Square roots

[Falcon[anti-yL]]

I took an algebra test today and the first part to solving the problem was to find the square root. I came across -100, at first I thought it was -10, then i realized that -10 squared is 100 and 10 squared is also 100. I entered it into the TI-83 and got a syntax error. I used -10 cause I didnt know what else to do. Anyone have some thoughts on this?

[dxoigmn]

I took an algebra test today and the first part to solving the problem was to find the square root. I came across -100, at first I thought it was -10, then i realized that -10 squared is 100 and 10 squared is also 100. I entered it into the TI-83 and got a syntax error. I used -10 cause I didnt know what else to do. Anyone have some thoughts on this?

sqrt(-100) = 10i

[Yoni]

If you're working in reals, -100 has no square root and you reached it either through a miscalculation, or because there's actually no solution to the equation you were trying to solve (example: x^2 + 100 = 0).

In complex numbers, every number has a square root (and any n'th root as well). The square root of -100 is 10i where i is an imaginary number defined as i = sqrt(-1).

[Adron]

Is i necessary for -100 to have a square root?

[Falcon[anti-yL]]

Bah, my teacher never taught me that, thats a 98% so far.  

[Yoni]

Is i necessary for -100 to have a square root?

No, you can use other methods like Eisenstein integers or quaternions, but complex numbers are much more common...

(You're probably thinking of something evil that I'm going to fall for. )

[Adron]

I was thinking of a minor evil. Just that of sqrt(-100) vs sqrt(pi) or any other number that we can't evaluate numerically, yet don't have a special symbolic constant for.

Sometimes I hear people saying that by defining i as sqrt(-1), sqrt(-100) becomes valid, but in my opinion that's not it. All you have to do is say that roots of negative numbers are valid. And you have to say that either way. Whether we have a name for it or not doesn't matter at all.

Just a matter of how you think about it.

[Yoni]

Oh, yes. That's what I said, you don't have a solution when you're working in reals, but in complex numbers you do ("every number has a square root"). In addition to that, assuming the common definition of sqrt(-1) as i, the square root of -100 evaluates to be 10i.

[Adron]

Yes... It was meant more as a question for everyone else to think about, not specifically to you

[hismajesty]

Bah, my teacher never taught me that, thats a 98% so far.  

I didn't learn i until Algebra II/Trig (this year.)

[Slaughter]

I didn't learn i until this year in Algebra II, so don't feel bad .

[Maddox]

Your teacher lets you use TI-83's on your test? Damn, I wish I was that lucky. Oh well, I've only had 2 tests this whole year I haven't received 100% on.

[iago]

Bah, my teacher never taught me that, thats a 98% so far.  

We were taught early that the square root of a negative number is undefined, and that would have been the correct answer on the test (when I was in math).  They didn't want us using imaginary numbers back then.

[ITAKal89]

This is a bit off topic, but don't you guys just hate that complex numbers are called, "imaginary numbers?" I mean, technically, they're as real as any other number gets, unlike the name implies. I just finished my Euclidean Geometry course for my Freshman year, so I don't know much about i, except that it works the same way as ratios do (don't immediatly disreguard what I'm saying), such that 3/1 isn't technically 3, since 3 is 3 times greater than one, it can be expressed as the ratio 3 to 1, or the real number 3, as i can be expressed in the real form of [sqrt]-1. Correct me if I'm wrong, but just to reiterate, I hate that they are called imaginary, just because the name was, "stuck."

[aDDicT]

Your teacher lets you use TI-83's on your test? Damn, I wish I was that lucky. Oh well, I've only had 2 tests this whole year I haven't received 100% on.

I'm in Pre-Calc this year and trust me when my teacher lets us use calculators, she makes the tests so difficult that it's impossible to solve any single problem without it. So I'd rather have problems that you could solve in your head such as the sqrt(-100) rather than having logarithms etc.