Square Root

The SquareRoot of a number is the number which, when multiplied by it's self, results in the number you are getting the square root of.

• sqrt(121)=11
• sqrt(-1)=i Oops. I accidently implemented an imaginary number.

You can NOT get the square root of a negative number. Ask any calculator that. Take, for example, the number -25.

 5^2 (5 squared) is 25

But -25 is negative. What about -5×-5? Isn't that -25?

Unfortunately, no. For some strange reason (which makes some sort of sense if you think about it REALLY hard I suppose), -5^2=25. Every positive number has TWO true square roots, but only the positive ones count.

That's weird! I thought square roots and squares were opposites.

So did I. So did everyone. But unfortunately, this is another example of NothingIsAsItSeems?. :|

--SimonMould

[ Not everyone. If you treat multiplication as repeated addition, you'll see -5 x -5 cannot be -25. (Yes, its not multiplication by scaling as espoused by the new CommonCoreMath? here in the U.S.) -5 x -5 expanded out as repeated addition looks like this: (-5)-(-5)-(-5)-(-5)-(-5). ]

 I haven't thought it for a long time. But almost everyone did at some point. --SimonMould

{Not quite. You get 0-(-5)-(-5)-(-5)-(-5)-(-5).}

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Multiplication by a negative number makes more sense when complex numbers are allowed; starting at 0, with the positive integers ahead and the negatives behind, getting to a negative number involves an about-face before counting off steps (or stepping backwards and turning at the end). -5 = (-1)×5 is an about face followed by five steps. -5 × -5 = (-1)×5×(-1)×5 = (-1)×(-1)×5× (because for multiplication to continue to behave as we've always known it to it must remain commutative when negatives are involved) and that's two consecutive about-face turns followed by making five steps five times = 25.

It makes more sense once complex numbers are involved because the "turn" becomes evident as soon as you ask what would happen if instead of an about-face you only turned half way... ComplexNumbersAreYourFriend

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"You can NOT get the square root of a negative number. Ask any calculator that. Take, for example, the number -25."

Erm, 5i. And (5i)^2 = -25. And every positive number has two *real* square roots. The negative one counts just as much as the positive one. Every negative number has two imaginary roots, and every complex number has two complex roots.

 In most sums you only EVER use the POSITIVE ONE. Yes it does have two, I said that, but USUALLY only the positive one counts. :-| --SimonMould

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The confusion comes from use of the word "root". "Square" is a geometric reference, "root" is an oblique reference to the "root of the problem" -- it's a mixed metaphor which is shunned in English grammar. The other point is the use of the word "square" which isn't immediately obvious from seeing the exponent (2). See also the discussion at BigInt regarding the square root and why it is an UnknowableNumbers, rather than irrational.

{It comes from the Latin "radix", meaning "base". As in exponentiation: x is the base in x^2. If a = b^2, b is the "square base" of a. No oblique references or mixed metaphors, just a meaning that has become unclear due to linguistic drift.}