Why are there more decimals between 0-1 than whole numbers between 0-∞?
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.
Note: This was the most intuitive way to I could figure out how explain why the infinite set of decimals 0-1 is bigger than infinite set of whole numbers between 0-∞.
But if you want to go deeper and learn about the actual mathematical proof that is the more commonly cited basis for bigger and smaller infinities, as well as other interesting explainers, I highly recommend you check out these videos to nerd out more.
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Sources
Dr. Trefor Bazett. (2018, March 26). Integers & Rationals are both infinite but is it the SAME infinity?. YouTube. https://www.youtube.com/watch?v=WQWkG9cQ8NQ&ab_channel=Dr.TreforBazett
Rayo, A., & Vazirani, G. (2013, June 10). Metaphysics: Sizes of Infinity, Part 2 (Getting Real). Wireless Philosophy. https://www.khanacademy.org/partner-content/wi-phi/wiphi-metaphysics-epistemology/wiphi-metaphysics/v/sizes-of-infinity-part-2-getting-real
Sid. (2023, July 17). Why Some Infinities are Larger than Others. Medium. https://www.cantorsparadise.com/why-some-infinities-are-larger-than-others-fc26863b872f
Strogatz, S., & Moore, J. (2023, April 19). How Can Some Infinities Be Bigger Than Others?. Quanta Magazine. https://www.quantamagazine.org/how-can-some-infinities-be-bigger-than-others-20230419/
Veritasium. (2021d, May 10). How An Infinite Hotel Ran Out Of Room. YouTube. https://www.youtube.com/watch?v=OxGsU8oIWjY&ab_channel=Veritasium
Vihart. (2014, June 19). Proof some infinities are bigger than other infinities. YouTube. https://www.youtube.com/watch?v=lA6hE7NFIK0&ab_channel=Vihart
WIRED. (2023, January 30). Mathematician Explains Infinity in 5 Levels of Difficulty | WIRED. YouTube. https://www.youtube.com/watch?v=Vp570S6Plt8&ab_channel=WIRED
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