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