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

= 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 $$\sqrt4$$, which always means just 2, not ±2, even though both 2 and &minus;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 &minus;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.--pm a 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 ax involves two numbers, each of which has its own name ("base" and "exponent") within the term ax, so that the exponential term ax can be read explicitly: "exponentiation of the base a, to the exponent x".

How about the logarithm logax ? 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 logax ? In other words: how should the logarithmic term logax 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 ax 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 logax, i.e. by saying something like: "Logarithm of the blablabla x, to the base a"...

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