What is a number that's bigger than infinity? Or is there one...

Last December, we took the time to really analyze infinity, and we ran into a little problem. We didn’t bring it up too much, but all of the infinite series were the same size! To be more specific, they were all equal to alef naught, we called it. Even the number of fractions couldn’t break this barrier. But we haven’t exhausted our toolbox yet!

Let’s look at the set of real numbers between zero and one. Say you had a list and said that it contained all of the real numbers on it. I’ll just put a few down here:

1) .6283185...

2) .2718281...

3) .1618034...

4) .3141592...

5) .1428571...

6) .4000000

7) .3333333...

I claim that there is a number that is not on your list. Of course, you can notice many off the top of your head, like .666666, .1, .2, ,3, and so on, but there is a systematic way to generate a new number. Take that first number, .6283185, the number tau over ten. Let’s change the first digit of the first number to something different. I’ll just keep it simple by adding one.

8. = .7...

Now, look at the second number on the list, .2718281. This happens to be the number e over ten, but that’s a complete coincidence. Let’s change the second digit of the second number by adding one.

8. = .78...

Let’s continue this process. Take the golden ratio over ten, .1618034. Add one to the one and get two. Keep going until you have gone through the full list.

8. = .7822614...

And we have a brand new number. Could it have been on the list? No. Why not? Well, let’s prove it by contradiction (actually, it is kind of a proof by logic if that actually exists, but whatever). Say it were the thousandth number on the list. Well, the thousandth digits of these two numbers must be different since that is how the process works, so they cannot be equal. Therefore, you cannot pair up the natural numbers (1, 2, 3, 4...) and the real numbers between 0 and 1.

In our other infinity post, or our alef naught post, I could say, we saw that you must be able to pair each number with a natural number in order for the size of the set to be equal to alef naught. You can’t do that with this list. Therefore, it is bigger than alef naught, and is its own number. We denote this number with the letter C, for continuum

What about the numbers between zero and two? That ought to be double of zero and one. Well, prove it. Look at the graph of y = 2x.

<graph>

Now, draw a laser beam from the origin up to the two line. This beam hits a point on the one line and then a point on the two line. Pair those two real numbers up. Now, draw another beam. Pair these two numbers up. Keep doing this, and you will find that no matter how many beams you do, there is always a number on the one line to match up with the two line.

With alef naught, we tried to pair everything up with the natural numbers. With C, we try to pair everything up with a real number between zero and one. Here, we have paired up the real numbers between zero and two with these numbers, so this set is size C.

What about all of the real numbers? That should do the trick! Well, same thing, graph any function that has a domain (x-coordinates) between zero and one and a range (y-coordinates) that goes from positive infinity to negative infinity, then you’ve paired up the real numbers between zero and one with all of the real numbers.

If that is equal to C, then what is more than C? Well, how about the points in the unit square; the square that has points (0,0), (0,1), (1,1), and (1,0). That must be more than C. But we actually can pair it up. Let’s say your points was the following:

x = 0.acegikmo...

y = 0.bdfhjlnpr...

Pretend that those letters are digits. Say they were the following:

x = 0.123456789

y = 0.987654321

We can pair this with a real number between zero and one, namely:

0.abcdefghijklmnop...

So, the example we had would pair up with this number:

0.192837465564738291

Basically, these two quantities are equal! We could go on forever, or to infinity I should say, with different quantities and test them to be equal to alef naught or C. There are some quantities that actually are bigger than C, but I won’t go into that now.

Infinity is such a cool concept to wrap your mind around, since you can’t define it or explain it, but you can actually start to grasp it after looking at some of these things. I think it even gives you a better handle on numbers, and understanding how many there really are. Considering that you learn about infinity (without actually saying you are) in first grade, it must be pretty cool!

Graphing Calculator Part 3: Games!!!

For a little while now, I've been mentioning how fun graphing calculators are. However, it's still all Algebra, and lots of people don't look at the fun side of things as much. Well, we have a solution, and it's all in one button.

The button is the "apps" button. Yes, graphing calculators have an app store, so to speak. The only difference is that everything is free! They are all different tools that will assist you in math class.

Click the "probability stimulator" button after you've clicked apps. This will bring up a number of choices; marbles, spinner, coins, cards, numbers, and dice. After you choose one, just hit enter and it will spin the spinner, roll the die, draw a card, flip the coin, etc.

You can use this for one of two things. You can use it to randomly generate data. While practicing the close-together method, or squaring two-digit numbers ending in five, it is always useful to be able to randomly generate numbers. It can also be used to practice graphing scatter plots and bar graphs.

The other thing you can use it for is to play games. It is a lot more fun than it seems to just gather a bunch of friends up and call out a bunch of numbers, and then have the calculator generate some numbers to see whose number comes up first. Though it isn't extremely mathematical, it is definitely a feature of the graphing calculator that I had to share.

Graphing Calculator Part 2: Data

Last week, we were using graphing calculators to do some graphing of equations, which was pretty simple. However, these things have way more capabilities than that!

We are going to try to use a graphing calculator to take a set of data, and not only graph it but show us an equation to approximate values of the equation.

The first thing to do is set diagnostics on. To do that, hit the catalog button (to do this, hit 2nd and 0), and scroll down with the arrows on the top right to "DiagnosticOn." To speed up this process, you can jump to the Ds by pressing whatever button has a small blue D to the upper right of it. The x^-1 button, also known as the reciprocal button, has the D next to it, so that will make things quicker. Click diagnostics on, then hit enter. It will say "Done" underneath after you have turned it on.

Now, we can put our data into the calculator. To do this, click the "Stat" button, then click edit. This will bring up a chart with L1 and L2 as the first columns. Fill in L1 with your x-values and L2 with your y-values. To keep things simple, we will use the same data:

L1: 1, 2, 3, 4, 5
L2: 1, 3, 6, 10, 15

Now, we will go fix up our window. Last week, we had the following:

Xmin = -10
Xmax = 10
Xscl = 5
Ymin = -10
Ymax = 10
Yscl = 5
Xres = 1

I'd like you to change it to the following, just for this data:

Xmin = 0

Xmax = 7

Xscl = 1

Ymin = 0

Ymax = 30

Yscl = 5

Xres = 1

Now, we will decide what we want our data to look like on the graph. To do this, we first hit statplot (2nd, Y=), then hit enter for plot one. Make sure that your type of graph is the first one, and Xlist is L1 and Ylist is L2. For mark, put whatever you prefer.

If you hit the graph button, you should see the data all graphed. I think that is pretty cool alone. However, we can do better!

This part is a little complicated and has to be done perfectly to work. First, hit stat again, and this time, move right to calc. Now, choose the type of equation it is. To do this, we can use some of the skills we learned from sequences in July and August, by finding common differences. In this sequence, the equation would be a quadratic.

1  3  6  10  15

  2  3  4    5

    1  1   1

If you don't know this process, don't worry. You can learn it at this post: http://coolmathstuff123.blogspot.com/2011/07/problem-of-week-day-2-week-of-717-723.html. However, it isn't necessary for the graphing calculator. You can either take an educated guess, or if you really don't know, hit PwrReg.

Since this is a quadratic, hit QuadReg, then enter. This should come up with this screen:

y = ax^2 + bx + c

a = .5

b = .5

c = 0

R^2 = 1

This looks like nonsense, but I'll bet we can interpret most of it. We have already mentioned that the base quadratic equation is ax^2 + bx + c. Well, we have that up top, then a =, b =, and c =. These are actually the values that are a, b, and c in the equation. If you want, you can plug them in to get the following:

y = 0.5x^2 + 0.5x

The R^2 button is a little complicated, but I will explain it briefly. It is called the correlation coefficient, and tells you how close the equation is to being a quadratic, or whatever regression you chose previously. In this case, your value is 1 because it is a quadratic. If it were something like .94386, this would be an equation that was close to a quadratic, but not quite there. The closer to 1 it is, the better the data is. If it were .32894, this would be bad data, and you might want to recollect your data.

But it can do better! For the next step, click stat calc again, and hit QuadReg. Now, we need to put in a little code to tell the calculator what we want it to do with the equation. First, hit L1 (2nd, 1). Then, put a comma, and hit L2 (2nd, 2). Now, put another comma, and hit Y1. To do this, you have to hit Vars, move over to Y-vars, then hit function, and Y1. Now, hit enter.

Wait, it just deleted it! It's okay, it gets better. Click Y=. You should see an equation present; the exact same one that it told us! Let's graph it. To do it, just hit graph. We already set ourselves up.

At this point, we should use this equation to approximate, or in this case determine, more data for the equation. Let's say we want to know the 6th value. To figure it out, do what we would have done before. Hit calc, just above trace, then hit value, and type in 6. It tells you that Y = 21.

In school, we learned about graphing calculators, but I would never have expected that they could do something as crazy as this! Considering that some of my friends who don't really like math refer to my graphing calculator as a "game," I think there must be something cool about it.

Since the algebra has been getting a little heavy over the past few weeks, I will bring it down a notch and prove to you that my friends are correct in calling it a game.

Graphing Calculator Part 1: Graphing

Mathematics is cool in itself with the proofs and patterns, but there are also tools in mathematics that are pretty cool as well. One of which is the graphing calculator. For the majority of February, I will be teaching how to use a TI-83 graphing calculator, and you will be able to use it and appreciate it as much as your iPods and iPhones.

First of all, these things only work on TI-83s and mostly on TI-84s as far as I know. I'm not sure about other types of calculators, so don't get confused with a different type. If you have either of those, take it out now so you can follow along. If you don't, you are missing out on some awesome mathematical technology, but you should be able to get the vive of what a graphing calculator is.

Okay, now to the fun stuff. What does a graphing calculator imply it should be able to do? Graphing, right! And it's as easy as ever with these devices. Let's say you want to graph 3x - 7.

First, we will turn on our calculators (in the bottom left corner). Now, two from the left on the very top is a purple button with the word window on it. Click that.

This screen enables us to choose what we want our graph intervals to be. First, we will choose what we want our X-minimum to be, called Xmin on the calculator. Let's put in -10. The negative button is one to the left of the Enter button, in white. Don't use the blue minus button, that won't work on the calculator.

Now, we have to choose an X-maximum. To keep things consistent, we'll use ten. Use the enter key as if it were a return key.

Next, we must look at the X-scale, labeled as Xscl. This basically makes a little mark at every interval you type in. I am going to put in five, just so there aren't too many of them on the graph. However, if you want to see intervals of two, feel free to put in two. Make sure you don't put a negative there!

For the y-values, we can change them up, but I will do the same as we did for the x-values. -10 for Ymin, 10 for Ymax, and 5 for Yscl. Leave Xres alone as one.

If you click the purple "graph" button now, you will see your Cartesian plane, with little marks at (-5, 0), (0, 5), (5, 0), and (0, -5). These are simply the intervals, which were correctly placed. If you see then at all of the even intervals, then you put 2 in for Xscl and Yscl.

Now, we will graph the line. To do it, click on the purple "Y=" button, which is right to the left of the Window button. It should come up with Y1, Y2, and so on. The curser should be next to Y1, where we will type in the equation. Type in a 3, then X. X will appear if you press the X,T,Θ,n button, which is a black button located one to the right of the green Alpha button. Now, press minus (not negative this time), and then seven. Now, hit graph again.


The calculator should have graphed the line right before your eyes. Pretty cool, huh!


Say you wanted to know what y was when x = 2. There's no need to approximate with your line. Simply  do the following:


1. Hit the calc button. To do it, click 2nd, then Trace. This will bring it up for you.
2. Click on the first choice of value. To do that, simply hit enter.
3. The graph should be brought up again with X= in the bottom corner. Type 2 into the calculator (since two is the value we want), then hit enter.
4. The calculator should have come up with Y=-1 in small print to the right of the y-axis. It has also put a little x over the point (2, -1) to show you where it's located on the graph.


The cool thing is that now that you have set up your window, you can graph anything you want. Say you want to graph -0.5x^2 + 5x - 5. Just click Y= and punch it in, and it will come up with the graph. To do the power, you can do one of the following:


1. Click the carrot just above the division sign, and then a 2
2. Hit the x^2 button to the left of the comma


Say you are curious as to what a quartic equation looks like; an equation with degree four, or an x^4 term in it. Try graphing x^4 - 5x^3 + 4x^2 + 6x - 8. It's pretty cool.


You can also look at trigonometric graphs. Try graphing the tangent of x, or tan(x). It's pretty cool!


Here's something I want you to try to graph:


Y1 = sin(x)
Y2 = sin(x+2)
Y3 = sin(x+4)


Those three look really cool together!


Tip: Remember not to use a minus sign for a negative. It will always mess up something that you were working on for a long time.


If you don't own a graphing calculator, but want to check out these graphs, I'd recommend www.graphsketch.com. The picture of the graph is better, but it doesn't have the same freedoms as a graphing calculator, as you will see next week.