So, on the previous page, we compared the set of natural numbers to many different infinite sets, and found that there was a bijection between the natural numbers and most infinite sets except for the real numbers. This is because there are actually two different sizes of infinity: countable (listable) and uncountable (unlistable). We will see just what we mean by that in this video.

Did you figure out why the real numbers are unlistable? I will go over it again just in case.

Instead of looking at the entire real number line, let us first simply look at the line segment from 0 to 1. Let's list some of the real numbers from 0 to 1:

Anyways, you come up with a completely different number than any number on that list. It can't be the first number, because the first decimal digit of our "missing" number is different. It's not the second number because the second decimal digit is different too. It can't be the seventeen millionth number because the seventeen millionth digit of our number is different. So we have found an entirely new number not on this list at all. And you can do this for every list of numbers anyone gives you, no matter how long it is. That is why the real numbers are considered "unlistable".

### Extra: (read this only if you want even more confusion to fill your brain)

For every list of real numbers, you can actually find infinite many different missing numbers by using infinite different methods of changing decimal digits. For instance, if we used James Grime's method, we would get a different missing number than if we just added 1 to each decimal digit, or subtracted 1 from each digit, or multiplied 7 to each decimal digit and don't carry, except if there is a 5 or a 0, in which case you add one. You can find some completely ridiculuous methods, but that still create unique missing numbers.Reveal Confusion

Instead of looking at the entire real number line, let us first simply look at the line segment from 0 to 1. Let's list some of the real numbers from 0 to 1:

- 0.098765421...
- 0.187894533...
- 1.000000000...
- 0.000200051...
- 0.134897128...
- 0.172384932...

- 0.098765421...
- 0.187894533...
- 1.000000000...
- 0.000200051...
- 0.134897128...
- 0.172384932...

Anyways, you come up with a completely different number than any number on that list. It can't be the first number, because the first decimal digit of our "missing" number is different. It's not the second number because the second decimal digit is different too. It can't be the seventeen millionth number because the seventeen millionth digit of our number is different. So we have found an entirely new number not on this list at all. And you can do this for every list of numbers anyone gives you, no matter how long it is. That is why the real numbers are considered "unlistable".

For every list of real numbers, you can actually find infinite many different missing numbers by using infinite different methods of changing decimal digits. For instance, if we used James Grime's method, we would get a different missing number than if we just added 1 to each decimal digit, or subtracted 1 from each digit, or multiplied 7 to each decimal digit and don't carry, except if there is a 5 or a 0, in which case you add one. You can find some completely ridiculuous methods, but that still create unique missing numbers.

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