Why 10/9 = 1.111… ; 9×1.111… = 9.999… ; but 9×10/9 = 10. Why is 10 equal to 9.999…?
In: Mathematics
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.
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