-50912987441723389382017-01-23T10:14:00.001-08:002017-01-23T10:14:20.215-08:00IXL Math - For Kids, Parents and Teachers <font size="4"> <b>IXL Math - For Kids, Parents and Teachers</b><br><br> IXL Math is a online math education resource for ages Pre-K through 12th grade. It covers everything from counting to Calculus.<br><br> IXL Math comes in two major versions - one for parents and one for teachers.<br><br> Parents use IXL Math at home to help their kids, and teachers use it in the classroom.<br><br> 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.<br><br> IXL Math is a product of IXL Learning, 777 Mariners Island Blvd., Suite 600, San Mateo, CA 94404 (USA)<br><br> </font> <b>Like this post? Please click below to share it.</b><br>Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>ken abbotthttps://plus.google.com/117501804565696782332noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-90823227233395429422017-01-14T06:32:00.002-08:002017-01-17T09:15:09.019-08:00Why I Write a Math & Physics Blog <font size="4"> <b>Why I Write a Math & Physics Blog</b><br><br> There's three reasons I write this math blog.. <br><br> 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. <br><br> Second, I want to get people interested in math and physics. Good teachers don't just teach, they create a lifelong desire to learn. <br><br> 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. <br><br> So that's my math. 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.<br><br> </font> <b>Like this post? Please click below to share it.</b><br>Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-4237344652332712932017-01-03T14:06:00.002-08:002017-02-04T07:37:32.014-08:00Differential Calculus Explained in 5 Minutes <font size="4"> <b>Differential Calculus Explained in 5 Minutes</b><br><br> 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.<br><br> Calculus is all about functions, so there's no point in studying calculus until you understand the idea of a function.<br><br> Let's take a simple function, say f(x)=x^2<br><br> 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..<br><br> 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.. <br><br> f(a+q)=(a+q)^2=a^2+2*a*q+q^2<br><br> 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..<br><br> f(a+q)=(a+q)^2=a^2+2*a*q<br><br> Notice the first term, a^2, is just the value of the function at a, f(a), so now we have..<br><br> f(a+q)=f(a)+2*a*q<br><br> Which means..<br><br> f(a+q)-f(a)=2*a*q<br><br> And so..<br><br> (f(a+q)-f(a))/q=2*a<br><br> 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..<br><br> Limit(f(a+q)-f(a))/q as q goes to zero is 2*a<br><br> Of course we could do this for any point a, so in general..<br><br> Limit(f(x+q)-f(x))/q as q goes to zero is 2*x<br><br> This is called the "derivative" of f(x) and is often written as df/dx, or sometimes as f'. So, to summarize..<br><br> 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).<br><br> Congratulations, you just did some calculus! You differentiated the function f(x)=x^2 and got the result 2*x<br><br> To generalize this example, the derivative of the function f(x)=x^n where n is any integer is..<br><br> df/dx=n*x^(n-1)<br><br> So for example, if f(x)=x^10 then the derivative is df/dx=10*x^9<br><br> 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!<br><br> 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.<br><br> </font> <b>Like this post? Please click below to share it.</b><br>Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-33911659091130081842016-12-12T04:41:00.001-08:002017-01-16T06:15:35.949-08:00Braille Explained in 2 Minutes <font size="4"> <b>Braille Explained in 2 Minutes</b><br><br> Braille was invented by Louis Braille in 1837 and was the first binary form of writing developed in the modern era.<br><br> 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.<br><br> Today, computer professionals will instantly recognize this as 6-bit encoding. Perhaps the first byte was 6 bits!<br><br> <div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-NDr26SxS3W4/WE6a_VIw95I/AAAAAAAAFpc/jZdwRTlmEq0Ner59FkJzNVUrPU8AMIi1QCLcB/s1600/braille-alphabet.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-NDr26SxS3W4/WE6a_VIw95I/AAAAAAAAFpc/jZdwRTlmEq0Ner59FkJzNVUrPU8AMIi1QCLcB/s320/braille-alphabet.jpg" width="304" height="320" /></a></div><br><br> </font> <br> <b>Like this post? Please click below to share it.</b><br>Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-42668608685139530522016-10-05T05:53:00.004-07:002017-01-16T06:15:53.076-08:00Prime Number Distribution and Shannon Entropy <font size="4"> <b>Prime Number Distribution and Shannon Entropy</b><br><br> The Prime Number Theorem is one of the most famous theorems in mathematics. It tells us something about the distribution of the prime numbers.<br><br> 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)<br><br> This is not an exact count, n/log(n) is only an approximation, but as n gets bigger the approximation gets better and better.<br><br> The Prime Number Theorem is also a statement about the Shannon Entropy of the primes! Here's how..<br><br> 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..<br><br> Shannon Entropy=(-1)*sum (pj*log(pj)) for j=1,2,3,...,n<br><br> 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..<br><br> Shannon Entropy=(-1)*n*(1/n)*log(1/n)=log(n)<br><br> Now imagine this is "distributed" equally across all numbers, so on average an individual integer has log(n)/n entropy.<br><br> If the integers 1,2,3,....,n contain m primes then the Shannon Entropy of the primes is simply m*log(n)/n<br><br> But the prime number theorem says that m=n/log(n) approximately. So the approximate Shannon Entropy becomes.. <br><br> Shannon Entropy=m*log(n)/n=1<br><br> and as n approaches infinity this approximation becomes exact. So we can say that..<br><br> "The Shannon Entropy of the primes is 1".<br><br> This statement is equivalent to the prime number theorem. How strange!<br><br> </font> <b>Like this post? Please click below to share it.</b><br>Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-80340704400245169272016-09-01T11:30:00.000-07:002017-01-16T06:16:12.341-08:00Do Black Holes Have Cores? <font size="4"> <b>Do Black Holes Have Cores?</b><br><br> To make a deep prediction about black holes and quantum gravity we first need to play with paper strips!<br><br> 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.<br><br> Now give the paper strip 1 half twist before joining the ends. This is our model for a spin 1/2 particle. <br><br> <b>But things get strange.. </b> <br><br> 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! <br><br> 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. <br><br> 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..<br><br> <b>1. Black holes contain a neutral spin 1/2 core </b><br>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".<br><br> <b>2. The Law of Conservation of Angular Momentum is violated</b> <br>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.<br><br> </font> <b>Like this post? Please click below to share it.</b><br>Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br>Learn Mathematics and Physics<br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-88220281026739636922016-05-24T05:11:00.000-07:002017-01-16T06:16:36.965-08:00Physics Explained in 5 Minutes <font size="4"> <b>Physics Explained in 5 Minutes</b><br><br> 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.<br><br> It was a remarkable achievement. Newton's Laws of Motion remained the cornerstone of physics for centuries.<br><br> 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! <br><br> 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.<br><br> 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!<br><br> 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.<br><br> 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.<br><br> 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!<br><br> So if you had to summarize Quantum Mechanics in one sentence try this.. "small objects behave very differently than large objects". Who knew!<br><br> What happened to Einstein? <br>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. <br><br> What happened to Quantum Mechanics? <br>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.<br><br> 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.<br><br> Where does physics stand today? <br>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. <br><br> Perhaps we need a new physics graduate - preferably one who can't find a job!<br><br> </font> <b>Like this post? Please click below to share it.</b><br>Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br>Learn Mathematics and Physics<br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-29227918631299702172016-05-16T08:04:00.003-07:002016-05-24T05:17:56.273-07:00Primes Within Primes <font size="4"> <b>Primes Within Primes</b><br><br> 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. <br><br> 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. <br><br> So, denoting a binary string concatenation operator by "+" we can say 7591=29+167<br><br> 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".<br><br> Let's do a simple example with our new operator..<br><br> In binary 3 is 11 and 5 is 101 so 3+5=11101 which is 29.<br><br> But 5+3=10111 which is 23.<br><br> By the way, notice that all these numbers 3, 5, 23, 29 are prime!<br><br> </font> <b>----> Read more posts <a style="text-decoration:underline" href="http://www.math-math.com">here.</a></b><br><br> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-42074612538443753662016-05-11T06:25:00.000-07:002016-05-24T05:18:04.993-07:00Collatz Conjecture as a Computer Program <font size="4"> <b>Collatz Conjecture as a Computer Program</b><br><br> One of the most famous unsolved mathematical conjectures totally lends itself to computer investigation.<br><br> It's the Collatz Conjecture, named after Lothar Collatz, who first proposed it in 1937. The great mathematician StanisÅ‚aw Ulam not only failed to prove it but said, "perhaps mathematics is not ready for such problems".<br><br> Here it is as a computer program..<br><br> Pick any positive integer n<br>INFINITE LOOP<br>If n is even replace it by n/2<br>If n is odd replace it by 3*n+1<br>If n=1 bail out of loop<br>LOOP<br><br> 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.<br><br> 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<br><br> 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!<br><br> </font> <b>----> Read more posts <a style="text-decoration:underline" href="http://www.math-math.com">here.</a></b><br><br> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-2052568720797194512016-05-09T04:46:00.004-07:002016-12-20T04:58:17.135-08:00Atomic Physics Explained in 2 Minutes <font size="4"> <b>Atomic Physics Explained in 2 Minutes</b><br><br> After a century of hard work physicists have established some impressive facts about atoms.<br><br> 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.<br><br> 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.<br><br> 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.<br><br> 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.<br><br> 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.<br><br> Given than the proton is amazingly small why would nature decide to give it such complex internal structure?<br><br> That's a philosophical question physicists can never answer. But they are working hard to document the internal structure of the proton. <br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-37016247498260326982016-05-06T08:48:00.003-07:002016-11-28T07:20:53.964-08:00Differential Calculus Explained in 5 Minutes <font size="4"> <b>Differential Calculus Explained in 5 Minutes</b><br><br> 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.<br><br> Calculus is all about functions, so there's no point in studying calculus until you understand the idea of a function.<br><br> Let's take a simple function, say f(x)=x^2<br><br> 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..<br><br> 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.. <br><br> f(a+q)=(a+q)^2=a^2+2*a*q+q^2<br><br> 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..<br><br> f(a+q)=(a+q)^2=a^2+2*a*q<br><br> Notice the first term, a^2, is just the value of the function at a, f(a), so now we have..<br><br> f(a+q)=f(a)+2*a*q<br><br> Which means..<br><br> f(a+q)-f(a)=2*a*q<br><br> And so..<br><br> (f(a+q)-f(a))/q=2*a<br><br> 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..<br><br> Limit(f(a+q)-f(a))/q as q goes to zero is 2*a<br><br> Of course we could do this for any point a, so in general..<br><br> Limit(f(x+q)-f(x))/q as q goes to zero is 2*x<br><br> This is called the "derivative" of f(x) and is often written as df/dx, or sometimes as f'. So, to summarize..<br><br> 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).<br><br> Congratulations, you just did some calculus! You differentiated the function f(x)=x^2 and got the result 2*x<br><br> To generalize this example, the derivative of the function f(x)=x^n where n is any integer is..<br><br> df/dx=n*x^(n-1)<br><br> So for example, if f(x)=x^10 then the derivative is df/dx=10*x^9<br><br> 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!<br><br> 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.<br><br> </font> <b>----> Read more posts <a style="text-decoration:underline" href="http://www.math-math.com">here.</a></b><br><br> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br> Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-53603944123158013592016-05-02T07:17:00.000-07:002016-05-03T19:33:44.649-07:00Simple Harmonic Oscillator - Using Difference Calculus<font size="4"> <b>Simple Harmonic Oscillator - Using Difference Calculus</b><br><br> 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!<br><br> Consider a function f(n) of an integer variable n defined by this..<br><br> f(n)=k*f(n-1)-f(n-2) <br><br> where n=2,3,4,5,.. and k is a constant.<br><br> If k, f(0) and f(1) are given then f(n) can be calculated for any n. <br><br> 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..<br><br> f(n)=sin(n*a)<br><br> 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. <br><br> 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.<br><br> 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. <br><br> Of course, you may have noticed that things are not exactly the same.. time is no longer a continuous variable!<br><br> </font> <b>----> Read more posts <a style="text-decoration:underline" href="http://www.math-math.com">here.</a></b><br><br> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-1596063367912457542016-04-30T08:14:00.006-07:002016-05-03T19:34:28.350-07:00AMS - American Mathematical Society<font size="4"> <b>AMS - American Mathematical Society</b><br /><br /> 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. <br><br> <a style="text-decoration:underline" href="http://www.ams.org/">AMS Website</a><br><br> </font> <b>----> Read more posts <a style="text-decoration:underline" href="http://www.math-math.com">here.</a></b><br><br> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-48576466536221744702016-04-28T07:42:00.001-07:002016-05-03T19:34:43.173-07:00MAA - Mathematical Association of America<font size="4"> <b>MAA - Mathematical Association of America</b><br /><br /> The MAA (Mathematical Association of America) is a professional society who's goal is to advance the mathematical sciences, especially at the university level.<br><br> It has a broad range of members, from high school students to professional mathematicians. Anyone is invited to join!<br><br> <a style="text-decoration:underline" href="http://www.maa.org">MAA Website</a><br><br> </font> <b>----> Read more posts <a style="text-decoration:underline" href="http://www.math-math.com">here.</a></b><br><br> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-50049584000303823732016-04-28T06:01:00.001-07:002016-12-21T12:18:47.225-08:00Algebra Explained in 5 Minutes<font size="4"> <b>Algebra Explained in 5 Minutes</b><br /><br /> Consider this problem, "what number, when added to 5, gives the result 21". <br /><br /> Instead of a sentence, this problem can be written much shorter and clearer as an equation, like this.. <br /><br /> 5+x=21 <br /><br /> where x denotes the number we are trying to find. <br /><br /> 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.<br /><br /> 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.<br /><br /> 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.<br /><br /> 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!<br /><br /> 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!<br /><br /> 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. <br /><br /> Bingo, we've solved the equation without any guessing!<br /><br /> 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!<br /><br /> Let's look at a slightly more complicated example..<br /><br /> 3*x+2=17<br /><br /> 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!<br /><br /> First move the 2 over to the other side. It was adding, so when it moves over it subtracts, like this..<br /><br /> 3*x=17-2=15<br /><br /> Now move the 3 over. It was multiplying, so when it moves over it divides, like this..<br /><br /> x=15/3=5<br /><br /> 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.<br /><br /> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br> Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-45379299271804040462016-04-28T05:48:00.001-07:002017-01-17T09:16:52.313-08:00Teaching Prime Numbers a Different Way <font size="4"> <b>Teaching Prime Numbers a Different Way</b><br><br> Let's consider the positive integers greater than 1, that is 2,3,4,5,.. <br><br> Suppose we are given the first integer and asked to make all other integers using only the multiply operation.<br><br> We soon run into problems because 2*2=4 and we have no way to make 3.<br><br> OK, we just add 3 to our set of given numbers g, so now g={2,3}<br><br> Can we make 4? Yes, 2*2=4<br><br> Can we make 5? No, all our tries fail, so we add 5 to our set of given number g={2,3,5}<br><br> Can we make 6? Yes, 2*3=6<br><br> Can we make 7? No, all our tries fail, so we add 7 to our given numbers g={2,3,5,7}<br><br> Can we make 8? Yes, 2*2*2=8<br><br> Can we make 9? Yes, 3*3=9<br><br> Can we make 10? Yes, 2*5=10<br><br> Can we make 11? No, so we add it to the set g={2,3,5,7,11}<br><br> Can we make 12? Yes, 2*2*3=12<br><br> Can we make 13? No, so add it to the set g={2,3,5,7,11,13}<br><br> Can we make 14? Yes, 2*7=14<br><br> Can we make 15? Yes, 3*5=15<br><br> Can we make 16? Yes, 2*2*2*2=16<br><br> 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. <br><br> 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. <br><br> </font> <b>----> Read more posts <a style="text-decoration:underline" href="http://www.math-math.com">here.</a></b><br><br> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br> Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-33495045683739229972016-04-26T05:37:00.001-07:002017-04-19T11:46:30.849-07:00A Number Bigger Than Infinity<font size="4"> <b>A Number Bigger Than Infinity</b><br><br> 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.<br><br> 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". <br><br> His work stands as one of the most elegant pieces of mathematics ever.<br><br> So what did Cantor do?<br><br> 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..<br><br> 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!<br><br> 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..<br><br> aleph1=2^aleph0<br><br> He even asked if there was an aleph number between aleph0 and aleph1. <br><br> 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.<br><br> </font> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-26521738083365787702016-04-24T08:02:00.002-07:002017-01-17T09:23:22.368-08:00A Strange Black Hole Prediction <font size="4"> <b>A Strange Black Hole Prediction</b><br><br> To make this prediction we first need to play with paper strips!<br><br> Take a strip of paper, join the ends, so you have a band. This is a model for a spin 0 particle.<br><br> Now give the paper strip 1 half twist before joining the ends. This is a model for a spin 1/2 particle. <br><br> Now give the paper strip 2 half twists before joining the ends. This is a spin 1 particle. <br><br> <b>But things get strange.. </b> <br><br> 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! <br><br> 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. <br><br> 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..<br><br> <b>Black holes evaporate </b><br>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. <br><br> <b>Black Holes are an intense source of neutrinos</b><br>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.<br><br> <b>The Universe is expanding </b><br>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.<br><br> <b>Intense gravitational fields are the source of dark matter </b><br>Could the neutral spin 1/2 fermion particle account for dark matter? i.e. dark matter is produced by the decay of black holes.<br><br> <b>The Law of Conservation of Angular Momentum is violated</b> <br>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.<br><br> </font> <b>----> Read more posts <a style="text-decoration:underline" href="http://www.math-math.com">here.</a></b><br><br> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br>Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.comtag:blogger.com,1999:blog-8920795989760428389.post-81019206173639821932016-04-20T04:53:00.000-07:002016-05-03T19:36:05.729-07:00Sets Explained in 5 Minutes <font size="4"> <b>Sets Explained in 5 Minutes</b><br><br> 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.<br><br> Let's consider a simple example, the set containing the first 3 letters of the alphabet..<br><br> S={a,b,c}<br><br> Can we say anything mathematically interesting about this set?<br><br> 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..<br><br> 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.<br><br> {a} is a subset, so is {a,c}, so is {b,c}<br><br> 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..<br><br> { }<br>{a}<br>{b}<br>{c}<br>{a,b}<br>{a,c}<br>{b,c}<br>{a,b,c}<br><br> 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..<br><br> 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!<br><br> 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.<br><br> </font> <b>----> Read more posts <a style="text-decoration:underline" href="http://www.math-math.com">here.</a></b><br><br> Content written and posted by Ken Abbott <a style="text-decoration:underline" href="mailto:abbottsystems@gmail.com">abbottsystems@gmail.com</a><br><br> Ken Abbotthttps://plus.google.com/101487824185426724709noreply@blogger.com