# is .9 repeating and 1 the same size?

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• If n = .999...

100n = 99.999....

100n - n = 99

99n = 99

n = 1

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

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• Yes. I can prove this.

Let n = .99999...

Let 10n = 9.99999...

10n = 9.99999....

n = .99999....

9n = 9

n = 1

• so if the universe is infinitely large, we still have 1 universe; not .999... of a universe

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• 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

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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• 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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• No. they differ by 0.000....00001

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• yes

.................................

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• Only if you round it up to 1

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• 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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• 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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• Not unless you are rounding up. Otherwise, they will never be the same no matter how many 9s come after.

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