# The Natural Numbers: How Big?

After learning about bijections, now we can compare sets with different infinities.

Count to Infinity (by way of natural numbers): 0

Comparison of Natural Numbers {1, 2, 3, 4, ...} to Other Infinite Sets
Natural Numbers Other Infinite Sets Guess If There Is a Bijection Bijection (hover over to see the answer)
{1, 2, 3, ...} {0, 1, 2, ...}
Whole Numbers
Yes
No
Don't Know

Yes
No
Don't Know

Yes
No
Don't Know

Yes
No
Don't Know

Yes
No
Don't Know

Yes
No
Don't Know
Yes. While this last set seems to have one more value than the first set, each natural number can be paired with the number 1 smaller than it in the whole number set. Because we will never run out of numbers to pair, this is a valid bijection, and the sets are equal.
{1, 2, 3, ...} {2, 4, 6, ...}
Even Numbers
Yes. Each natural number gets paired with the number twice itself in the even number set. As before, because we will never run of numbers to pair, this is valid. Think of it like this: Each and every even number can be assigned a natural number. Because both sets go on forever, we will never run out of natural numbers to assign to the even numbers. This is a bijection, and this means that the sets are equal.
{1, 2, 3, ...} {... -2, -1, 0, 1, 2, ...}
Integers
Yes. Click on the text to go to an image that depicts the bijection between natural numbers and integers. As you can see, each natural number can be assigned to an integer, and because we will never run out of natural numbers, this is a bijection, and they are equal.
{1, 2, 3, ...} Set of All Integer Coordinate Points
e.g. (0 , 0), (1 , 4),
(23 , 374),
(198173450 , 578942098)
Yes. Click on the text to go to an image that shows the bijection. Draw a graph with every integer coordinate point on it, and draw a line through it as shown in the image. As you can see, every integer coordinate point has been assigned a natural number. This is a bijection, and the two sets are equal.
{1, 2, 3, ...} Set of All Rational Numbers Yes. Click on the text to go to an image that shows the bijection. Write down all the rational numbers in a grid like the one shown in the image. Put the number on the x-axis over the number on the y-axis. For example, at the point (2,3), the fraction would be 2/3. However, because you will have many repeating fractions (e.g. 1/2, 2/4, 3/6, ...), just ignore fractions you have already seen. You can now draw a line through every single rational number, labeling each non-repeated rational number with a natural number. (All the points on the y-axis are missed because you can't divide by 0. There is only one point on the x-axis because 0 divided by any number is 0, so it is necessary to mark only one.
{1, 2, 3, ...} Set of All Real Numbers No. We will see why we can't create a bijection between the natural numbers and the real numbers on the next page.

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