We left off with Bombelli's discovery that if he allowed the square root of -1 to be its own number,
he could solve problems that have been stomping mathematicians for decades.
Despite the usefulness of his discovery, Bombelli and other mathematicians generally regarded it as a hack.
After all, what could it possibly mean to take the square root of a negative number?
Just like our friends 0 and negative numbers before, the square root of -1 was generally regarded with suspicion,
because it didn't correspond to anything people could think of in the real world.
For this reason, the square root of minus 1 was given the terrible names "imaginary" or "impossible."
A century or so later, Euler or begin using the simple 'i' to indicate the square root of negative 1, making the algebra less clunky.
Unfortunately, the name imaginary stuck around, and that's still we call these numbers today.
In response, everything on the original number line gets the name "real."
When we put together a real and imaginary number, we obtain what we today call a complex number.
What is remarkable about this time period is that although imaginary and complex numbers were used in calculations and derivations,
the deeper meaning behind these numbers left undiscovered for over 200 years after Bombelli's death.
Before we dive into this deeper meaning, let's think about 'i' algebraically for a moment.
If we raise 'i' to higher and higher powers, it doesn't get bigger as other numbers would.
We know i^2 is negative 1 from the definition, and if we keep multiplying i by itself, we see a pattern that repeats every four multiplications, over and over and over and over.
Hold onto that fact for 90 seconds!
Let's return to our friend the number line. Remember that all the numbers we know about show up here except imaginary numbers. They are nowhere to be found.
If we think back to our original problem with the roots of negative numbers, we can visualize this using the number line.
Remember the issue we had was finding a number that when multiplied by itself would yield a negative.
To see this more clearly, we'll use arrows instead of dots to indicate numbers.
Multiplying a positive by itself maintains direction on the number line. It stays positive.
If we multiply by a negative, we flip directions, or rotate 180°.
Squaring a negative lands us in the positive numbers because we start on the left side with our first negative and rotate 180° when we multiply by the second negative, so there's no way to land on negative number when squaring.
A positive square results in a positive, and a negative square requires starting in the negative territory, and when we multiply by the other negative, we arrive back in the positive numbers.
So what we need is something in the middle; a number that when we multiply by it only rotates 90°, not 180° as negatives do.
This is exactly what imaginary numbers do. i^2 is negative 1, meaning that the first i puts us 90° from the positive real numbers, and multiplying by i rotates 90° further, exactly where we wanted to be; firmly in negative number territory.
Back to that fact you were hanging on to. Since multiplying by i corresponds to one 90° rotation, if we place our imaginary axis at a right angle to our number line, our algebra will perfectly fit with our geometry.
If we start with the real number 1 and multiply by i, algebraically we get i, which geometrically corresponds to a 90° rotation from 1 to i.
Multiplying by i again results in i^2, which by definition is minus 1, and again matches a 90° rotation from i.
As we keep raising i to higher and higher powers, we keep rotating around with our values repeating every 4th power, just as they did algebraically.
So the big insight here is that imaginary numbers do not exist apart from the real numbers, but right on top of them; hiding in a perpendicular dimension.
This is the deeper meaning behind imaginary numbers. They aren't just some random extra number or hack, they are the natural extension of our number system from one dimension to two; numbers are two-dimensional.
And what's even more remarkable is that if we accept this, that numbers have a hidden dimension, we end up not only with a more complete mathematics, but with incredibly powerful tools for science and engineering.
Next time, we'll show how and why thinking about numbers this way is useful.