0.999... = 1

10 1 0

by MP_431

by MP_431

You've probably read the title, and you'd think this is an estimation. But no, it's not. Yes 0.999 is exactly equivalent to 1. And here are the proofs.

Let's take 0.999... and make it a variable. Maybe x

0.999... = x

Ok that's good, now multiply both sides by 10.

0.999...*10 = 9.999...
x*10 = 10x
9.999 = 10x

Still makes sense. Now subtract 0.999... from both sides

9.999...-0.999... = 9

And because 0.999... is equivalent to x:

10x - 0.999... = ?
10x - x = 9x
9 = 9x (simplify)
1 = x

Wait, but isn't 0.999 already x? Yes! Here's another reason:

Every repeating decimal can be expressed as a fraction.

0.222... = 2/9

0.237... = 237/999

0.12345678987654321... = 12345678987654321/99999999999999999

Ergo:

0.999... = 9/9 = 1

Not enough to make you believe it's true? Here's another:

Take 0.333... which we all know is 1/3

0.333... = 1/3

Multiply both sides by 3

0.333...*3 = 3*[1/3]
0.999... = 3/3 (simplify)
0.999... = 1

There you go! 2.999... reasons why 0.999... = 1

Sources:

Vi Hart: 9.999... Reasons Why 0.999 = 1 - https://www.youtube.com/watch?v=TINfzxSnnIE