Is 1/3 the same as 0.33 recurring?

Question

I've always understood that 1 divided by 3 is 0.333333333... and so on. i.e. 0.33 recurring.

But it's been pointed out to me that 0.33 recurring multiplied by 3 is only 0.99 recurring.
1/3 multiplied by 3 is 1.

So does that mean 0.33 recurring is not the same as 1/3?

And if so, how can 1/3 be expressed as a decimal number?

Answer 1

1/3 is the exact value of 1 divided by 3.

0.33333 is the decimal representation of this value (not exact)

The 3 in this value goes on and on.



1/3 is represented in decimal value 0.33 recurring

though not mathematically speaking exactly the same.

Answer 2

0.33 recurring is the same as 1/3

If you multiply 0.33 by 3 you do get 0.99 recurring, but that is the same as 1. If you think about it, what's 1 - 0.99 recurring? Well the 9s go on forever so the answer is 0. It's a bit of a weird concept!

So yeah they're the same, but 1/3 is a much more accurate way of writing it, makes it much easier to deal with and use in equations.

Answer 3

Yes, they are the same.



You are right that 0.33 recurring * 3 is only 0.99 recurring. But, rounding 0.99 recurring to the nearest whole number you get 1. 0.33 recurring is the closest you can get to being 1/3.



1/3 as a decimal number is 0.33 recurring. You'd write it as "0.33" with a line over the "33" part of that.

Answer 4

0.33 recurring is 1/3. It means that 0.99 recurring is equal to 1.



Let x be 0.33 recurring

10x = 3.33 recurring

10x -x = 3.33.... - 0.33....

9x = 3 (recurring cancels)

x = 3/9 = 1/3

But we defined x as 0.33 recurring so it must be that 0.33 recurring is 1/3



To expand on that let's multiply both sides by 3

0.33 recurring * 3 = 0.99 recurring

1/3 * 3 = 1



Therefore 0.99 recurring must equal 1