# is .9 repeating and 1 the same size?

### 15 Answers

- KrishnamurthyLv 710 months ago
If n = .999...

100n = 99.999....

100n - n = 99

99n = 99

n = 1

Therefore 1 = .999999999999.........

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- 10 months ago
Yes. I can prove this.

Let n = .99999...

Let 10n = 9.99999...

10n = 9.99999....

n = .99999....

9n = 9

n = 1

- Richard10 months agoReport
so if the universe is infinitely large, we still have 1 universe; not .999... of a universe

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- PuzzlingLv 710 months ago
They have exactly the same value. They are both equal to the integer 1.

People conceptually seem to have a problem seeing this wanting to say,

- "it is *almost* the same", or

- "it is just less than 1 by 0.0000...0001", or

- "it *approaches* the same value but never reaches it".

But those are all people that misunderstand a repeating decimal representation or can't wrap their head around there being a second representation that is equivalent to the whole number 1.

Take the example of 1/3. If you use long division, you'll get 0.3333... You can't ever finish because it will just continue having a repeated digit of 3. That string of 3s will go on forever. Mathematically 0.3333... (repeating forever) is exactly equal to 1/3. It's not *almost* 1/3. It's not *approaching* 1/3. It *is* 1/3.

So now look at this:

1/3 + 1/3 + 1/3 = 0.3333.... + 0.3333.... + 0.3333...

1 = 0.9999...

It's pretty obvious this way that they are the same thing. 0.9999... (repeating) *is* exactly the same value as 1. We don't usually write it that way, but technically we could.

Here's another pattern you can look at. Look at the representations of 1/9, 2/9, etc.

0.1111... = 1/9

0.2222... = 2/9

0.3333... = 3/9 (or 1/3)

0.4444... = 4/9

0.5555... = 5/9

0.6666... = 6/9 (or 2/3)

0.7777... = 7/9

0.8888... = 8/9

0.9999... = 9/9 (or 1)

Here's yet another way to show it:

x = 0.99999...

10x = 9.99999...

Subtract the two:

9x = 9

x = 9/9

x = 1

Hence 0.99999... = 1

Answer:

They are exactly the same. 0.999... (repeating) = 1; it's not *almost* or *approaching* the same value. It *is* the same!

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- Jeff AaronLv 710 months ago
What if x = 0.99999999999999999999999...

Then 10x = 9.99999999999999999999999...

Subtract the first equation from the second one:

9x = 9

x = 9/9

x = 1

So yes, 0.99999999999999999999... = 1

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- nineteenthlyLv 710 months ago
Yes. Any rational number can be expressed with trailing zeros or as the number with the last digit in the decimal expansion before those zeros minus one followed by trailing nines.

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- davidLv 710 months ago
YES they are the same

//// think. 1/3 = 0.3333333333... repeating

multiply both sides by 3

3x(1/3) = 3x(0.3333333...)

1 = 0.999999999... Yes they are the same

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- David B.Lv 710 months ago
Not unless you are rounding up. Otherwise, they will never be the same no matter how many 9s come after.

- ...Show all comments
There are an *infinite* amount, so yes, they are the same. There is no rounding required.

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