Adding decimals...?

Question

A decimal with an endless series of 3's added by itself should be a decimal with an endless series of 9's. That is, 0.333... times 3 should equal 0.999..., but it equals 1.

Why?

I meant: A decimal with an endless series of 3's added by itself THREE TIMES should be a decimal with an endless series of 9's.

Answer 1

.99999.... is the same as "1".

Seems hard to believe, right?

The easiest way to understand why the two are the same is to use this fact about the real numbers:

If there is no number *between* two real numbers, then those two numbers are the same number.

In the same way, 1/3 = .3333 ..... So .3333....*3 is the same as 1/3*3 = 1.

Answer 2

The decimal point representation is a very close representation of the fractional notation.

1/9 is one ninth
1/3 is one third
1/9 * 9 =1
1/3*3 = 1
but 1/9 = 0.11111111111111111111111111111111
and 1/3 = 0.333333333333333333333333333333333
all the way to an infinite place and the more places you use the closer you get to 1 if you multiply by 9 or 3 respectively. The point is, you are right we will never reach 1 but we get so close to it that the difference becomes negligible and we say it is close enough to be 1.
Consider this your first lesson in calculus because you will see this again when you take calculus and talk about limits.

Answer 3

The repeating decimal .999999..... IS equal to 1!
Proof:
1/9 = .111111111.......
multiply by 9 and you get .99999999....
but (1/9) * 9 = 1.
QED

Answer 4

.3333..... = 1/3

so 1/3 * 3 = 3/3 = 1