# Thread: Does 0.9r = 1?

1. ## Does 0.9r = 1?

Was having an argument with a friend today about whether 0.9recurring was equal to 1.

I said that it was equal to 1 when you use it in arithmetic, but by itself, it isn't equal to 1. He disagreed.

Could a maths geek (Manan, Neil, Hakon) clear this up for me. Many thanks.

2. Sounds like a chicken and the egg style question.

3. Originally Posted by PhoenixFire
Was having an argument with a friend today about whether 0.9recurring was equal to 1.

I said that it was equal to 1 when you use it in arithmetic, but by itself, it isn't equal to 1. He disagreed.

Could a maths geek (Manan, Neil, Hakon) clear this up for me. Many thanks.
My initial thought would be that for engineers it's the same, for math and physics, it is not.

But something nagged at me, so I thought about it some more, and I think it is actually equal to 1. Because both 1 and 0.99999 repeating have the same 'real' value. So yea, you could say they are equivalent.

To further what I said above, I looked up the proof and the mathematical proof is:

So my initial intuition is wrong, and yes, 0.9999...... does equal one. I am a bit dissapointed as I really should have seen it right away, as real numbers do not work the way you intuitively think they do.

Intuitively, using just basic math, you can say that 0.3 repeating is the same as 1/3. So if 1/3 * 3 = 1, then 0.3 repeating * 3 must also equal 1. SInce 0.3r*3 is 0.9r, then 0.9 repeating must equal one

4. I will be reviewing my bank statements in light of this evidence

5. When they make a coin with a value of 0.9 repeating, you should.

6. Originally Posted by silentstriker

Any chance of explaining those symbols please?

7. The big E is summation, the lim is limit. I think I've explained limits before. It's relatively simple proof, but I don't think you've had calculus so it might be hard to work through it. You are first defining what you mean by 0.9 repeating. Then you are using a formula to get that 0.9r, and then you are showing why that formula = 1. And since that is true 0.9r must equal 1.

You can probably understand the arithmatic proof much better, as I said:

Intuitively, using just basic math, you can say that 0.3 repeating is the same as 1/3. So if 1/3 * 3 = 1, then 0.3 repeating * 3 must also equal 1. SInce 0.3r*3 is 0.9r, then 0.9 repeating must equal one.

8. Yeah, it was the 1/3 thing that made perfect sense to me. First time I gained something from your scientific dross SS

9. The words make more sense than the weird symbols.

10. Originally Posted by GeraintIsMyHero
Yeah, it was the 1/3 thing that made perfect sense to me. First time I gained something from your scientific dross SS
Haha ouch. I must have written tens of pages of science dross on this site.

11. haha, nah tbh, science doesn't interest me, except for Maths, I appreciate your passion for it, and your knowledge is impressive, but I could never bring myself to read through it all. I've never regarded Maths as a science because I was always good at Maths at school and abysmal at the sciences, including Physics which I was always told made me a walking anomale.

12. Probably more to do with lack of effort or not finding it interesting (not trying to judge you BTW) as I have never come across anyone who is good at Maths and crap at Physics. They are one and the same really.

13. SS has explained it reasonably well already, but in case you need another argument.
If 0.9 recurring truly was smaller than 1, then you are able to imagine a number in between the two.
So consider a number, larger than 0.9 recurring, but smaller than 1.
Then attempt to write that number with decimals.
That number can not have 1 before the comma, as it is less than 1
The largest imaginable number that starts with 0 is 0.9 recurring - which is, again, not larger than 0.9 recurring. Hence no such number can exist, and as 0.9 recurring is definitely not larger than 1, the two must be equal.

14. Originally Posted by GeraintIsMyHero
haha, nah tbh, science doesn't interest me, except for Maths, I appreciate your passion for it, and your knowledge is impressive, but I could never bring myself to read through it all. I've never regarded Maths as a science because I was always good at Maths at school and abysmal at the sciences, including Physics which I was always told made me a walking anomale.

Well, most people don't like Physics 1 & 2 (Mechanics, E&M, Sound, etc). Only after Physics 2, when you learn about Einstein's stuff, quantum mechanics, etc does it get real interesting for most people. No one gives a **** when I talk about friction, but most people are fascinated (or at least feign facination) when I talk about how time slows down for you when you move fast. But most people never get past the basic high school physics 1 & 2. So that's a real shame.

15. I understand all of the proofs and everything, as I am saying to Manan ATM, I just have dificulty comprehending that two different real numbers can have the same value.

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