Why 10/9 = 1.111… ; 9×1.111… = 9.999… ; but 9×10/9 = 10. Why is 10 equal to 9.999…?
In: Mathematics
Because you are using different notations. 9X 1/9th has to be equal to 1. If you write everything out in fractions, there’s no confusion:
9/1 x 10/9 = 10.
Because 9.99… = 10. There’s a proof for it, but instead, here’s a better ELI5:
10 / 9 = 1 and 1/9th.
9 * 10/9 = 9 * 1 and 1/9th = 9 and 9/9ths = 10
Because there is no number, no matter how small you try to imagine, between 9.999…. and 10.
> Why 10/9 = 1.111…; … ?
Because 10 is one set of 9 with 1 left over, 1 is one set of 0.9 with 0.1 left over, 0.1 is one set of 0.09 with 0.01 left over, …
It’s just long division in base-10. Do the division in a different base and it’ll look different. In base-9, ten is written as 11 (i.e a 1 in the nine’s place and a 1 in the one’s place) and nine is written as 10 (i.e. a 1 in the nine’s place and a 0 in the one’s place). In base-9, the result of this division is 1.1 (i.e. one and one-ninth).
Even though the numerals are different between these two representations, *the quantity they describe is the same*. Think of it like converting between feet and meters. Yeah, the numbers are different, but it’s not like the *physical distance* has changed just because you switched units. It’s still the same underlying thing, you’re just writing it down differently.
> Why is 10 equal to 9.999…?
Because the **value** of a number is not the same thing as **how we write the number**. In order to write a number, you have to choose a base, and as mentioned before, the choice of base affects what the result will look like. However, the base does not affect the **value** of the number, only what it looks like when you write it down.
The equals sign doesn’t care about representation, it only cares about value. 2+2 and 4 are just two different ways of writing the same value. *That* is why 2+2 = 4; because both sides are representations of the same *value*. Likewise, 10 and 9.999… are just two different ways of writing the same underlying value, thus they’re equals.
Numbers are only different from each other, if the distance between them is greater than zero.
What’s the distance between 9.9999 and 10? If you name any nonzero number, the real distance is smaller than that.
If the distance between two numbers is smaller than any number, then you have actually just found two different names for the same number.
Now, how repeating digits work in general, is a little bit more subtle. To understand those properly, you need to approach using the concepts of ‘partial sums’ and ‘limits’. It will help to think a ilttle bit about Zeno’s Paradox (that old story where the runner runs across half the racetrack, then half of the remaining distance, then half of the remaining distance, then half the remaining distance…) Happy to get further into that if you like.
We’ve got a proof in the expressions you’ve given.
We have:
10/9 = 1.1…
9*1.1… = 9.9…
and 9×10/9=10
So lets rewrite the last one in terms of the first two:
9×10/9 = 10/9 * 9 = 1.1… * (9.9…/1.1…)
We can cancel out the two 1.1… terms as one is a multiplication and one is a division.
So we *must* have 9×10/9 = 9.9…
This means we have both 9×10/9 = 10 _and_ 9×10/9=9.9…
Therefore we’ve proven that 10 = 9.9…
9.99999… = 9 + 0.9 + 0.09 + 0.009 + …
= 9 * (1 + 0.1 + 0.01 + 0.001 + …)
= 9 * ((0.1)^0 + (0.1)^1 + (0.1)^2 + (0.1)^3 + …)
= 9 * sum of ((0.1)^k) with k from 0 to infinity
= 9 * (1-0)/(1-0.1) ***
= 9 * 1/0.9
= 9/0.9
= 90/9
= 10
the part marked *** is from the known formula :
sum of (r^k) with k from 0 to n = (1 – r^(n+1)) / (1-r) (if r is not 1)
Particularly, if |r|<1 (which is the case with r=0.1), we know that r^n tends toward 0 when n tends to infinity. So this formula towards infinity becomes
sum of (r^k) with k from 0 to infinity = (1-0) / (1-r) if |r|<1
IMO all of the answers here (as of when I’m posting this) really miss the point, which is the meaning of the ellipsis (…) in those expressions you write (e.g. 9.999…). Without precisely defining what those ellipsis mean, we can’t actually discuss these numbers because we don’t know what they really are! Sure, you might say that it’s the number 9.999 with infinitely many 9s following, but we have to formalize that concept in order to discuss it accurately.
In general, when you write a decimal number like 23.18, what you’re really writing is the number 2 * 10^(1) + 3 * 10^(0) + 1 * 10^(-1) + 8 * 10^(-2). That’s how the decimal system works, and is how we define the value of a decimal number.
In this context, then, the way we interpret the ellipsis is as a *limit*: for instance, we **define** 9.999… to be equal to the limit
limit(N -> infinity) Sum(n = 0 to N of 9/(10^(n))).
which, written another way, is the infinite sum
9 + 0.9 + 0.09 + 0.009 + … = 9 * 10^(0) + 9 * 10^(-1) + 9 * 10^(-2) + 9 * 10^(-3) + …
This limit is *precisely equal* to 10 by the definition of the limit, which is a little hard to ELI5, but essentially says that we can always pick some N (i.e. some upper bound for that sum) that will get the value of the sum as close as we want to 10—if I want to get within 0.00000001 of 10, then there’s an N that’ll get me there; if I want to get within 0.00000000000000000000000001 of 10, there’s another N that will get me there; and likewise for any distance from 10, however small it may be.
So when you write something like 9.999…, you’re really just writing 10 in a different way, because of what the ellipsis mean—you’re writing a complex limit expression that’s actually equal to 10. It’s similar to how, when you write the fraction 4/2, you’re really just writing 2 in a different way.
The algebraic “proofs” discussed here are fine, I suppose, but they aren’t technically justified because you can’t manipulate some expression without first defining it precisely and confirming that the definition is valid (i.e. that the definition gives an actual number). These sorts of unjustified arguments with infinite series are what can lead to the bogus arguments, for instance, that 1+2+3+…= -1/12 (you can’t manipulate infinite series just like normal numbers—particularly if they don’t converge!). So while they happen to work here, and may be helpful for explaining the ideas in a simple way, they aren’t actually precise. If you want to know really *why* 9.999…=10, the key is that the ellipsis denote taking a limit, and that limit is precisely equal to 10.
This might be getting a bit beyond ELI5 territory, but what you’re noticing here is actually the key to one of the constructions of the real numbers from the rational numbers, namely, the construction of reals as equivalence classes of Cauchy sequences of rational numbers (look up “constructions of real numbers Cauchy sequences” for details). There are many Cauchy sequences of rationals that all converge to the same real number, which is why we can write things like 9.999… = 10.000… without there being any contradiction—in some sense, the number 10, considered as a real number rather than an integer, is defined as the equivalence class of sequences that includes 9.99… and 10.0….. as well as infinitely many other infinite sequences of rational numbers that all converge to 10.
If you substract 9.999… from 10, the result is literally 0.000… with nothing but zeroes forever after.
Since the difference between these numbers is zero, these numbers mean the exact same thing.
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