How do we know that there are infinitely many transcendental numbers?

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How do we know that there are infinitely many transcendental numbers?

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Well, a simple proof would be that any number has an infinite amount of multiples, and the multiple of a transcendental number is still transcendental.

Also, we know that the real numbers are uncountably infinite, but the algebraic numbers are only countably infinite. And since the real numbers are the algebraic numbers *plus* the transcendental numbers, the transcendental numbers would also have to be uncountably infinite.

We can prove that the real numbers are uncountable and the algebraic numbers are countable. If the transcendental numbers were countable, then the union of the transcendentals and the algebraics (i.e. the real numbers) would also be countable. Since the real numbers are uncountable, that assumption must be false, so the transcendentals must be uncountable.

Imagine the number line, all values of numbers that could be written as a list of infinitely many decimal digits with a decimal point somewhere in the list. This number line includes
69.420420…
1.000000…
123.123123…
3.141592…
You get the picture. These are called the real numbers and are basically what everyone thinks of when someone says “number”.

Now imagine just the counting numbers. 1,2,3,4, and so on. There are obviously infinitely many of both types of numbers, but as it turns out, one of these types is, in a sense, MORE infinite.

To see why, think of what it means for one set of things, it doesn’t have to be numbers, to have more things than another. Say I have 4 oranges and 3 apples. The set of oranges is bigger because we can pair each orange with an apple and still have an orange leftover. The same is true even if the sets of things have infinitely many objects.

For a fun mind-blower, think of just the even counting numbers. 2,4,6,8, so on. As it turns out, there’s a way to pair each even counting number with each counting number and have none left over. One way is like this: 2 pairs with 1, 4 pairs with 2, 6 pairs with 3, and so on so that each even counting number is paired with the counting number that is half it’s value. Every number in both sets will be part of a unique pair and so mathematicians say that both of those sets are the same size (the math word is cardinality).

This is a bit counterintuitive, as many things are when it comes to infinite sets of things, but there are as many even counting numbers as there are even and odd counting numbers.

The real numbers are different. Suppose someone said that they had a way to pair each real number with a counting number. They must be lying or mistaken and to prove it, imagine what a pairing, if it existed, would look like.

1 pairs with say 0.333333…
2 pairs with say 123.456789…
3 pairs with 2.346129…
And so on. This is just an example and their pairing could be different. As it turns out though, it’s possible to find a real number that hasn’t been paired in their list, no matter what the list looks like. This is how it’s done:

To start, our number will begin with 0. and we’ll use their list to figure out what to put after the decimal. If we look at the first number on their list, the first digit after the decimal is 3, so if we pick the first digit in our new number to be different from 3, say 8, then our number will not be the first number on their list since the first decimal digit is different. Now we look at the second number in their list. The second digit after the decimal point is 5 and so we pick our number to have a different second digit after the point, say 2. Because of this choice, our new number is guaranteed to not be the second number in their pairing list. We do the same for the third, forth, fifth, and so on numbers and we’ll end up with a new number that looks like 0.823… this number has to be different than all of the real numbers in their pairing because it differs in at least one place from all of their listed real numbers.

So, no matter how someone might try to pair real numbers to counting numbers, there will always be a real number left over. In fact there will be infinitely many left over since we could have made many choices when picking each digit different from their’s. So the real numbers are a bigger set of numbers than the counting numbers, by a LOT.

To answer your question specifically, as it turns out there is a way to pair non-transcendental numbers with counting numbers, though it’s a bit more involved so I won’t go into it here. Because of this, the non-transcendental numbers are the same size (cardinality) as the counting numbers. We just worked through showing that there are infinitely many more real numbers than counting numbers and so it must be the case that there are infinitely many more real numbers than non-transcendental numbers. If we took all of the non-transcendental numbers off of the number line, we would still be left with infinitely many numbers and so there must be infinitely many transcendental numbers.

If you double a transcendental number, you get a transcendental number. This is because if you half an algebraic number (not transcendental), you get an algebraic number. Why? An algebraic number x satisfies a polynomial equation with rational coefficients ax^n +bx^n-1 +…=0, so x/2 satisfies the polynomial (2^n a)(x/2)^n +(2^n-1 b)(x/2)^n-1 +…=0, so it also satisfies a polynomial with rational coefficients.

So keep multiplying a transcendental number by 2 to get infinitely many transcendental number.