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 equationsIn the 16th 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 16th century mathematician would respond by saying that the equation
is impossible because
had their origin in cubic equations.
Scipione del Ferro was a 16th century Italian mathematician working on cubic equations of the form:
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:
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:
To solve the above equation set
and so we have
Del Ferro knew how to solve quadratic equations. We have
which means that:
From Eq. (4) we had
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:
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:
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:
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:
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.