Wikipedia:Reference desk/Archives/Mathematics/2010 February 15

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February 15

Definition of ${\sqrt {-1}}$

Why do we typically define ${\sqrt {-1}}$ to be $i$, instead of $\pm i$? Yakeyglee (talk) 03:11, 15 February 2010 (UTC)

Both $i$ and $-i$ are square roots of $-1$. Often the expression ${\sqrt {-1}}$ is avoided, because it's ambiguous; you're right when you say there isn't any particular reason it should be $i$ rather than $-i$. But when this expression is used, it is typically meant to stand for a single value, not two different values. The same goes for the expression ${\sqrt {4}}$, which always means just 2, not ±2, even though both 2 and −2 are square roots of 4. (The square root symbol ${\sqrt {\ }}$, when applied to a nonnegative real number, always indicates the principal square root, not just any square root. However, there is no compelling reason to say that $i$, rather than $-i$, should be the "principal" square root of −1.) —Bkell (talk) 03:26, 15 February 2010 (UTC)

To add to that, there's really nothing to distinguish the two square roots of -1. We arbitrarily pick one of them to be the principle square root ${\sqrt {-1}}=i.$ Then the other one must be $-i$. However, if we switched them, the complex numbers would look the same. Rckrone (talk) 05:08, 15 February 2010 (UTC)

Also notice that defining ${\sqrt {1}}$ to be $\pm 1$ isn't a great notation in computations, for then e.g. ${\sqrt {1}}+{\sqrt {4}}$ should mean either $\pm 1$ or $\pm 3$, and so on.--pma 18:09, 15 February 2010 (UTC)

The radical square root sign ${\sqrt {}}$ indicates the positive root by convention. For example, $x^{2}=2$ has two solutions, $x=\pm {\sqrt {2}}$. Of course, two numbers square to 2. Similarly, three (complex) numbers cube to 2, and so on. Often the notation $(a+bi)^{z}$ denotes the set of all z powers of the complex number $a+bi$. When z is an integer, there is only one power (or none, as in 1/0). When z is fractional or complex, there can be multiple powers. Tbonepower07 (talk) 04:13, 16 February 2010 (UTC)

What's the name for the argument of the logarithmic function?

The exponentiation a^x involves two numbers, each of which has its own name ("base" and "exponent") within the term a^x, so that the exponential term a^x can be read explicitly: "exponentiation of the base a, to the exponent x".

How about the logarithm log_ax ? Note that it involves two numbers as well, a being the base - as before; however, does x have - also here - its own name within the term log_ax ? In other words: how should the logarithmic term log_ax be read? "Logarithm of the...(???) x, to the base a...? HOOTmag (talk) 20:50, 15 February 2010 (UTC)

Actually I'd say "a to the x" for the exponentiation. I may be wrong but I don't believe there is any special name for the argument of log. And I'd say something like "the log of x" or just "log x" where there's lots, or "log a of x" when the base needs to be stated. Dmcq (talk) 21:15, 15 February 2010 (UTC)

I was taught "log x, base a", for what that counts. - Jarry1250 ^{[Humorous? Discuss.]} 21:31, 15 February 2010 (UTC)

Yes, I know that, just as you can say "exponentiation of a, to the exponent n". However, when reading the exponential term a^x you can also use the explicit name "base" for a, and say: "exponentiation of the base a to the exponent n". My question is about whether you can also use any explicit name for x - when reading the logarithmic term log_ax, i.e. by saying something like: "Logarithm of the blablabla x, to the base a"...

HOOTmag (talk) 08:53, 16 February 2010 (UTC)