Richard asked in Science & MathematicsMathematics · 9 months ago

# is .9 repeating and 1 the same size?

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• 9 months ago

If n = .999...

100n = 99.999....

100n - n = 99

99n = 99

n = 1

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

• 9 months ago

Yes. I can prove this.

Let n = .99999...

Let 10n = 9.99999...

10n = 9.99999....

n = .99999....

9n = 9

n = 1

• Richard9 months agoReport

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

• 9 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

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

• 9 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

• 9 months ago

No. they differ by 0.000....00001

• 9 months ago

yes

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

• David
Lv 7
9 months ago

Only if you round it up to 1

• 9 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.

• david
Lv 7
9 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