Music Banter - 1 = .999..... Right?

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1 = .999..... Right?

So I had an argument with a couple friends about this. They can't be convinced that 1 = .999 repeated

I showed them the Wikipedia proof:

1/9 = 0.11111....

9 * 1/9 = 9 * 0.1111....

1 = 0.99999....

They say that Wiki isn't a reliable source :banghead:

Anyone want to back me up on this? Just for ****s and giggles.

They kept trying to argue that .9999 doesn't equal one because if you have .99999... it's not the same as having 1 of something. What they don't seem to grasp is that .9999 is a limit.

Any math wizards here?

And crap, I wanted a Yes/No poll on this!

They're confusing math with intuitive reasoning.

Quote:

Originally Posted by Freebase Dali (Post 909052)

They're confusing math with intuitive reasoning.

Yeah they get hung up on the fact that it's .9999.... so that it must end on a 9 right? They obviously can't grasp the concept of infinity.

I hate stubborn people. Oh, and Wikipedia is a great source 95% of the time.

Yes, 0.9 recurring (0.9...) is equivalent to 1. It's related to the idea that every number has a non-truncating decimal equivalent (i.e. a decimal which doesn't end in heaps of zeros). Here's what I remember from an informal 0.9... = 1 proof I saw a few years back:

Consider an infinite set S of numbers {0.9, 0.99, 0.999, ...}. Each element of S has a finite number of 9s and is marginally smaller than 1. 0.9... doesn't belong to S as it has an infinite number of 9s (and hence is bigger than every element of S).

Now imagine a number which is just smaller than 1 - let's call it 1-ϵ, where ϵ is an infinitesimally tiny number. Since S is an infinite set, there inevitably exists a number in S which is bigger than 1-ϵ. Hence 0.9... is also bigger than 1-ϵ. This leads us to the corollary that 0.9... is larger than every number smaller than 1. Now obviously every number bigger than 1 is also bigger than 0.9...

So if every number smaller than 1 is smaller than 0.9... and every number bigger than 1 is bigger than 0.9..., then 0.9... must equal 1.

Technically .999... is not 1, but for all practical purposes it is = 1. But the fact remains, .999 technically Isn't 1.

Quote:

Originally Posted by Samm (Post 909100)

Technically .999... is not 1, but for all practical purposes it is = 1. But the fact remains, .999 technically Isn't 1.

It is actually. See my proof, and Seltzer's (who's knowledge far surpasses my copy-paste skills).

Technically, .9999... isn't a number. It's a limit, and the limit equals 1.

Quote:

Originally Posted by Samm (Post 909100)

Technically .999... is not 1, but for all practical purposes it is = 1. But the fact remains, .999 technically Isn't 1.

Explain.

Quote:

Originally Posted by http://mathcentral.uregina.ca/QQ/database/QQ.09.99/andrew1.html

Name: andrew
Who is asking: Student
Level: All
Question:
is 1.9 repeating the same as 2?
explain.

Hi Andrew,

One way to look at this is to see what the difference between the two numbers is. If the difference is 0, then we will say that the two numbers ARE the same.

Now try subtracting:
2-1.9 = .1
2-1.99 = .01
2-1.999= .001
2-1.9999 = .0001
2 -1. 99999 = .00001

If you go out, say 20 places, the difference will be .000 000 000 000 000 000 01
So whatever the real error is (when you don't approximate but go on repeating), it is SMALLER that any positive number you can name. What number is smaller than any positive real number? 0.

A SECOND point of view is a little less direct:

Let S = 1.999999999 (repeating)
Then 10 S = 19.99999999 .... (repeating)
Subtract the first from the second:
9 S = 18.00000 ( repeating) = 18

Divide by 9
S = 2

This is actually the same reasoning (in disguise). However, this principle for figuring out what any repeating decimals are as fractions is widely used as you continue in mathematics. You will see it again around sequences (adding geometric sequences in calculus).

Walter Whiteley

I'm not sure I agree with this, because it is TECHNICALLY a different number. But the differences are so minuscule that it might as well be the same number.

Quote:

Originally Posted by Kirby (Post 909210)

I'm not sure I agree with this, because it is TECHNICALLY a different number. But the differences are so minuscule that it might as well be the same number.

But it isn't. 1.999999.... = 2, TECHNICALLY.

Quote:

Originally Posted by Kirby (Post 909210)

I'm not sure I agree with this, because it is TECHNICALLY a different number. But the differences are so minuscule that it might as well be the same number.

You're not realizing what 0.9999 means. It means 1. It's a mathematical representation of a quantity defined by a limit of not logically being able to be anything but 1.

It's not a matter of numerical semantic or intuitive disbelief. If you use math and logic, as is shown in the thread, it is unequivocally 1.
Why? Because it actually works. TECHNICALLY.