β Why are there more decimals between 0-1 than whole numbers between 0-β?β
|
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-β.
To list out the whole numbers between 0-β, 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-β is a countable infinity; it goes on forever and you could theoretically count all the numbers if given infinite time.
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 thatβs 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.
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-β), 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, |
βSources for this week's newsletterβ
P.S. Shoutout to Saurav (my β€οΈ 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.
|
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!
β
βCheck out the full list of books I've recommended here.
P.S. π€ Do you have a question for the newsletter? Click here to submit your question (or reply to this email!)
|
Learn something new every Friday. Join 2,000+ nerds and satisfy your curiosity one question at a time with topics including science, technology, history, and more.
Why donβt woodpeckers get brain damage? π The short answer Woodpeckers avoid brain damage thanks to special adaptations in their anatomy. Their skull, jaws, eyes, and tongue are all uniquely designed to help absorb the impact of the constant pecking and prevent concussions and retinal damage. π The long answer To us humans, banging your head against a wall sounds like a regrettable moment of frustration. To woodpeckers, it sounds like a Tuesday. Source: GIPHY Woodpeckers rapidly peck trees so...
Why does a wooden spoon stop water from boiling over in a pot? This question came from a reader submission (thanks Thomas!). Has a curious question stumped you lately? Feel free to submit your own question here: Submit your question π The short answer Placing a wooden spoon across the top of a pot of boiling starchy foods, like pasta or potatoes, prevents spills by disrupting the bubbles as they rise. The wood's lower temperature and rough surface bursts the bubbles, keeping the foam from...
Why do we say "make a beeline"? This question came from a reader submission (thanks Georgia!). Has a curious question stumped you lately? Feel free to submit your own question here: Submit your question π The short answer The phrase "make a beeline" comes from how honeybees fly directly back to their hive after gathering nectar or return straight to a bountiful food source. π The long answer In the Merriam-Webster Dictionary, you'll find this entry for the word "beeline":...