Some years back I sat with my son as he did his math homework.
Looking over his shoulder, I saw that he was working on the function

After he finished plotting it, I thought, let’s try another one,

I began by plotting the function for various integers of x and got the black
dots in the graph below. Then, naively, I
connected the dots:

Looking
at the plot, I realized that this could not possibly be right. How could this function cross the
x-axis? That would mean that there are
some values of x for which f(x) = 0. But there were no such values. After all, you could
not raise a number to a power and get zero.
So what’s going on?

What is going on is that our function

spends most of its time outside of my piece of paper, in an ‘imaginary’ world
that lies above and below the plane of my paper. The points that I had plotted in the figure (the
black dots) are the points that I can see when this function crosses the plane
of the paper. The rest of the time the
function is outside my plane. Here is
what our function really looks like:

In the above figure, the plane of the paper is colored
green. Our function is like a
cork-screw, winding itself around the x-axis.
I wondered, how did humans discover that in addition to the “real” world
(the plane of my paper), there must also exist an “imaginary” world? What was the origin of the idea of imaginary
numbers?

**Roots of equations**

^{th}century (and for a century or two after that), mathematicians were very much concerned with geometric meaning of equations. So if you asked one what is the root of the following equation

he
or she would think about it in terms of the function

and
ask where this function crosses the x-axis.
Here is what this function looks like:

Our
quadratic function never crosses the x-axis, and so our 16

^{th}century mathematician would respond by saying that the equation
is
impossible because

had
their origin in cubic equations.

**Depressed cubics**

Scipione del Ferro was a 16

^{th}century Italian mathematician working on cubic equations of the form:
(1)

These
are called depressed cubics because they are missing the quadratic x term. His objective was to find the roots of this
equation, which translates into finding the value or values of x for which this
equation is true. This means finding the
value of x for which the function

crosses the x-axis. A
cubic function will always have at least one location at which it will cross
the x-axis, so del Ferro knew that there must exist at least one value of x for
which this equation is true.

If we put this into our cubic equation we
get:

(2)

Expanding
it we have:(3)

We can pick u and v arbitrarily (as long as x = u + v), and so del Ferro picked u and v such that

This implies that the second term in Eq. (3) is zero, and so
we have:

(4)

To
solve the above equation set

and so we have

Del
Ferro knew how to solve quadratic equations.
We have

which
means that:

(5)

From
Eq. (4) we had

and so

(6)

The
way to understand Eqs. (5) and (6) is as follows: u can take on two values, one
given by the plus term, and the other given by the minus term. When u is given by the plus term, v is given
by the minus term, and so on. Now if we
write the solution x = u + v, we end up with the expression:

(8)

When
p and q are positive the right side of Eq. (8) will become the third root of a
negative number, which can be uncomfortable to deal with, and so let us
re-write it by noting that

Using this we can re-write Eq. (8) as:

(9)

del Ferro had found a solution to a cubic, something that
had eluded man for 2000 years, ever since Babylonian times.

This was a remarkable achievement
indeed. However, del Ferro knew that in
his equation lied a deep mystery: when p and q were both positive his equation
gave the correct answer, but when one or the other was negative, his equation
gave an impossible answer. He did not
know why this formula seemed to fail in some cases. This, it turns out, is the key mystery that
led to discovery of imaginary numbers.

**The impossible equation**

Consider the cubic equation

When
we plot this equation, we have:

The equation crosses the x-axis at x=2, and so 2 is one of
the roots of this equation (indeed, 2 is the only real root). Using del Ferro’s formula (Eq. 9) and a
calculator we find that the rather hairy calculation produces an answer that
is, remarkably, exactly 2. So far so
good.

Next, let us try the cubic

When
we plot this equation, we have:

We see that x=4 is a solution. In fact, our cubic crosses the x-axis three
times, and one of those times is at x=4 (this cubic has three real solutions). But now let us try del Ferro’s formula. From Eq. (8) we have:

(10) But if del Ferro’s formula is correct, then the following must be true:

(11)

And so we arrive at the mystery: we know that x=4 is a
solution to this cubic, and we know that del Ferro’s formula is correct. Yet, when we use it, we get what appears to
be an impossible equation (Eq. 11): we have two instances of a square root of a negative
number, which at del Ferro’s time were thought to be meaningless, and yet when
these two numbers add, they produce a real number! How could that be?

It took another 50 years of thinking, and the result was a
book entitled Algebra (1572), by Rafael Bombelli, a mathematician that received
no college education. He was the first
to see that Eq. (11) required existence of a whole new set of numbers, called
imaginary numbers.

He proposed that perhaps Eq. (11) is true because each of
the third roots produce something that is partly real, and partly imaginary,
and the sum causes the two imaginary parts to cancel, leaving only a real
part. That is, he proposed that:

(12)

We proceed by cubing the two sides of Eq. (12):

To
solve for a and b, we set:

(14)
And we find that the solution is a = 2, and b = 1. So Bombelli showed that:

(15)
And therefore Eq. (11) is true because the imaginary parts
of the third roots cancel, leaving a real number.

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