Nib asked
Why does 0.999 repeating equal 1?
0. 999 repeating is not a number creeping up on 1; it is a second way of writing 1, because no gap at all can survive after endless nines.
7 steps, drawn by Nib. Read it here, or watch him draw it. Updated .
1the myth
Many people picture 0.999… as a number that keeps getting closer to 1 without ever arriving. But the three dots do not describe a journey. They name one fixed number, the one with nines in every decimal place, and that number is exactly 1.
0.999… and 1 are two spellings of one number.
a common mix-up0.999… is not just below 1, and it does not creep upwards: it is exactly 1.
2zoom in
Look at the number line just left of 1. The point 0.9 is a tenth short of 1. Zoom in on the stretch from 0.9 to 1 and 0.99 sits a hundredth short. Each extra nine puts the point ten times closer to 1.
Each extra nine makes the gap to 1 ten times smaller.
3zoom again
Zoom again: 0.999 is a thousandth short. With nines forever, any gap would have to be smaller than one over ten to the n, for every n.
After n nines the gap is 1 over 10 to the n; endless nines leave no gap.
4the algebra
Let x = 0.999… Multiplying by 10 shifts every digit one place, giving 10x = 9.999… Both numbers have nines in every decimal place, so subtracting x from 10x leaves exactly 9: 9x = 9, which gives x = 1.
10x − x = 9, so x = 1.
In the algebra proof, why does 9.999… minus 0.999… leave exactly 9?
Show the answer
Both have nines in every decimal place, so the endless nines cancel and only the 9 before the point is left.
5thirds
Dividing 1 by 3 by long division gives 0.333… with threes forever. Three of those thirds make 0.999…, but three thirds also make one whole pie, so 0.999… = 1.
Three thirds are 1, and three times 0.333… is 0.999…
6no gap
If 0.999… were less than 1, the difference g would be a positive number. But after n nines the gap is 1 over 10 to the n, and 0.999… has nines beyond every n, so g is smaller than 0.1, 0.01, 0.001 and so on. No positive number is smaller than all of those, so g is 0.
A gap smaller than every positive number is no gap at all.
Can a positive number be smaller than 0.1, 0.01, 0.001 and every one after?
Show the answer
No: for any positive number, a large enough n makes 1 over 10 to the n smaller. So g must be 0.
7the sum
Write 0.999… as 0.9 + 0.09 + 0.009 + … The sum after n terms is 1 − 1/10ⁿ, and as n grows without end the amount missing from 1 shrinks to nothing, so the total is 1. This is not a quirk of 1: every decimal that stops, like 0.5, has a twin ending in endless nines, like 0.4999…
Added digit by digit, the nines total exactly 1.
The dots do not mean almost: they mean nines forever, and forever leaves no gap, so 0.999 repeating and one are the same number.