I'm Settling This Math Debate Forever (.99 repeating = 1)

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"girl you denser than the set of real numbers."

I always liked the demonstration like this: x = .999... 10x = 9.999... 10x -x = 9.999... - .999... 9x = 9 x = 1 = .999...

I remember as a kid i came up with a similar solution when my dad told me that 0,999... and 1 are the same! :D I just thought by myself "what would the difference between these two numbers be?" and my answer was 0,000.... with infinite zeroes and a 1 attached to it, but the 1 is literally nowhere since the 0s go on forever! This is way more sound tho hahahah

Plot twist: 1 is actualy equal to 0.999...

But it has so many cool rich nines how can it be the same as just a lousy poor one? That's right, it cant! (Proof by segregation)

I'm still a believer that most of the difficulty arises from the concept of what is a real number.

When I first saw this equality, I knew it was true as I can never think of a number between 0.99... and 1.

Wikipedia mentions the debate and some reasons why people can't grasp it. One of them is people imagine 1 as an asymptote, where 0.9... approaches it but never crosses. As if it was a process, not a number. Like imagining someone writing a series of 9s, on and on and on, never truly reaching 1. However, 0.9... isn't something approaching 1. It's a constant value. The number of 9s isn't increasing, they all already exist.

This video really squeezes my theorem.

Phenomenal channel really really like the way you explain things not just this but things in general especially regarding integrations and infinite sums. My only wishbis that you go deeper into more difficult math

I was debating on bprp's channel and now I am here

Another incredible video. I really prefer these types of videos that explore a concept more in-depth (denser information, you could say) and try to address different perspectives. Thanks for sharing and keep ‘em coming!

I feel this proof wouldn't change anyone's mind about the problem. If one believes 0.999... is not equal to 1, the same person doesn't believe 1/(10^n) converges to 0

This "debate" was something settled long ago when developing the rudiments of real analysis, and was one of the introductory demonstrations the professor performed for us when I took real analysis.

Among mathematicians, this has never been a debate. The fact that 0.9999... = 1, as well as 17.563999999... = 17.564 and 1.29999999... = 1.3 is just a redundancy of the decimal notation. We just decided to stick with the form that ends with an infinite string of 0's, instead of the one that ends with an infinite string of 9's. More formally, every real number is defined to correspond to a certain equivalence class of cauchy successions of rational numbers, where two rational cauchy successions {a_n} and {b_n} are defined to be equivalent if the succession {a_n-b_n} converges to zero. This definition obviously includes all the pre-defined subsets of real numbers; e.g., if a class of cauchy successions converges to a rational number in Q, then define that class of cauchy successions to represent that number. Of course, given an arbitrary real number, there exist infinite cauchy successions that represent it; we can pick the one we prefer the most from its equivalence class when we want to make maths with it. The decimal notation is just choosing the succession to be in the same form for every real number: the number a⁰.a¹a²a³... is the one represented by the class of rational successions that, among the other infinite ones, includes the succession: {a⁰, a⁰ + a¹/10, a⁰ + a¹/10 + a²/100, a⁰ + a¹/10 + a²/100 + a³/1000, ...} In easier-to-grasp terms we say that a⁰.a¹a²a³... = a⁰ + 0.a¹ + 0.0a² + 0.00a³ + ... Now let's take a rational number p/q; it can be shown that its decimal expantion is either periodic or finite, depending if the prime factorization of the denominator q (after simplifying every common factor between p and q) has only 2's and 5's or not. Let's stick with the 'problematic' case, when the decimal expantion is finite. Let's make it even simpler, lemme just take the star of the video, the number 1. How can we write it as 1 = a⁰ + a¹/10 + a²/100 + ... for a certain set of digits a⁰,a¹,...? Well, of course choosing a⁰ = 1 and a¹=a²=...=0 works fine, so the decimal expantion of 1 is 1.000000... But is it the only possible choice? Let's try to change some of the digits, starting from a⁰. We clearly can't plug in it a 2 or a 3: since the succession a⁰, a⁰+a¹/10, ... Is strictly increasing, we can't hope to make it converge to 1 if already the first term is bigger than 1. So we can try putting in a⁰ = 0. What about a¹? Well, we know that a²/100 + a³/1000 + ..., for every choice of digits a²,a³,... Is surely smaller than (or at most equal to) 0.1. if we hope to build a succession that converges to 1, we have to set a¹ = 9, otherwise the limit would be smaller than (or at most equal to) 0.(a¹+1). The same goes for all the other digits: if we hope to build a succession that converges to 1, we have to set them all to 9. So we understood that 0.9999... is the only second possible candidate to represent 1, together with 1.00000... but do they represent the same number? That is, does the difference between the succession {1, 1+0, 1+0+0, ...} And the succession {0, 0+0.9, 0+0.9+0.09,...} Converge to zero? Well, of course it does: the difference gives the succession: {1, 0.1, 0.01, ...} Which clearly approaches zero. Tldr the definition of decimal expantion is ambiguous for the numbers with a finite decimal expantion, as the definition is compatible both with a finite sum of terms and with a neverending sum of 9's. But everyone who knows what we decided the meaning of "real number" to be has no doubts about the fact that 0.999.... = 1. Maths always comes down to definitions. Many people think that there's a sense of 'trueness' behind some mathematical identities, that some things are truer than others and that there's room for debate. But it all always boils down to the definitions we gave for the stuff we work with. "0.9999... = 1" because of what mathematicians mean with "1", of what they mean with "0.99999..." and of what they mean with "=". There's no mystical and inconceivable truth waiting for us at the end of that infinite string of 9's.

What are other math topics that are true but hard to accept?!

This is a super weird but clever approach that I have never even bothered to think or question. Super cool.

This is a cool proof that I haven't seen before! Thanks!

I just used my own "rule of ninths": 1/9 = 0.111... 2/9 = 0.222... etc., so then 9/9 = 0.999... and pretty clearly 9/9 = 1. Slept like a baby ever since.