i. How Could Something 'Imaginary' Be So Important?

Factual Questions
1

i. It is the square root of negative one. It also is called imaginary, because it basically doesn’t exist. You can’t take the square root of a negative. Yet it is apparently a very important concept in mathematics. In fact, as I understand it, modern aeronautics relies a lot on it.

My question is simply this: How could something imaginary, be so important to modern science?

Leprechauns are imaginary. And Big Foot probably is. Charlie Brown is a fictional cartoon character. But none of them have any place in math or any other science. So why i?

:):):slight_smile:

2

“Imaginary” (and its counterpart “real”) in this sense are technical terms, not a judgment on whether these numbers actually “exist.”

Check out A Visual, Intuitive Guide to Imaginary Numbers at BetterExplained.com.

3

Indistinguishable to the customer service counter, please.

Don’t read anything into the name “imaginary”. That’s just one of those perjoratives from the stubborn idiots of the day, that happened to stick. “Imaginary” numbers are just as valid and, er, real, as “real” numbers.

Really, all i means is “rotate by 90 degrees”, and all that i^2 = -1 means is that if you rotate by 90 degrees and then do it again, you’re now facing the opposite direction from where you started.

4

As others note, it’s just terminology. If it bothers you, then stay away from

– Transcendental numbers, which have nothing to do with Ralph Waldo Emerson, or

– Perfect numbers, which might not meet your standards of perfection, or
– Irrational numbers, which nonetheless make perfect sense, or

— *Triangular * numbers, and a host of other “shaped” numbers – pentagonal, etc., and even Trapezoidal numbers, which don’t look like those shapes at all, or
– Polite numbers, which don’t obviously have any more social skills than other numbers, or

–Prime numbers, which don;'t seem any more choice than other numbers.
There are lots of other types of numbers with equally odd appellations, but they have to call them something. It’s all part of the marriage of math, terminology, and typography. I still don’t understand what’s so exciting about factorials! myself.

5

That. You might as well call them “red” and “blue” numbers. Real and imaginary are just very unfortunate and arbitrary labels that have stuck.

To the extent that the number 3 exists, the number i exists just as much.

6

Yes and no.

I will have 3 apples.

I will have i apples.

It’s not so much a matter of whether they “exist”, but whether they allow us to do meaningful calculations.

7

And the conjugation of a Real and an Imaginary is a Complex. Which is perhaps best understood in terms of personal relationships than mathematics. :smiley:

I always find it interesting that engineers and physicists become trivially comfortable with *i *as a representation of angle or phase. Whereas many others never become comfortable at all. It is an easy way of explaining away the issues of “exists”

We could of course mention Quaternions and Octinions. They make complex numbers look quite sensible. Just how many forms of imaginary would you like?

8

What does this mean? That calculations involving i are not meaningful? Can you have -6 apples, or π oranges?

9

The best thing to remember about math is that it is all made up. The rules that are around that you get taught in school are rules that people have found useful. If you go very far in math you might study extensions of the rules that have not yet found use beyond the academic pursuit of math.

Imaginary numbers got there name because it was a notational trick to get the answer to some problems in that resulted ultimately in regular (real) numbers. People found use for that notation beyond it just being a trick. But is was too late for a better name.

10

Ah but there are mathematics reasons for i being the square root of -1…

Euler realised trigonometry can be made into an exponential function, if there was a value for the base… which he worked out to be e… and e^(i * radians ) would then give the Cartesian x value in the real part, and the Cartesian y value in the “imaginary” part.
So it really is trigonometry using exponential instead of sine curves,
(well, you do some sin/cos/tan stuff, as needed, but you can avoid it where the exponential can be worked with directly.)

11

It makes sense to say that you own -6 apples if you possess 0 apples, but have promised to give 6 apples to a friend. π oranges doesn’t make so much sense, because you can’t measure parts of an orange to that accuracy, but you could have 22/7 oranges: 3 oranges plus one part of an orange cut into 7 equal pieces.

12

If you can’t have π apples, you can’t have 1/7 of an orange.

13

The simple answer is that imaginary numbers let you perform certain basic equations. The imaginary numbers cancel out and disappear leaving you with a normal number as your answer. But without imaginary numbers, you couldn’t perform the calculation to get to that answer.

14

Obligatory XKCD link.

15

I can imagine an orange containing an exact multiple of 7 of each kind of molecule that an orange contains, and I can imagine cutting off a piece containing exactly 1/7 of the number of molecules of each kind in the whole orange. That would be exactly 1/7 of an orange. (In practice, it might be very hard to identify an orange that is exactly divisible by 7, and then to cut off 1/7 of it, but it remains a theoretical possibility.

Of course, you cannot have a piece equal to exactly (π - 3) of an orange, because π is irrational.

16

If real numbers were actually the only ones existing in reality and imaginary numbers actually only existed in your imagination, then surreal numbers only exist in the imagination of someone who’s just dropped acid:

17

The square root of two is irrational, and yet I can imagine a geometrical construction that would yield precisely √2 of a fruit.

The point I’m trying to make is that there aren’t any special, privileged numbers. No numbers are things that actually exist. They are ideas that we invented that let us communicate and figure out things about the world.

From counting numbers to integers to rational numbers to real numbers to complex numbers, every time there was a problem that we couldn’t solve, we invented a new kind of number to solve it. Counting numbers let us reason about quantity and addition and multiplication. Integers let us figure out things like debts and subtraction. Rationals let us deal with division in a self-consistent way, and real numbers did the same thing for roots. And complex numbers solved all of basic algebra.

So you can hold three rocks and three cookies. And you can reason that these two piles have a feature in common: their quantity. But you can’t hold a three. Three is a notion, just like 5/9 or √2 or i. And since all of these things exist only in our brain, none of them is more or less “real” than any other.

18

And in turn, computer graphics programmers (at least those of us who have written our own math primitive functions) become comfortable with quaternions, because they end up being computationally cheap ways to do 3D rotation.

19

A recent Numberphile video titled Fantastic Quaternions.

20

Imaginary numbers are used quite a bit in electrical engineering, and actually the “imaginary” part of it is quite applicable.

In AC power, you have watts and vars. Most people have heard of watts, since that is what the power company charges you for. For typical residential service, the power company doesn’t charge you for vars, so most folks haven’t heard of them.

Var stands for volt-amp-reactive. A “reactor” in electrical terms is something like an inductor or capacitor. A simple inductor is a coil of wire. A simple capacitor is two metal plates close together, but not touching. When you apply electricity to a reactor, it stores energy. An inductor stores the energy in a magnetic field (you can kinda think of it as an electromagnet) and a capacitor stores energy in an electric field. Remove the electricity, and the magnetic or electric field collapses, releasing the energy back into the system. So in that respect they are temporary energy storage devices. In AC systems, the electricity is a sine wave, which means that these reactors are constantly charging and discharging.

In electrical engineering, we use j instead of i, because i already stands for current (from the French “intensitie”). We use real numbers for the watts, and imaginary numbers for the vars (as well as for the imaginary or reactive current). The watts end up being the “real” power, i.e. the power that is converted into heat and is actually used up (like in a light bulb), and the vars end up being the “imaginary” power, or the power that is just wasted by charging up those reactors. Even though that power is later put back into the system when the reactors discharge, the extra current required to charge up the reactors means an extra load on the generators.

So we might say for example that the current is 15+j2 amps. That means 15 amps of “real” current and 2 amps of “imaginary” current.

Most homes tend to be slightly inductive due to the coils in motors for things like hair dryers, clothes dryers, refrigerators, etc. The thing about inductors and capacitors is that they work opposite of each other. When one is charging during the AC cycle, the other is discharging, and vice-versa This means that you can use capacitors to balance out the inductance. So inductors will add vars and capacitors will subtract vars. The power company tries to balance out the vars, so when the vars are equal to zero, the generator only has to supply the “real” power and the reactive power essentially ping-pongs back and forth between the inductors and capacitors during the AC cycle. Since the generator doesn’t have to supply any of the reactive current, this makes the power generation and transmission much more efficient.

A simple resistor has an impedance of R. An inductor has an impedance of jwL, where L is the inductance. A capacitor has an impedance of -j/wC, where C is the capacitance. w = 2(pi)f, where f is the frequency (60 Hz in the U.S., 50 Hz in some other countries). The w is actually a lower case omega, not an English W, but it’s fairly common to type it as a w when using English letters. Anyway, if you know L, you can calculate the C you need to balance it. If you know the voltage and current then you can calculate the impedance, which again allows you to figure out the L and C values. So, lots of fun with complex math, basically.

Var balancing capacitors are located in power system substations, or may be mounted on power poles. The power company just includes the cost of the capacitors as a general equipment cost (like the generators and wires, etc) and only charges you for the watts that you use.

Industrial and commercial customers are charged for vars. Industrial users in particular often have big motors (which means a big inductance), and the power company isn’t quite so happy to pay for the huge capacitors needed to balance those out. So, the power company charges them for vars, and they charge them out the wazoo for them too. This gives those types of customers a big incentive to install their own var correcting capacitors. Those big capacitors aren’t cheap, but when the power company’s rate for vars is on the order of the “bend over and squeal like a piggy” magnitude, the capacitor banks end up being much cheaper in the long run. It’s definitely in the company’s best interest to eliminate the “imaginary” vars so that they only pay for the “real” power that they use.

Var balancing is usually called power factor correction, in case you are curious. Power factor is another way of expressing the phase angle between the voltage and current, which is something that you can calculate if you know the watts and the vars. It’s just different ways of expressing the same thing.

Anyway, the overall point here is that in AC power, real and imaginary numbers definitely make a lot of sense with real and imaginary power.