Making Sense of Imaginary Numbers

Several recent questions (including last week’s post) involved complex numbers, and made me realize we haven’t yet talked about them here. So let’s start a series on the topic, beginning with how we talk about them to students who are just meeting the idea for the first time, or are troubled by it.

What are imaginary numbers?

What Are Imaginary Numbers?

A fellow 6th grade teacher and I were lamenting the amount of math knowledge we have forgotten since our own high school and college educations.  Her son is currently having trouble with imaginary numbers.  I can't even remember what they are, their purpose, and how to use them.  Can you give me a brief refresher summary?  Thanks!

Imaginary numbers were conceived in response to the question of whether or not we could think about the square root of negative numbers, or equivalently whether or not there existed a value that satisfied the equation x^2 + 1 = 0.  If we decide that this equation _does_ have a solution, then we can give that solution a name: let's call it i.  Then it's not too hard to show that i^2 = -1, and that -i also satisfies the equation x^2 + 1 = 0.  The reason mathematicians chose "i" as this new number's name is that they still questioned its validity as a number, and questioned its right to co-mingle with the real numbers.

So imaginary numbers started as just that: Imagining (contrary to fact!) that there was a number whose square is $$-1$$, giving it a name, and seeing what would happen.

After all, they had a lot to worry about - what physical significance does "i" have?  (Several, in particular, in the physics of electric circuits.) Does its introduction lead to logical inconsistencies? (No - as a matter of fact, it opens up vast fields of study from abstract algebra to complex analysis.)  But as time went on, people began to realize that "i" was a generally good idea - though the term "imaginary" stuck.

Mathematics not infrequently starts with a “what if” like this, and it is only after much study that we can be sure the idea will bear fruit. But this one did … eventually!

How does one use "i" in calculation?  Well, i follows most mathematical conventions, though care needs to be taken occasionally during multiplication and root extraction. For example, i+i = 2i, and (1+i)+(3-i) = 4.  i^2 = i*i = -1, from the definition; note Sqrt[-4]*Sqrt[-1] is not equal to sqrt[(-4)(-1)] = 2, but rather Sqrt[-4]*Sqrt[-1] = i*Sqrt[4]*i = -2.  This apparent contradiction arises simply because the rule where Sqrt[p]*Sqrt[q] = Sqrt[p*q] is only applicable when p and q are non-negative.

So we just treat i like any number (or a variable), with the added fact that $$i^2=-1$$, and a little caution about square roots.

Thus we have two sets of numbers: "real" and "imaginary." They are mutually exclusive (except perhaps for 0, which could be considered as both real and imaginary, though nearly always it is thought of as belonging to the former). The union of these sets forms the *complex* numbers; these are numbers of the form a+b*i, where a and b are reals.  As can be verified, addition and multiplication are well-defined (non-ambiguous), there is an *additive identity* (zero), a *multiplicative identity* (one), there is always an *additive inverse* (i.e., for every complex number a+bi there exists a unique c+di such that (a+bi)+(c+di) = 0), and a *multiplicative inverse* (i.e., for every a+bi except 0 there is a unique c+di such that (a+bi)(c+di) = 1).  Finally, addition and multiplication are associative, commutative, and distributive.

So the complex numbers, formed by adding real and imaginary numbers, can be shown to have all the properties we are used to.

The most common purpose of the imaginary numbers is in the representation of roots of a polynomial equation in one variable.  For example, what are the roots of x^2 + 2*x + 5 ?  Using the quadratic formula, we find

-2 + Sqrt[4 - 4*5]   -2 - Sqrt[4 - 4*5]
x = ------------------ , ------------------
2                     2
or
x = {-1+2*i, -1-2*i}.

An important theorem in algebra (which I stated for a particular case at the beginning of this letter) states that for a polynomial in one variable of degree n, there are exactly n roots, counting multiple roots.  So a cubic equation has 3 roots, a quartic 4, and so on.

By working with complex numbers, rather than real, it turns out that every polynomial equation has a solution (this is the Fundamental Theorem of Algebra that was discussed last week), and in fact exactly as many as its degree.

Do negative numbers really have a square root?

The next question is from 2001:

The Imaginary Number J

One of my maths teachers says that you cannot find the square root of a minus number, especially minus one. I say that the square root of minus one equals J and is an imaginary number. My other maths teacher agrees with me and I would just like to check with somebody else.

It’s interesting that Sam has somewhere learned about i, but saw it called j instead. This is traditional in physics, where i already has a meaning (“current”, which is called i for “intensity of current”!), so a different letter is used.

I answered, without commenting on that, or the misuse of “minus”:

Hi, Sam.

They're both right. At your level, if you haven't officially learned about imaginary numbers, or in general when you are working only with real numbers, you can't take the square root of a negative number. But when you have learned about imaginary and complex numbers, and are working in a setting where complex numbers are allowed, then you can.

Everything depends on context.

It's just like what happens with negative numbers. Young children learn that they can't subtract a larger number from a smaller number. That's still true, even after you've learned about negative numbers, when you are working on a problem for which negative numbers don't make sense, such as calculating how many cookies are in the cookie jar, or subtracting digits in a large subtraction problem. But once you know about negative numbers, you can solve problems you couldn't solve before - IF you check that negative numbers make sense in the problem you are solving.

So, just as you can’t use negative numbers for some things, but you can for other things, some quantities can’t be imaginary, but others can. The trouble is that you need some pretty advanced knowledge to even know what those things are! (Look through any engineering textbook and you’ll see them everywhere.)

Where do imaginary numbers fit in?

A bigger question came to us in 2017, from a student who had not been taught about imaginary numbers, but ran across them in her reading:

Making Sense of the Imaginary

In class, we learned about square roots of numbers, and how positive numbers have a negative and a positive square root. So I understand that, for example, the square root of 25 is 5 or -5.

But what happens if you want to find the square root of a negative number? For example, what is the square root of -25? How do you solve this problem?

There are certain rules for finding the correct sign for a number (++ = +, +- = -, -- = +), but there is no rule that states that multiplying two of the same signs gives a negative. So I do not understand how you can find a square root of a number with a sign that is virtually impossible to make using two of the same signs. This is confusing, since in math there are few of these exceptions. Numbers are usually in balance.

While researching about Computer Generated Imagery for a project, the term i was briefly mentioned for square rooting negative numbers, but this was not explained and I do not understand it. I understand why i is there, but not what exactly it does to a number -- if that is even defined -- and why you are allowed to use it in maths.

It seems to me like the concept of square rooting lies more in fabricating something out of whole cloth to solve the problem rather than backing it up mathematically. I think this problem of negative square roots is impossible, mathematically speaking; but it would be solvable, logically speaking, if you invent something new. There must be a term, kind of like x, that makes an exception and states that the square root is [something] i -- a term that makes an exception to the rules and states that something is something else when it is actually not.

Are we just talking about things that don’t really exist? Sophia is not the first person to feel that way! She needed a gentle introduction.

Hi, Sophia.

You've done some good thinking!

You're right that it is impossible for a real number (the numbers you've used all your life) to have a negative square; and it does seem like things ought to balance out better than that. So it seems that we could somehow invent a new kind of number in addition to the real numbers -- an extension of the concept of number -- for which this would be possible.

But can you just invent a number to fill in a gap like that?

You've seen this same sort of thing earlier in your life.

At one time, you had only worked with whole numbers, and you found that you couldn't divide every pair of numbers and get a number for an answer; 1 divided by 2, for example, was not a (whole) number. But then you were introduced to a new kind of number that would fill the gap: fractions (or, more generally, rational numbers).

Similarly, you had found that you couldn't subtract every pair of numbers; 2 - 3 was not a (positive) number. And you were introduced to a new kind of number, the negative numbers, that made this possible.

The whole story of numbers is one of starting with numbers you understand, and extending the concept. From counting numbers, we’ve extended to integers, and rational numbers, and real numbers (which I didn’t mention, but we need them to do roots and calculus). But we need another leap, literally into another dimension!

Now, having had success solving those two roadblocks, you expect to be able to extend the number system yet again with a new number that can be the square root of negative numbers.

And this is where "i" fits in: square roots of negative numbers are multiples of this new number called "i".

For example, the square roots of -25 are ±5i:

sqrt(-25) = sqrt(-1)*sqrt(25)
= i * ±5
= ±5i

Just like positive numbers, negative numbers also have two square roots.

So where does i "live" in relation to the numbers you know?

The whole numbers you started with were just dots on the number line:

o   o   o   o   o   o ...
1   2   3   4   5   6

When you learned about fractions, they filled in the gaps between them:
+---+---+---+---+---+---+-->
0   1   2   3   4   5   6

When you learned about negative numbers, the line extended in the other direction:

<--+---+---+---+---+---+---+---+---+---+---+---+---+-->
-6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6

So where are these "imaginary numbers"? Well, they're off to the side:

^
|
3i+
|
2i+
|
i+
|
<--+---+---+---+---+---+---+---+---+---+---+---+---+-->
-6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6
-i+
|
-2i+
|
-3i+
|
v

There was nowhere to go on the number line, so we had to go somewhere else!

In fact, this makes a strange kind of sense, because multiplying by -1 rotates the whole number line by 180 degrees, and multiplying by i rotates things by 90 degrees -- so that multiplying by i twice is the same as multiplying by -1.

And when we add a real and an imaginary number, we get a point elsewhere on the plane; the number 2 + 3i is at coordinates (2, 3). These are called "complex numbers" -- real and imaginary numbers combined.

Imaginary numbers themselves are just real numbers times i, and “live” on the y-axis; complex numbers fill out the entire plane.

We've gone from one dimension (a line) to two dimensions (a plane). This gives us a whole new expanded world of numbers in which, as it turns out, there are no more gaps -- no more operations we can't do (except special cases like division by zero, which can't be fixed without messing up the rules by invoking odd things like infinity).

All this can be backed up mathematically. As the following pages explain, we can just define complex numbers as the points on the plane, state what the operations on them will be, and then show that this "new" thing has all the properties we need in order to call it an extension of the real numbers:

I gave links to pages like what we’ll look at below, and next week. Then I closed with this:

I can tell you have what it takes to enjoy the discovery of new mathematical worlds, since you asked all the right questions -- and asking questions is what mathematicians do best.

Have fun exploring all this!

Why invent new numbers?

I’ll close with a question from 2006:

History and Discovery of Imaginary Numbers

How do mathematicians decide on creating imaginary numbers such as the square root of -1 or (i)?  It just seems weird to have a number that is the square root of a negative one.

Is it not easier to say that we deal with only "real" numbers instead of these "imaginary" ones?  It would make life, and math class, a lot easier!

What is the motivation? I answered:

Hi, Lisa.

There are two things that drive mathematicians to invent something new, and then to accept it as worth keeping: curiosity and usefulness.

Curiosity is what makes us wonder, "What if there were a square root of -1?", and then go ahead and see what happens.  We're interested in new possibilities, because much of mathematics today is really a big game of "what if".

Usefulness is what makes us say, "I'm stuck trying to solve this, so I'm willing to try anything!"  It's also what later leads us to say, "Weird as it seems, this 'i' business actually works, so we'd better keep studying it and see if we can make sense of it."

We can invent something “just for fun”, or “just in case”!

Both are involved in the history of imaginary numbers, but especially the second, to begin with.  I don't know that anyone tried out the idea from mere curiosity at first; that attitude is a more recent development.  What started it all is that, in the 1500's, people were trying to work out ways to solve more complicated kinds of equations, having completely tamed quadratic equations.  They wanted a formula for cubic equations.  In the process, they found that for some equations (which they knew had solutions), they ran up against a brick wall: they had to take the square root of a negative number.  Some of them decided to go ahead and pretend that there was such a number; they just wrote sqrt(-16) as 4 sqrt(-1) and kept going as if sqrt(-1) were a number.  To their surprise, they found that the weird numbers canceled out and left them with the correct solution!  Pretending the number existed turned out to work.

I’ve described this as taking a portal into another dimension, and ending up in the real world with a valid answer. Was it just a fluke that doing illegal things led to a legal result?

It took a couple hundred years before mathematicians got comfortable with thinking of the so-called "imaginary numbers" as really existing; they thought of them as a cheap trick to get to the right answer. (Oddly enough, during the same period, and even beyond, negative numbers were likewise treated with suspicion, as if using them were cheating.)  As they tried out more and more things with complex numbers, they found that they were useful in many more ways: that complex numbers could be solutions to polynomial equations, and that including them actually simplified that subject, rather than making it more complicated; that they unified the subjects of trigonometry and exponents, so that the rules of one turned into mere applications of the rules of the other; and so on.  They are now routinely used to describe very real things like alternating electric currents.

You can read about the controversy over negative numbers in How Real Are Negative Numbers? The usefulness of complex numbers, as of negative numbers, expanded greatly after they first gained acceptance.

Also, mathematicians became more comfortable, along the way, with the idea that they could make any definition and see what would happen; they found that complex numbers were not only useful, but also interesting in themselves.  Having done this with complex numbers,  they found themselves inventing many other new kinds of objects that behaved sort of like numbers, and had interesting properties worth studying.  Some of these, eventually, turned out also to be useful in solving real problems.

Much of modern math is more curiosity than usefulness, to the extent that there are now many fields of math that have no known use (yet) – but new uses of “pure math” keep being discovered.

So as it turns out, complex numbers actually make things easier rather than harder, in the long run; and they make math beautiful in some surprising ways.  Yes, it takes some getting used to, and some effort to learn how to work with them, but once you have them, you can do things you couldn't have done without them, or do familiar things much more easily.  Mathematicians were almost forced into inventing them in the first place, but now consider them one of their most valuable creations (or discoveries, if you think of them as a world we didn't know existed, but found ourselves in by accident).

Next time: more on history, and how mathematicians used the idea of the complex plane to put all this on a firm foundation.

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