Okay, I failed math. Crashed and burned, and noone in a five foot radius of me in Math class survived. But I’d always known (as it were) that a negative number can’t have a square root, because any number multiplied by itself will make a positive. But I’ve heard rumours, whisperings late at night, that through some kind of bizzare, way-over-my-head mathematic aerobatics, it can be done. I can’t remember where I heard that, or even why this has suddenly resurfaced in my mind, but now I’m real curious.
So how about it? Can a negative number have a square root? And can it be explained to me in a matter which won’t cause my head to implode?
No real number can be the square root of a negative number, because, as you said, any negative multiplied by another negative yields a positive.
However, just for the hell of it, mathematicians wanted to talk about these hypothetical square roots, and called them imaginary numbers. They are in a different set than real numbers and don’t have much to do with reality, except they are defined as being square roots of negatives. For example, the square root of [sup]-[/sup]9 is written 3i.
Which is why most mathamticians don’t like the name and often prefer to think of all numbers as “complex numbers” which have a real and so-called imaginary component.
(When you hear a math major talking about complex geometry, he may not be referring to how complicated it is)
IINM, imaginary numbers were simply invented as a “tool” more-or-less for helping simplify certain equations. I also had a prof tell me that any problem (currently) utilizing the imaginary domain can be solved without it. Apparently it’s just a bitch to do so.
If you mean a physical application, then yeah, you’re probably right. Imaginaries are used to reduce the number of variables and to simplify some relations.
On the other hand, certain mathematical problems do require imaginary numbers; for instance, x[sup]2[/sup] + 1, while a polynomial with real coefficients, has no real roots.
Up until the 16th century or so (maybe the 15th, I can’t remember offhand), square roots of negative numbers (SRNN) were readily dismissed as being non-existent. At about that time, Cardano, in particular, noticed that when playing around with the (top secret) cubic formula, sometimes SRNN would pop up during the intermediate steps, only to cancel out later, and the formula would still provide the correct solutions to the cubic. He was very puzzled that the algorithm could go through some “bogus” mathematical steps, yet still work (he called SRNN the “fictitious numbers”).
He never did quite figure out how the imaginary/complex numbers worked, but it was he who began the investigation.
Well, all numbers are imaginary. I have never seen a negative number in nature any more than I have seen an imaginary number or a trascendental number.
Pi or the square root of two are pretty imaginary as we cannot calculate them exactly. We just define them and use them. Heck even dollars are imaginary things and yet pretty useful.
Try this: You’re familiar with the number line, right? Going off to positive infinity on the right, negative infinity on the left, zero in the middle - sound familiar?
OK, those are the real numbers. Now picture a 2D graph, one of those things with X and Y coordinates. Now put the old number line along the X axis. Presto, you have the domain of complex numbers! That’s just a fancy name for numbers with the ordinary “real” part and and “imaginary” part.
A complex number can be anywhere on that plane, not just on the X (real) axis, and it’s just as legitimate a number if it’s not on the real axis as if it is. A complex number can be described by 2 coordinates, usually in the form “a+bi”. The X coordinate, or “a” in that format, is the real part, and the Y coordinate, or “b”, is the imaginary part. To tell them apart, you put a little i (or, if you’re an electrical engineer, j, because i means current to them) next to the imaginary part. The i simply means the square root of -1. Which, by the way, is located on the Y (imaginary) axis, 1 unit up from the origin, or at coordinates (0,1) - that’s “0+1i” in complex notation.
So why do EE’s care, especially? They’re constantly working with sinusoidal functions to describe voltage and current in alternating-current systems. It’s easy to think of sines and cosines as being the X and Y, or real and imaginary, components of vectors originating at the origin (where else?), and extending to the point representing the complex voltage or whatever. Instead of picturing a sine wave going endlessly up and down, it can be more useful to picture a vector sweeping around the complex plane like a hand on a clock.
Well, maybe a little formalism will clear things up (I know, that’s not what you expect, but it really does work).
Let’s suppose we have ordered pairs of real numbers. So (a, b) is what we’re looking at, when a and b are real numbers. We say that (a, b) = (c, d) if a = c and b = d. We’ll define addition in the intuitive manner: (a, b) + (c, d) = (a + c, b + d). Because we know what we’re looking for, we’re going to define multiplication in a weird way: (a, b) * (c, d) = (ac - bd, bc + ad). This is completely arbitrary, but it’s well-defined (whatever that means), so it’s OK.
So what does this have to do with square root of negative numbers? Well, let’s consider (0, 1) * (0, 1). Using the definition above, we see that this is equal to (-1, 0). In fact, (0, a) * (0, a) = (-(a[sup]2[/sup], 0) for any real number a. Because we don’t like writing parentheses and commas, we’re gonna write (a, b) as a + bi. And those are the complex numbers.
I hope this is clear. Basically, the complex numbers are an invention that happened to be useful. Note that this completely ignores historical context, but this is the modern understanding of them (minus a few details…just a few ;)).
Waddya gonna do. You wind up with the solution to a problem being sqrt(-1). Since this is not a real number, but it is an answer, you just call it something… ( i ).
i = i
i[sup]2[/sup] = -1
i[sup]3[/sup] = -i
i[sup]4[/sup] = 1
And so on, and then you just keep going and develop a whole theory of complex numbers.
This is such a beautiful relation that I had to take a crack at it in vB code. The best I could do was this:
e[sup]i pi[/sup] + 1 = 0
Apparently, there is no surefire way to get a ‘pi’ symbol…although perhaps someone with more experience in vB codes can do it. (Does anyone know where to find more info on these codes? The link offered when you post a reply here only lists a few of them)
Cardano’s 1545 Ars Magna gave birth to i, but he dismissed the concept as useless. Bombelli did calculations utilizing it but also dismissed the invention as resting “on sophistry.” Only at the end of the 18th century did Wessel, Argand, and Gauss develop the geometric interpretation of complex numbers summarized so well above. Then the explosion of analysis utilizing complex numbers began.
