and also ELI16 what a an imaginary number is and how it works in real life
In: Mathematics
Of the numbers you’re used to, like 1 or 2 or -2.7, none of them can be multiplied by themselves to get a negative number. A negative times a negative is a positive, and a positive times a positive is a positive.
So, what times itself is negative? None of the “real” numbers that you’re used to, that’s for sure. So, let’s make up a number and cal it “i”. This number has no “real” value that you can write down, so we’re stuck calling it “i” forever. But, we can say that i times itself is -1. Or, put another way, i is the square root of -1.
There are a lot of times that it can be useful to find the square root of negative numbers, for instance in differential equations it shows up a lot, and can suggest periodic functions (like sine and cosine).
A lot of this comes from the fact that we can use real and imaginary numbers to represent two dimensions in “one number”. Something like 2+3i corresponds to (2,3) but we can treat it as one number, streamlining math.
Imaginary numbers are numbers that cannot be real numbers on the number line, yet are extremely useful for resolving certain mathematical expressions in physics and engineering, and in advanced mathematics.
For a five year old – It’s literally a number you made up, taking a flergle and adding a flergle. Flergles do all kinds of weird things, but you don’t need to care what they are exactly, they are just weird numbers.
For a sixteen year old – Imaginary is like Pi. The value of i is root minus one, same way that the value of pi is circumference divided by diameter. Just pretend i is a fancy greek letter, and it doesn’t matter too much whether i as a concept makes sense.
It might be best to look at it first from a historic viewpoint: at some point, mathematicians found that they can solve specific equations if they temporarily assume such a number, i.e. one that has a square root of -1 existed. They only needed it for one step in a longer mathematical proof, and in the next step it could be taken out again, so that’s why it was called “imaginary”, as in “let’s just imagine such a number existed”.
it was only later that (other) mathematicians found that this “imaginary” number *i* is very, very practical for a lot of other cases as well. For example, a lot of complicated physical properties can be calculated only if we assume such a number. And thus it was integrated into general mathematics.
Let’s not forget: most maths is not just done to come up with interesting formulas and properties of numbers (though that can actually be fun, if you are into it), but to describe reality. And the imaginary number *i* has proven to help describe reality.
To add to the other responses, there isn’t anything actually “imaginary” about imaginary numbers. Descartes (who coined the term) did think of them that way:
> the true roots as well as the false [roots] are not always real; but sometimes only imaginary [quantities]; that is to say, one can always imagine as many of them in each equation as I said; but there is sometimes no quantity that corresponds to what one imagines..
…but note that he is talking about “false” roots as well; those are the negative ones. To him negative numbers were false as well, and for imaginary numbers you had to imagine them.
Now most of us are pretty happy with negative numbers, but to Descartes they were almost as weird and silly as imaginary ones.
An “imaginary” number is one that, if you square it, the result is negative. The other kind of number, “real” numbers, cannot work like this; if you square them, you get a positive value, because a negative number times a negative number is a positive number.
“Imaginary” numbers are called that because, as a matter of something you can actually directly *measure* like a length or a temperature or a current, no such number can be measured. But they are not imaginary in the usual sense of being “just made up, completely fantastical” (or, at least, not any more than any other number.) Instead, the difference between “real” numbers and “imaginary” numbers is that they tell us different things. “Real” numbers tell us raw data, the direct observable stuff. “Imaginary” numbers, on the other hand, are a way to talk about the *phase* of something. This is extremely useful because a lot of our universe can be described using waves, and waves can affect each other depending on their phase.
Two waves are perfectly in phase when their peaks exactly line up with each other, same for their troughs. Two waves that are exactly in phase will have “constructive interference” and thus add all of their amplitude together, so (for example) two sound waves that have the same frequency and amplitude, and are perfectly in phase, will be twice as loud as they were individually. On the other hand, two waves could be exactly reverse: where one has a peak, the other has a trough, every time, making them perfectly out of phase (aka 180 degrees out of phase). When that happens, it’s called (complete) “destructive interference” and it causes the smaller wave to cancel out part of the bigger wave. If the two are perfectly identical other than their phase, then they will entirely cancel out, leaving it seeming like there’s no waves at all. Most of the time, waves are only partly out of phase, somewhere between 0 and 180 degrees out of phase, meaning they partly add and partly subtract.
A “complex” number, which has both a real part and an imaginary part (usually written “a+bi,” where a is the real part and b is the imaginary part), can encode this phase information alongside the actual amplitude of the wave. This allows us to do very quick calculations in a simple way (using exponents), without needing to faff about with angles and cosines and sines and such. As a natural consequence of this approach, we can easily determine the actual physical situation (e.g. places where waves will cancel out or amplify each other).
This has a lot of uses. Lasers, for example, are coherent light beams. Or for a very practical example, the design of a concert arena’s speakers needs to account for places where the waves from two speakers would cancel out. You don’t want your audience to be left with big silent spots because their seats happen to be in a dead zone! Quantum physics uses this all the time, and various imaging, electronics, and sound applications exist that make use of waves in one way or another.
The terms “real” and “imaginary” are a bit misleading. i is a number, just like 1 or 2. There was a time where people didn’t think pi really existed, but we accept it today, and people should accept i the same way. The only reason sqrt(-1) feels so icky is that you’ve grown up with every teacher in school telling you you can’t do that. But given a bored enough mathematician, anything is possible in math! There’s even cases where we choose to define dividing by 0, but these turn out to not be very useful or nice, so we don’t teach them in school.
Imaginary numbers on the other hand are quite useful, so we do teach them in school! They’re really great at representing 2D rotations, so these pop up all the time in electrical engineering and physics. In fact, there’s even a step above complex numbers called quaternions that are good for representing 3D spin, which physicists use a lot. But complex numbers are complicated enough, so we don’t bother teaching quaternions in school.
In general, none of the rules that you learned are “required” in math are actually required, and mathematicians choose to break these rules all the time to see what happens. When this leads to something cool and useful being discovered, we simply change the rules. You may think this would “break” math or the universe, but math is simply a set of rules we *choose*, and we just typically choose the rules that help us describe our universe. i does help describe our universe since it helps us describe rotating things easily, and so we changed our rules to allow this.
Squaring a number (x^(2)) means taking a number x and multiplying by itself. So 3^(2) is `3 * 3 = 9`. The square root of a number is the opposite, find what number multiplied by itself will equal it. The number 9 for example has a square root of 3, but also -3. `-3 * -3` gives you 9. Imaginary numbers kick in when you want the square root of a negative number.
What’s the square root of -9?
It’s not 3 since that gives 9, it’s not -3 since that also gives 9. We need some way of breaking out that negative. What we can say is it’s the square root of 9 * the square root of -1 (i) so our answer is 3i. We use i to represent the square root of -1. It’s imaginary.
Now to the real question, what can we do with that? Well, you might have some formulas that use square roots and the values might end up negative. That might be OK though if you can eliminate the i. So lets say you end up with:
`x = (2 + 3i)(2 – 3i)` which becomes
`x = 4 + 6i – 6i -9i`^(2) which becomes
`x = 4 – 9i`^(2)
Now we’ve already said i is the square root of -1. If you square that, that means you have the real -1!
`x = 4 – (9 * -1)`
`x = 13`
Basically even if you need imaginary numbers, your algorithms might be able to get rid of them or make them not matter BUT it allows you to do the math on paper.
You know how when you first learned about addition and subtraction, you learned that you can only take a smaller number from a bigger number? That is, 5 – 3 is 2, but 3 – 5 is not allowed. Because if you have a basket with three apples in it, it doesn’t make any sense to take more than three apples out of that basket.
But then, later on, you learned that, actually, you *can* take a bigger number from a smaller number, you just end up with a *negative* number. And while a basket can’t contain a negative number of apples, negative numbers can still be useful for describing things like debt, or downward motion, or a bunch of other things.
There’s another rule in math that says you can’t take the square root of a negative number. That’s because when you square a negative number, you get a positive number, so no number, positive or negative, can be squared to get a negative number.
But, just like with subtraction and negative numbers, it actually is possible to take the square root of a negative number. It’s just that the answer is a new *type* of number, like how negative numbers were a new type of number.
These numbers are called imaginary numbers for historical reasons, but they’re no more imaginary than negative numbers. Again, a basket can’t contain an imaginary number of apples, but imaginary numbers are still useful for describing real life things like electrical current or quantum mechanics.
First, imagine the number line containing the set of real numbers. Let’s focus on zero for a moment.
Now imagine another number line containing the set of imaginary numbers. This line runs perpendicular to the real number line and intersects at 0.
Geometrically, these two lines form a number plane containing the set of complex numbers. Multiplying a number by i rotates a number 90 degrees counterclockwise on this plane.
Now, I’m not clever enough to put this knowledge to good use in a way where only algebra would be needed when normally higher math is required, but I’ve heard of examples involving preserving direction.
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