Why 0.9 recurring is equal to 1

In other words, " 0. There will always be another " 9 " to tack onto the end of 0. So don't object to 0. Yes, at any given stop, at any given stage of the expansion, for any given finite number of 9 s, there will be a difference between 0. But the point of the "dot, dot, dot" is that there is no end; 0. There is no "last" digit. So the "there's always a difference" argument betrays a lack of understanding of the infinite.

That's not a "criticism", per se; infinity is a messy topic. Proof by geometric series. The number " 0. In other words, each term in this endless summation will have a " 9 " preceded by some number of zeroes. This may also be written as:. Since the size of the common ratio r is less than 1 , we can use the infinite-sum formula to find the value:. So the formula proves that 0. Note: Technically, the above proof requires that some fairly advanced concepts be taken on faith.

If you study "foundations" or mathematical philosophy way after calculus , you may encounter the requisite theoretical constructs. Other pre-calculus arguments. Reasonably then, 0. But 3 0. In the case of an infinite decimal, again standing in for the kind of infinite sequence of terminating decimals we saw above, we identify the sequence with its limit. This is what we mean when we say that 0.

The same idea works for any rational number with a repeating infinite decimal expansion. Something similar happens with irrational numbers that have non-repeating decimal expansions.

In general, any infinitely long decimal expansion is thought of as the limit of the sequence of terminating decimals that make up the infinite expansion. So, the reason why 0. For you. World globe An icon of the world globe, indicating different international options. Get the Insider App. Click here to learn more. A leading-edge research firm focused on digital transformation. Good Subscriber Account active since Shortcuts. Account icon An icon in the shape of a person's head and shoulders.

It often indicates a user profile. You have to define "1" first. If you're dealing with countries, 0. Generally speaking, taken in isolation, numbers are abstractions of "pure" or "pepfect" values which don't exist in the real world.

Don't try to apply "real-world logic" to numbers, it doesn't work. Mathematics is a codified system of conventions, you can work within the conventions or you can ignore them. Pentarctagon wrote: but if they converge at infinity, doesn't that mean that they never actually converge or at least that's what my math teacher said? The Fires of Pride 0. On hold while I try and finish my book. Post by Velensk » July 2nd, , pm I've always settled for the answer that. The kind who didn't go to war and who say that they should have lived fast died young and left a handsome corpse and the old men who did go to war and who say that there is no such thing as a handsome corpse.

Is an apple with any size bite taken out of it still and apple? If a country looses land, people, etc And do you see the same colour as blue that I see as blue? Answer to all 3: "if you say so". You wouldn't be able to tell a shopkeeper that your apple lost a cell and now it's not a whole apple and so you demand a price reduction, you'd probably get punched in the mouth if you start telling people "actually, that's not pure blue becasue there is 0.

Numbers are created to make our lives easier: one coral reaf for example is a very complex notion which includes billions of organisms, as well as geographical features, etc I hope I have explained myself better here.

If something is infintely approaching a value, it is that value; for example, we can see this kind of thing in calculus, where there are two accepted ways of writing an expression for the area under a curve. You have standard Leibnitz notation with integrals, or you can use Riemann sum notation where you take the sum of an infinite amount of infinitely small rectangles under the curve.

From a mathematical standpoint, we know that when we allow the number of rectangles to approach infinity the approximation becomes infinitely accurate.

Another way to look at it is if you consider. Post by PsychoticKittens » July 2nd, , pm Infinity is a principle. Principles are not numbers. Therefore incapable of being equal to a number. We just round because its really freaking close to 2, and simplifies things.

Infinity doesn't really occur in day to day things, so you can't really say you'd ever have to worry about something literally being 1. Someones probly gonna bring up something, or divide, to prove me wrong.

But who divides outside of school without rounding to the nearest something? Generally whole numbers. Thats my thoughts on the matter anyway. No matter how many people talk about this we'll probably never get a conclusive answer due to how many different ways this is taught. Teachers would argue about it too. Uh, well at least, they saved Soarin's apple pie. Post by Jetrel » July 3rd, , am Alink wrote: This is a common pattern, people arguing that 0.

In the other hand, proofs that 0. And nobody argues against that. Sorry, there is also people who accept the truth but are disturbed by its counter-intuitive aspects. To them, I suggest to check the underlying points used by their intuition like "there is only one way to write a number".

Post by pauxlo » July 3rd, , pm PsychoticKittens wrote: Infinity is a principle. PsychoticKittens wrote: So 1. Secondly, I'd like to suggest that without context. I say this because we have another infinitely procedural calculation being scientifically considered its own distinctive value all on its own.

It's known as pi, which is defined as 3. As it's linearly impossible to approach one number without approaching the others in the same direction, which I'll submit a ruler and a pencil as proof of, I don't think this presumption should have too much difficulty in being accepted. Thirdly, I'd suggest our mathematical proofs are inherently flawed because they rely upon converting fractions to decimals.