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In the history of mathematics, there were several times when mathematicians encountered problems that they didn’t know what to make of.
It wasn’t a case of a problem with a very difficult solution so much as it was a problem that didn’t seem to make any sense.
In one such case, the resolution of the problem led to an entirely new branch of mathematics.
Learn more about imaginary numbers, aka complex numbers, on this episode of Everything Everywhere Daily.
There are certain things in mathematics that go beyond the realm of “hard problems”. There is any number of these which are the type given at elite mathematics competitions which are difficult to solve but solvable.
Then there are other problems that are more philosophical in nature. Take for example dividing by zero.
Most of you know that you can’t divide by zero.
When the number zero was created by ancient Indian mathematicians, they were able to create rules for using zero in normal mathematical operations. You could easily add, subtract, and multiply by zero.
But when they tried to divide by zero, it didn’t make sense.
Some of the first Indian mathematicians to encounter this problem said that dividing by zero equaled zero. Other mathematicians said that dividing a number by zero didn’t change the number.
Neither group was right. The answer is that you just can’t divide by zero. It can’t be done. It doesn’t even make sense.
For example, take 6/0. For this to make sense, then there must be some number that you can multiply by zero to make six. The problem is there is no number you can multiply by zero to make six because any number multiplied by zero is zero.
Likewise, 0/0 is also impossible, even though anything times 0 is 0. Avoiding dividing by zero takes precedent.
Another example of this is 0!.
A factorial is just a number followed by an exclamation point. You calculate it by just multiplying all the numbers together from the number one. So 4! Would be 1x2x3x4.
This then brings up the question, what would 0! be?
Unlike dividing by zero, this has an answer. The answer is 1. Mathematicians define it as 1 because the factorial of any number is that number times the factorial of the number before it. Therefore for 1! to be equal to 1, 0! 1as to be equal to 1.
In calculus, there are tons of these cases of two functions divided by each other which are of the form 0/0 or ?/? or 00 or ?? . These are not actual numbers but limits, and there are techniques to solve these types of problems, which I am not going to get into detail here.
This leads me to other such philosophical problems encountered by mathematicians, which brings me to the subject of this episode.
Before I do that, just a quick refresher.
A positive number multiplied by another positive number is a positive number.
A negative number times a negative number is a positive number.
A square is a number multiplied by itself, and a square root is a number that when multiplied by itself, is the number in question.
For example, take the square root of 4.
2×2 equals 4, so 2 is the square root of 4.
However, -2 x -2 also equals 4, so -2 is also the square root of 4.
So, the square root of a positive number will have two correct answers. A positive number and a negative number.
This then raises the interesting question, what happens if you take the square root of a negative number?
Both a positive and a negative number when multiplied by itself will be positive. So what does it even mean when you try to find the square root of a negative number?
This problem has been around a really long time.
The first time we know that someone encountered this problem was our buddy Hero of Alexandria in the first century.
You may remember him as one of the first people to develop an early version of a steam engine.
He was working on calculating the volume of a pyramid cut by two parallel planes. The answer he came up with was the square root of 81-144 or the square root of -63.
The square root of -63 made no sense to Hero and he just assumed he had made an error, so he just switched it to the square root of 144-81 and left it at that.
The next person we know of who dealt with the problem was another person who has been mentioned on this podcast many times. The great Islamic mathematician Al-Khwarizmi.
Al-Khwarizmi’s solution to the problem was pretty simple and, to be totally honest, did make sense. He simply said that only positive numbers are squares, so the square root of a negative number makes no sense.
His solution was similar to the divide by zero problem. Just get rid of it.
However, the negative square root problem wasn’t the same as the divide by zero problem.
The problem took a big step forward in the 16th century with the Italian mathematician Gerolamo Cardano. He was working on solving cubic equations, which were variables raised to the power of three.
He found that even if he just wanted positive results, he would have to manipulate the square roots of negative numbers. His discovery was that working with negative square roots, even though they made absolutely no sense, was totally necessary to solve real problems.
This was very much unlike dividing by zero.
In 1637, the French philosopher and mathematician Renee Decartes coined the term “imaginary numbers”.
The next huge breakthrough occurred in 1748 with one of the greatest mathematicians of all time, Leonhard Euler. He discovered a relationship between trigonometry functions and the exponential function.
The exponential function is the number e raised to some variable.
The relationship he discovered only works if you use the square root of a negative number.
He also created a convention that is still used today. He used the lower case letter “i” to represent the square root of -1.
In fact, his famous equation, known as Euler’s Equation, can be simplified to ei? + 1 = 0.
It is one of the most elegant equations in all of mathematics and unifies all of the fundamental constants.
While mathematicians had these imaginary numbers appear in equations they were solving, there was a big problem. It was more of a metaphysical problem than it was a mathematical problem.
The number “i” didn’t exist anywhere on the number line, yet it clearly fit into mathematics, and the equations which used it worked. But what was it??
A huge step towards clarifying this problem was made by the Danish mathemetician Caspar Wessel in 1799. He expressed these imaginary numbers geometrically by thinking of numbers as a plane with two axes.
The x-axis was the regular old number line. The y-axis was the imaginary numbers. So going up from 0 you would have 1i, 2i, 3i, 4i, etc. Likewise, you could go down and have -1i, -2i, -3i, etc.
You could then pick a point on that plane to create a number with a real part and an imaginary part. So, you could have a number like 3+4i.
This sort of numbers, which was used back by Cardano, are known as complex numbers, and the plane is known as the complex plane.
Wessel’s publication of the complex plane didn’t get much attention and it was rediscovered several times in the 19th century.
With this new tool and a better understanding of complex numbers, a new mathematical field known as complex analysis developed in the 19th century.
Most of the greatest mathematicians of the last 200 years have used complex analysis for their discoveries, and now complex analysis is a core part of mathematics as a discipline.
The philosophical angst suffered by early mathematicians because of imaginary numbers is gone, and they are considered as normal as real numbers.
All the normal mathematical operations of addition, subtraction, multiplication, and division can be used on them.
The term “imaginary number” is one that is seldom encountered in mathematics today.
If you are just going about your everyday life, you probably aren’t going to encounter many complex numbers. Even number heavy jobs like accounting don’t need to use them.
However, they are important in fields of science and engineering, and of course math. Complex numbers are critical for any field studying waves which includes anything to do with radios, wifi, sound, fiber optics, GPS, and MRI machines.
Even though these numbers might be imaginary, they are very real in their use and in their practical applications.
Everything Everywhere Daily is an Airwave Media Podcast.
The executive producer is Darcy Adams.
The associate producers are Thor Thomsen and Peter Bennett.
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