Maths.. thing. .9999... = 1

Someone told me about this irl after discovering it online a few months back, I thought it was pretty nifty tbh.

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The argument goes something like this:

0.333… is 1/3, right? Well 1/3×3=1. But surely 0.333…x3=0.999…! Therefore, by one or another form of the transitive property, 0.999…=1!

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Basically,the argument is; if 1/3 = 0.3333 recurring, 2/3 = 0.66666 recurring and 3/3 = 0.99999 recurring, yet 3/3 is a "whole" (1), does that make 0.99999 recurring = 1 without any form of rounding up. I've read through a few approaches to it online and it has caused some interesting debates haha.

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• http://boards.straightdope.com/sdmb/...9&pagenumber=1
• http://www.straightdope.com/columns/...y-doesnt-999-1
• http://sciencedefeated.wordpress.com...11/05/09999-1/
• http://www.askamathematician.com/201...ly-equal-to-1/
• http://www.xamuel.com/why-is-0point999-1/

Was hard to get my head around when I first found out :hmm:

Thread moved from 'Discuss Anything' by Matts (Forum Moderator)

This is really confusing but I do at least half understand. I've never even thought of that before ;o

I understand this and I asked my math teacher in GCSE. He says 0.999 recurring is basically so infinitely close to 1 that they call it 1. It's literally INFINITELY close to 1.

0.999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999 is nowhere near the number it truly is.

Basically, can you think of a number between 0.99... and 1? No you cannot, therefore they are the same. Its the same with 0.4999..., there is no number between 0.499 and 0.5 therefore 0.499 and 0.5 are effectively the same thing.

Its like saying there's no number between 5 and 5, therefore they must be the same. There's no number between 106 and 106, therefore they must be the same etc.

I'm sorry but my brain has just turned to mush.

You can mathematically prove that 0.999... recurring is equal to 1.

Example:

Lets say we have x = 0.999... recurring.
Therefore 10x = 9.999... recurring.
10x - x = 9, so 9x = 9
Divide through by 9, and x = 1
If x = 0.999... recurring and x = 1, then 0.999... recurring = 1

This is very basic and simple really.

Heard about this from a video @0.37 http://www.youtube.com/watch?v=xM3_ltur0lo

It's not exactly 1 but it's close enough to be considered enough, the difference is that minimal.

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Originally Posted by Kardan

You can mathematically prove that 0.999... recurring is equal to 1.

Example:

Lets say we have x = 0.999... recurring.
Therefore 10x = 9.999... recurring.
10x - x = 9, so 9x = 9
Divide through by 9, and x = 1
If x = 0.999... recurring and x = 1, then 0.999... recurring = 1

This is very basic and simple really.

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Hey, not to start with :(

I like your explanation though haha, still confuses me (not "confuses", I can't think of a word) as to how one number can directly equal another.

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Originally Posted by Rixion

It's not exactly 1 but it's close enough to be considered enough, the difference is that minimal.

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It IS exactly 1. That is the point of the whole confusion surrounding it.

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Originally Posted by Intersocial

Hey, not to start with :(

I like your explanation though haha, still confuses me (not "confuses", I can't think of a word) as to how one number can directly equal another.

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Try not to think of it as 0.999... = 1, but that 0.999... is just another way of writing 1, as they are the same.

Like a stroller and a pram are the same thing.

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Originally Posted by Kardan

It IS exactly 1. That is the point of the whole confusion surrounding it.



Try not to think of it as 0.999... = 1, but that 0.999... is just another way of writing 1, as they are the same.

Like a stroller and a pram are the same thing.

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So by that logic, does 1.999 recurring = 2?

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Originally Posted by Intersocial

So by that logic, does 1.999 recurring = 2?

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Yes.

x = 1.999...
10x = 19.999...
10x - x = 9x = 18
9x / 9 = x = 2
x = 1.999... and x = 2

Works with every number that has infinite 9's after the decimal point.

Can also work in other examples...

E.g: 0.2499999... = 0.25

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Originally Posted by Kardan

Yes.

x = 1.999...
10x = 19.999...
10x - x = 9x = 18
9x / 9 = x = 2
x = 1.999... and x = 2

Works with every number that has infinite 9's after the decimal point.

Can also work in other examples...

E.g: 0.2499999... = 0.25

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Out of 11 years of maths at school, this is the only thing in maths that has genuinely interested me haha. It's still such a wonderful occurrence though how 1 number directly equals another without needing alteration. At least you managed to explain it simply as opposed to those sites full of debates over it haha.

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Originally Posted by Intersocial

Out of 11 years of maths at school, this is the only thing in maths that has genuinely interested me haha. It's still such a wonderful occurrence though how 1 number directly equals another without needing alteration. At least you managed to explain it simply as opposed to those sites full of debates over it haha.

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This is the thing though, you say "How one number equals another"... But they're not different numbers. They are the same number. Like saying 1/2 = 0.5 = 5/10. It's 0.999... = 1 = 10/10 = 456/456

A number cannot equal a different number :)

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Originally Posted by Kardan

This is the thing though, you say "How one number equals another"... But they're not different numbers. They are the same number. Like saying 1/2 = 0.5 = 5/10. It's 0.999... = 1 = 10/10 = 456/456

A number cannot equal a different number :)

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This is driving me round the bend ;l. So it's basically 1 number written in a different format? Dear God just say yes, or I'll be confused as to how you can write "0.9999" and then "1" and say you have 1 number :P I get the rest though haha.

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Originally Posted by Intersocial

This is driving me round the bend ;l. So it's basically 1 number written in a different format? Dear God just say yes, or I'll be confused as to how you can write "0.9999" and then "1" and say you have 1 number :P I get the rest though haha.

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Yes :)

People are fine accepting that numbers have numerous forms as a fraction (E.g: 2 = 4/2, 8/4), but people don't seem to accept that numbers can have numerous forms as a decimal :)

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Originally Posted by Rixion

It's not exactly 1 but it's close enough to be considered enough, the difference is that minimal.

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No, it is exactly 1. There is NO difference at all.

0.999... is exactly the same as the number 1. As I said earlier, try and think of a number between 0.99... and 1. There is no such number therefore they are the same.

There is no number that lies between 6.2 and 6.2, therefore they are the same number.

0.333 = 1/3. 1/3 * 3 = 1, therefore 0.333 * 3 also equals 1.

The difference between 0.999.. and 1 is SO minimal that it is non existant 8)

But there is a minimal albeit tiny difference but it's that small it isn't considered.

There isn't a difference, they are exactly the same thing (h)

The gap between 0.99999... and 1 is so infinitely small that it is non existent I guess.

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Originally Posted by Rixion

But there is a minimal albeit tiny difference but it's that small it isn't considered.

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There is no difference, that's the whole issue that people have with this statement :P