Why are there more decimals between 0-1 than whole numbers between 0-∞?

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📓 The short answer

The set of numbers between 0-1 contains a bigger infinity than the set of whole numbers from 0-∞. This is because the numbers between 0-1 form an "uncountable" infinity, while the whole numbers make up a "countable" infinity.

📚 The long answer

To be infinite is to mean something goes on forever. All the decimals between 0-1 and all the whole numbers between 0-∞ go on forever. Both of these sets are infinite.

But one is a bigger infinity than the other, and we can prove it.

How do you prove one set of numbers is bigger than another set?

One way to determine if a set of numbers is bigger than another set is to try to pair each number in one set with a unique number in the other. If we can’t establish a one-to-one pairing between all the numbers of the two sets, then we know that one set has more numbers.

As an example, let's say there are two sets, and we want to prove that Set A is bigger than Set B. Set A has Leah, Meer, and Stephen. Set B has Ari and Saurav.

Leah (A) pairs with Ari (B), Meer (A) pairs with Saurav (B), and that leaves Stephen without a pair buddy (sorry, Stephen!). Therefore, we have proved that Set A is bigger than Set B.

So to find out if one set is bigger than another, we can start by listing the numbers of both sets to see if we can find one-to-one pairings. Let's start with whole numbers between 0-∞.

What is a countable infinity?

To list out the whole numbers between 0-∞, you start with 0. Then it's obvious what comes next, 1.

You keep adding 1 to the last number forever: 0, 1, 2, 3, 4, 5.... It's clear which number comes next, even if you count all the way up to a number as big as 1,203,439,303,304,596,930,227. Just add 1 and you'll have the next number in the set. This means whole numbers between 0-∞ is a countable infinity; it goes on forever and you could theoretically count all the numbers if given infinite time.

What is an uncountable infinity?

Next up, time to list out the numbers (specifically the decimals) between 0-1. What's the first number that comes after 0 in this set? It's not 0.1. Why? Because you can think of a number that's closer to 0. Okay fine, 0.0000000000001. Well, that doesn't work either. No matter how close you get, you can always find another decimal that’s closer to 0.

We haven't even started counting and there's already an infinity contained within this infinity. The set of numbers between 0-1 forms what mathematicians call an uncountable infinity. Like a countable infinity, it goes on forever but the difference is you can't count the numbers one by one.

Why are there more decimals between 0-1 than whole numbers between 0-∞?

Since the uncountable infinity (decimals between 0-1) will always have more numbers than can be matched with the countable infinity (whole numbers between 0-∞), it is the bigger infinity.

--- Thanks for reading this week's newsletter! If you have any thoughts, questions, or favorite GIFs, my inbox is always open. Just hit reply to send me a note! :) ​ All my best, Caitlin ​Click here to read five facts about Caitlin​

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​Sources for this week's newsletter​

P.S. Shoutout to Saurav (my ❤️ and in-house math major) and Rodrigo (a long-time subscriber and Python teacher) for all the help in getting my mind to wrap around this topic.

📖 Book of the week

​The Joy of X by Steven Strogatz​

Whether you aced advanced calculus or nearly flunked eighth grade algebra (me), I think you'll really enjoy this book. Each chapter explains a key concept of math starting with the basics (e.g. what numbers are) all the way up to the tricky stuff (e.g. differential equations). It's written in an impressively easy-to-digest manner with fun examples, diagrams, and stories. I would highly recommend it, especially if you found today's newsletter interesting!

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​Check out the full list of books I've recommended here.

👀 Catch up on other curious questions

• ​Why does x⁰ = 1?​​
• ​Why is Shakespeare still taught in high school today?​​
• ​What do our recyclables become?​​

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