Sophia, pick what goes with your brow color, not your hair color. In my case, my brows match my natural hair color, which is a light brown. So taupe works perfectly.

Thanks Kristen!

Can anyone here outline the proof for why the square root of 8 is irrational? I've written one, I'm just not sure it's right.

Emily, is there a reason it's 8, of all numbers? I've only done it with 2.

With 2, you assume that its root is rational, m/n, and that both m and n have no common factors. Therefore, its square is m²/n²=2, so m²=2*n².

Now there are several possible cases for m and n: If both m and n are odd, their squares, both odd as well, can't have a factor of 2 between them, and we get a contradiction. If m is odd and n is even we get the same contradiction.

If m is even then its square is a multiplication of 4. n can't also be even (because if it were, 2 would have been a common factor between m and n and we ruled that out already), so n can only be odd. But a square of an odd number is an odd number and multiplying it by 2 won't give a number that is a multiplication of 4, so we get a contradiction again.

All possibilities led to a contradiction, when we assumed that the square root of 2, m/n, is rational, so it has to be irrational. QED.

With 8 I think there should only be slight correction in the last part, the "m is even" part, right?

[Edit: geekily enough, I think I wrote the whole thing down just so that I could write "QED" at the bottom. We only ever got to write the Hebrew initials of "what was to proove", so I never got to write "QED" on anything. Until now. Thanks, Emily!]

Ah yes, Nilly knows it.

Good times... good times....

Can you just prove that the square root of 2 is irrational, and since the square root of 8 = 2 * square root of 2 that means that the square root of 8 is also irrational?

OK, I bitch about my cats when they get noodgy, obnoxious or pissy, but at least the are under 20 lbs. Pissy Pet

I'm happy with my proof that sqrt(5) and sqrt(6) are irrational. With 8, I've gone with the following:
a/b=sqrt(8)
a=sqrt(8)b
a*a=8b*b=2*4b*b
Now the 4 in there means a*a is divisible by 4, so a is divisible by 2. But the other 2 means that a*a/4 is also divisible by 2, and since it's not a square, a must be divisible by another 2, so a must be divisible by 2*2, so a=4c
4c=sqrt(8)b
16c*c=8b*b
2c*c=b*b so b must be divisible by 2 as well, so a and b weren't respectively prime. I'm just not sure if the part where I explain why a must be divisible by 4 is clear enough.

Then again, she may just mark me down for not doing it in the Greek style (oh god, she can't, can she? Because that's a totally different proof, involving lengths of lines, and... and she didn't teach us that, so no. Whew).

[Edit: geekily enough, I think I wrote the whole thing down just so that I could write "QED" at the bottom. We only ever got to write the Hebrew initials of "what was to proove", so I never got to write "QED" on anything. Until now. Thanks, Emily!]

Ooh! What are the Hebrew initials and what are the words they stand for? (My Abstract Algebra teacher was explaining on Tuesday about the use of a little box to signal the end of the proof, but couldn't remember what "QED" stood for. Shocking!)

The thing is, she showed us the proof for sqrt(2) (which I already knew and used to use as a sort of meditation to get to sleep oh god am I geeky enough yet?), but wanted us to do it for these three others.

Now I'm imaging George Clooney living with that pig. It's kinda cute.

Now I'm imaging George Clooney living with that pig.

Wha...how did you get there?