A while back I was driving in my car listening to the radio and was gobsmacked to hear a song (What's My Name? by Rihanna and Drake) in which the singer's rap involved accurately estimating a square root. Unfortunately it was in the context of a rather vulgar play on words ("the square root of sixty-nine is eight-something..."), and in fact the local station is now censoring everything after "square root". But still, mathematics on the pop charts! Who'd have guessed?
So today's function is the square root function, familiar to adults and school children alike:
As the kids are told by their teachers, a square root of a given number is the number which when multiplied by itself produces the original number. The square root of 4 is 2, the square root of 100 is 10, the square root of 69 is 8.306623863... and so forth. Clever children notice two problems with this. First, they can't find square roots for everything - negative times negative is positive, and so there's no such thing as the square root of -1. Second, the square roots that we do have are not unique - why can't we say that the square root of 4 is -2, since (-2)*(-2) = 4?
Good teachers and/or their textbooks usually answer the first question by saying that negative numbers simply don't have real-number square roots, though they might mention that in high school we'll just define a number i to be the square root of -1 and explore its many useful and important properties. In a similar way, the second question is usually answered by appeal to definition: "Taking the positive square root is the most intuitive and useful choice, so we define the square root to be the positive one."
That last answer is more subtle. Consider trying to take the square root of a complex number which itself involves i, say, this one:
You can take my word for it that the two possible roots are
1.09868 + 0.45509 i
-1.09868 - 0.45509 i
One has positive coefficients and one has negative, and you might think you could just so with the same sign convention as for real numbers. But if we want to find something like:
The possible solutions are
-1.09868 + 0.45509 i
1.09868 - 0.45509 i
And there it's not entirely clear which might make more sense to pick. To work on a plan for systematically picking a systematic convention for which root to chose, I propose we start with the number 1. We pick its square root to be 1. Then we head around the complex plane in a counter-clockwise circle. We'd start with a number very close to 1 and pick the root that (through continuity) matches the sqrt(1) = 1 convention we started with. The procedure is pretty simple, let your complex number z = cos(θ) + i*sin(&theta); with θ running between 0 and 2π. I'll graph the real and imaginary parts of the resulting sqrt(z):
Argh! The value of z at θ = 0 and 2π is the same, z = 1. But from the graphs, the square root is not the same. So if we choose our roots for consistency and continuity, we find that we're still stuck with a discontinuity on either side of the positive real axis. For instance, sqrt(1 + 0.0001i) = 1 + 0.00005 i, but sqrt(1 + 0.0001i) = -1. + 0.00005 i.
And there's no way around it. This is a branch cut in the complex plane. Whatever root convention you pick is going to involve a discontinuity along a cut in the plane (though you have a lot of freedom in selecting where it is). But having picked a definition, you absolutely must stick with it. If you don't, you run into contradictions like this (from Wikipedia):
The third equality is invalid essentially because -1*-1 lands on the 2π side of the branch cut while 1 is by our definition on the 0π side of the branch.
The square root is far from the only function with a branch cut, and the whole issue is simultaneously one of the tremendous aggravations and subtle beauties of complex analysis. It's worth getting to know in some detail.
Now if we can just get Taylor Swift to write something like "You're The Branch Cut In My Heart" I think we can declare victory...
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Great post but I lost you at "But the real part of the square root is discontinuous about the line Im(z) = 0"
Much of the trouble comes from defining i = sqrt(-1). The proper definition is i^2 = -1, which has two solutions, sqrt(-1) and -sqrt(-1). That Wikipedia contradiction goes wrong already at the second "equality".
@Anonymous: Possibly because I worded it badly - it should probably say "on either side of the positive real axis" or something along those lines. I'll try revising it.
@2: No, that's not a problem. Essentially, it doesn't matter whether we choose i or -i as the "right one". You get (-i)*(-i)=i*i either way.
complex analysis is only fun until someone loses an i.
"Math and Emotion: the square root of one" by Klangstabil.
Somewhat daggy industrial anthem (though from a very competent duo).
mr. Nonsense on stilts
complex analysis is only fun until someone loses an i.
How about the lyrics to As Time Goes By?
[This day and age we're living in
Gives cause for apprehension
With speed and new invention
And things like fourth dimension.
Yet we get a trifle weary
With Mr. Einstein's theory.
So we must get down to earth at times
Relax relieve the tension
And no matter what the progress
Or what may yet be proved
The simple facts of life are such
They cannot be removed.]
You must remember this
A kiss is just a kiss, a sigh is just a sigh.
The fundamental things apply
As time goes by.
Tora Bora in particular was a cluster fuck and should simply not have happened.