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[Rolan7]

What I meant is that they simplified (-1)^2 on the left side to -1 on the next line down, which is wrong.  The right side with the scratched-out bit is simplified accurately, yeah.

[Frumple]

Is... isn't rooting both sides to remove the powers simplification? I'm probably missing something 'cause the picture was bugger off huge on this computer and I didn't look at it so I got sod all idea of what y'all are talking about, but rol saying they simplified and then you saying simplified in more words is kinda'...

[TheBiggerFish]

Is... isn't rooting both sides to remove the powers simplification? I'm probably missing something 'cause the picture was bugger off huge on this computer and I didn't look at it so I got sod all idea of what y'all are talking about, but rol saying they simplified and then you saying simplified in more words is kinda'...

You can't do that.

[Frumple]

Sure you can! It'll just be wrong.

[Rolan7]

What I meant is that they simplified (-1)^2 on the left side to -1 on the next line down, which is wrong.  The right side with the scratched-out bit is simplified accurately, yeah.

No, they square rooted both sides to remove the powers.

Yeah, but you just can't do that.  ((-1)^2)^(1/2) isn't 1, it's ±1 (thanks Felissan).
It's like asking "What value, squared, is 1?"  There are two valid answers.  So you can't simplify it by saying it's the positive one.

And to go further:  What valued, squared, is -1?
i
The imaginary 1.

[Putnam]

Also, -i. Every complex number has two square roots.

[scrdest]

Also, -i. Every complex number has two square roots.

Forgive me if that's like complex numbers 101 thing, but this post just made me realize...

Complex numbers are essentially an extension of the principles of negative numbers to another type of operation, aren't they?

What I mean is it seems to me you could essentially view all negative numbers as complex-ish numbers of positive numbers and -1. -1 is in its own way an 'imaginary' construct - when have you last seen negative-one apple on the table? - that acts weird in respect to a 'basic' form of the operation of addition, i is just the same thing for powers.

[Neonivek]

Why don't you have to dry brakes anymore?

[Sheb]

Why don't you have to dry brakes anymore?

When did you dry brakes?

[Neonivek]

Why don't you have to dry brakes anymore?

When did you dry brakes?

in the 80s or even earlier you had to dry brakes. (I know from an informative video I had to watch for driver school... but how outdated it was, was obvious)

But you don't have to do the dry brake maneuver today... So why not?

[Frumple]

... I'm actually not finding any mention of having to dry brakes, even from twenty or thirty years ago. Can't recall older relatives talking about it, either. Closest to it is wanting to after driving through water or very heavy rain, and even that's kinda' iffy. Gods know I've been in cars that drove through low level floodwaters before and no one mentioned the need or did anything but kept on driving.

There's auto-drying mechanisms for that nowadays, though, have been for a while, and mostly aren't needed to begin with. You can still do it more or less manually, too. Not something that comes up often.

When exactly did you watch this mentioned video, anyway?

[McTraveller]

Forgive me if that's like complex numbers 101 thing, but this post just made me realize...

Complex numbers are essentially an extension of the principles of negative numbers to another type of operation, aren't they?

What I mean is it seems to me you could essentially view all negative numbers as complex-ish numbers of positive numbers and -1. -1 is in its own way an 'imaginary' construct - when have you last seen negative-one apple on the table? - that acts weird in respect to a 'basic' form of the operation of addition, i is just the same thing for powers.

What do you mean by "an extension of the principles of negative numbers"?  Negative numbers simply flow from the rules of addition.  Countable numbers are natural numbers, negatives are still real numbers.

Or put another way, what do you mean by "i is the just the same thing for powers"?

[scrdest]

What do you mean by "an extension of the principles of negative numbers"?  Negative numbers simply flow from the rules of addition.  Countable numbers are natural numbers, negatives are still real numbers.

Or put another way, what do you mean by "i is the just the same thing for powers"?

They do flow from addition, and irrational numbers flow from exponentiation in a very similar way.

Negative numbers were conceptualized to allow the rules of addition to hold in an unusual case - to put it very weirdly, we had a defined function for addition, 'repeat X times: yield next number', and someone essentially asked what would yield the starting number of zero.

Similarly, irrational numbers plug a hole in the exponentiation, but... that's absolute basic stuff. That IS irrational numbers 101, so I don't see how that was unclear.

What I found interesting is that they both expand the number-space, they both can be treated as a weird thing stuck to a natural integer as a modifier of its behavior, and were derived by a similar process - 'hey, what DOES that function do when we feed it something outside its usual range?'.

[Sheb]

So, if irrational are the expension of number for exponentiation and can be respresented on a 2D space (one for the real component, one for the imainary component), just as real can be represented on a 1D line... Is there an equivalent adding a third dimension?

[Helgoland]

Nope. You can get 4D (quaternions) and 8D (octonions) though, but that's all - this is a very deep result.

You lose some nice properties at every step though. From reals to complex numbers you lose the ordering, from complex to quaternions you lose commutativity of multiplication, and from quaternions to octonions you lose associativity. Wikipedia even tells me that the word 'associativity' was invented to say that the octonions are not.

(I implicitly assumed that you're talking about division algebras, ie things where you can multiply and add, and also divide and subtract. If you generalize in another direction, there's many more possibilities.)