Pi, Lie, Same Thing Part 3: Tau 2000

This week, I wanted to take a break from the hardcore “Cool Math Stuff” and talk to you about something a little different. Since we have just been talking about the Pi vs Tau argument, I wanted to bring up an event that I am hosting that involves Tau.

On May 6, 2012, I will be establishing a new world record by reciting from memory 2000 digits of Tau. This event will be a fundraiser for the Bethel Public Library. The event will also include magic, raffles, contests, and more.

If you would like to find out more about this event and see how you can help out, please go to www.Tau2000.com.

Bonus: When I was teaching at Westport Minds in Motion this past Saturday, I heard a puzzle that isn’t super mathematical, but does involve some mathematical logic and reasoning. It was a really cool one, so I thought I would share it.

You are in your basement and there are three light switches. Each switch controls a bulb upstairs, but you do not know which switch controls which bulb. You also cannot see any of the bulbs, or have any mean of finding out if the bulb is on or off (helpers, really well trained dogs, face time, etc.). You are also feeling kind of lazy, and will only go up the stairs once. With all of this in mind, how do you figure out which switch controls which bulb.

I will post the answer to this puzzle in about a month for anybody who wants to try to figure it out, which I definitely suggest trying.

Pi, Lie, Same Thing... (Part 2)

Last week, I did some explaining on why tau is better than pi. However, that post was relating it to past posts more and using the popular reasons less. Let me bring up some of the more popular reasons that are a little easier to follow.

First off, we looked at trigonometric reasons last week, but the “homeland” so to speak of both of these numbers is the circle. This week, we will only look at circles.

People measure angles in degrees usually. There are 360° in a circle. Mathematicians, however, measure angles in radians. There are 2π radians in a circle. No wonder. In a third of a circle, there are 2π/3 radians. A third of a circle corresponds to this mess!

What about a quarter circle. This is equivalent to 1/2π. Still, a quarter corresponds to a half. That’s also crazy.

What if we used tau? A third of a circle is a third of tau. A quarter circle is a quarter of tau. 28/53 of a circle is 28/53 of tau. Does it get any easier?

You might be thinking that I am only showing you one side of the story. Obviously, this is partly true, but let’s look an argument that a pi person might bring up. What about the area formula? This is just a plain old pi.

Yes, but one of the most common proofs for the area formula (I will definitely post this at some point) ends up with the following equation, which you have to simplify:

1/2 x 2πr^2

So basically, there was a 2 there, but it got cancelled out. The new area formula is:

1/2τr^2

So, the 1/2 does us a favor. It helps us remember the proof.

This might not be good enough for you, so let me bring up how the 1/2 makes it more natural as well. What is the area formula for a triangle:

1/2bh

What about a trapezoid?

1/2h(b₁ + b₂)

Even the universal polygon area formula starts with a 1/2. So, the new formula just makes the circles fit in more with the other geometric shapes. I think this is a good thing, not a bad thing.

Next week, I won’t be talking as much about tau, but about what I have done with tau over the past few months. I’ll bet some of you might already know what it will be about. For a pretty good hint, check out www.Tau2000.com.

Pi, Lie, Same Thing...

Last week, you might remember that I mentioned something about Euler’s Identity getting better. In fact, it makes it more natural too, as I will show you. But before we go all the way there, let’s begin with the basics.

First off, let’s start with pi. Pi is approximately equal to 3.14 and is the ratio of a circle’s circumference to its diameter. It is probably considered to be one of the most important numbers in geometry and trigonometry. You might hear pi referred to as the “circle constant.”

The true circle constant should be a number that comes out of the definition of a circle, correct? And what is the definition of a circle? According to dictionary.com, this would be:

A closed plane curve consisting of all points at a given distance from a point within it called the center.

This definition just defined two lengths in the circle:

The circumference: A closed plane curve

The radius: A given distance from a point within it called the center

In other words, the circle constant should be the ratio between the circumference and the radius, right? This number comes out to the number tau. Tau is in fact 2π, but it is called tau since it is quicker and easier to say and write it that way. There is no doubt that tau should be the circle constant.

What about all of pi’s trigonometric significance, and other properties involving circles. Pretty much all of them are made better with tau. Next week, I will show you that even things that are made more complicated with tau are still more natural or helpful.

However, I didn’t want to keep you waiting too long about Euler’s Identity. Here it is again:

e^iπ = -1

Its rewritten form that we call God’s Equation is below:

e^iπ + 1 = 0

What if we use tau? It looks like this:

e^iτ = 1

Pretty good! Tau just took the little ugly thing and fixed it. Some people are actually upset that we lost a little zero, since zero is significant too. Well, no worries!

e^iτ = 1 + 0

I actually found out something about the new version of Euler’s Identity that was really surprising, and mixes in another past cool math stuff post which can never be bad. Remember the very first post on imaginary numbers back in October when we learned about the cube roots of one? If you haven’t definitely check it out. It was a really awesome post. Just search “cube root” and it will come up.

This post taught that there are two square roots of one, three cube roots of one, four fourth roots of one, and so on. Well, turns out that these roots can be found with the new Euler’s Identity.

e^iτ/1 = 1

e^iτ/2 = -1

e^iτ/3 = -1/2 + i√3/2

e^iτ/4 = i

e^iτ/6 = 1/2 + i√3/2 

e^iτ/8 = √2/2 + i√2/2

e^iτ/12 = √3/2 + i/2

When I saw that, all I was shocked that that actually worked, but it also makes tau such a beautiful number. Don’t get me wrong, e and i definitely get credit, but they don’t have such promising numbers to take their place.

Since tau has so many cool things about it, I am turning it into one of my little three week series, like I did with graphing calculators and at CTY. Even though it’s kind of late, have a happy half-tau day!

A very interesting post - literally

We have talked about many different numbers and their fascinating patterns, but there is one number that is extremely important that we have never touched upon. In fact, it is the most important number in calculus. This number is known as e.

Before I tell you what e is equal to, I want to try something first. Let’s do the following problem:

(1 + 1/10)^10

Multiply it out on your calculator and you will get something like 2.5937... What if we substituted that ten with one hundred? Would it be higher or lower? Let’s see:

(1 + 1/100)^100 = 2.7048...

What about if we use 1000? Then, it becomes:

(1 + 1/1000)^1000 = 2.7169...

What about if we use a million? Well, let’s see how big this gets.

(1 + 1/1000000)^1000000 = 2.7182804...

A billion? That would be giant!

(1 + 1/1000000000)^1000000000 = 2.7182814...

It is getting closer and closer to a number that is about 2.71828. You guessed it. This number is e.

What about if we did this:

(1 + 2/1000000000)^1000000000 = 7.389055...

Turns out that that is getting closer and closer to, or converging to e². If we were to put a three there, then it would be converging to e³. A basic formula for this would be:

(1 + x/N)^N converges to e^x

Now that we know a little background on e, let’s start looking at some applications. Let’s look at compound interest for a moment. Say you had 5000 dollars in the bank and earned 5% interest. After one year, you would have:

5000(1.05) = 5250

Pretty good. Let’s say that instead of just giving you the 5% interest at the end of the year, the bank gave it to you in parts. Say after 6 months, they gave you 2.5% and then another 2.5% at the end of the year. Then, you would have:

5000(1.025)^2 = 5253.12

What if your interest was compounded quarterly, meaning that you would get 1.25% four times throughout the year. Let’s see:

5000(1.0125)^4 = 5254.73

It’s inching up there. What if your interest was compounded monthly. Then, you would have:

5000(1+.05/12)^12 = 5255.81

Okay, you earned an extra buck. How about daily. This would be:

5000(1+.05/365)^365 = 5256.34

Again, just a little bit more. How about secondly. It’s a lot of work for the bank, but it might earn you a little money. It would give you:

5000(1+.05/31536000)^ 31536000 = 5256.35

Just a cent more. It seems like this is getting closer and closer to one number, or converging to one number in the 5256.3 area. Why could this be? What is our formula for e^x?

(1 + x/N)^N converges to e^x

This is just like what we have been doing! x is acting as the percent interest and N is the amount of pieces we are cutting the year into. So, to find the total amount if your interest was compounded constantly would be:

e^r

If r is the rate that we are compounding the interest, this would be your formula if it were just one dollar in the bank. For more money, you would do:

Pe^r

With P being the original amount of money you had in the bank. If you want to know it for 3 years, you just multiply the rate by three. Our universal formula, which is very easy to remember as you will see, is:

Pe^rt

This is a very cool formula, but nowhere close to the one I will show you right now. In fact, this formula, or equation I should say, is sometimes titled as “God’s Equation.” Since we have learned about e now, I can present it to you without any confusion. Here it is:

e^iπ + 1 = 0

Look at this for one second. It looks complicated, but we know all of the components very well. We have e, which we know to be 2.718... and the most important number in calculus. We have i, which you may remember back from the cube roots of one post or the geometric addition post. i is the square root of negative one, and is one of the most important numbers in algebra. We have π, which is 3.14... and one of the most important numbers in geometry. We use it for circle and ellipse areas and circumferences, as well as other things in subjects such as trigonometry. We also have one and zero, the foundations of arithmetic and mathematics itself.

If you look closer, you will find three operations: addition, multiplication, and exponentiality, the three most important operations in mathematics. We also have equality, the primary relationship in mathematics. How much better does it get? Next week, you will see that it does get better, but we’ll save that for then. It takes a little bit of  trigonometry to prove this, and I don’t want to go into it, but it involves properties of the number i. Anyways, this equation is pretty awesome, and I’m so happy I could share it now that we have covered e.

Fibonacci Day: Combining Two Famous Sequences into One Cool Math Stuff Post

Finally, we have reached another Fibonacci day. It is March 3, and 3 is the fourth Fibonacci number. Let’s look at the Fibonacci numbers again, as they may have faded from our memories over the past month.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,...

There is also another series of numbers that we did look at back in August called the prime numbers. If you don’t remember them, they are natural numbers that are only divisible by one and itself. For a quick review, 5 is prime because it is only divisible by 1 and 5. 6 is not prime because it is divisible by 1, 2, 3, and 6.

Here are the prime numbers:

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 41,...

These are two famous infinite series that are both very significant in science and mathematics. Let’s see what numbers are in both series.

2, 3, 5, 13, 89, 233, 1597, 28657, 514229,...

That’s interesting. There is one very cool thing about these numbers though, that is kind of hard to find at first glance. It isn’t about the numbers themselves, but their position in the Fibonacci sequence. Here are their positions:

3, 4, 5, 7, 11, 13, 17, 23, 29

Do you see a pattern? They are all prime numbers, except for that 4 at the beginning. The four is the only exception as far as I know to this rule. If you notice another exception, then please comment and let us know.

If you look at it vice versa though, you don’t see this beautiful pattern though. If you look at the second Fibonacci number, we have 1, which is not prime. It is actually called a universal number.

Also, the nineteenth Fibonacci number isn’t prime. While 4181 looks very prime, it is actually 113 x 37.

There are a bunch more of these, which I will list below:

31. 1346269 = 557 x 2417

37. 24157817 = 10877 x 2221

41. 165580141 = 2789 x 59369

53. 53316291173 = 953 x 55945741

Though there are many exceptions, I find it pretty cool that almost every prime Fibonacci number is in a prime position.

Bonus: Here is a challenge for you. We know that there is an infinite quantity of Fibonacci numbers. In August, we proved there is an infinite quantity of prime numbers. But are there infinitely many prime Fibonacci numbers?

This is a question that mathematicians still have not figured out. If you were to figure this out, you would have eternal fame in the world of mathematics. I’d say that’s a pretty interesting fun fact. (You don’t have to do the challenge)