All about imaginary numbers


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Podcast Transcript

In the history of mathematics, there have been many times when mathematicians ran into problems they didn’t know what to do with.

It wasn’t so much a problem with a very difficult solution as it was a problem that didn’t seem to make sense.

In such a case, solving the problem led to a whole new branch of mathematics.

Learn more about imaginary numbers, aka complex numbers, in this episode of Everything Everywhere Daily.


There are some things in math that go beyond the realm of “hard problems.” There are a number of them that are of the type given in elite math competitions that are hard to solve but solvable.

Then there are other issues that are more philosophical in nature. Take for example division 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 can easily add, subtract and multiply by zero.

But when they tried dividing by zero, it made no sense.

Some of the first Indian mathematicians to encounter this problem said that dividing by zero equals zero. Other mathematicians said that dividing a number by zero didn’t change the number.

Neither group was right. The answer is that you simply cannot divide by zero. It cannot be done. It doesn’t even make sense.

For example, take 6/0. For this to make sense, there must be a number that you can multiply by zero to make six. The problem is that there is no number you can multiply by zero to make six because any number multiplied by zero equals zero.

Similarly, 0/0 is also impossible, even though anything multiplied by 0 equals 0. Avoiding dividing by zero takes precedence.

Another example of this is 0!.

A factorial is just a number followed by an exclamation mark. You calculate it by simply multiplying all the numbers together starting from number one. So 4! Would be 1x2x3x4.

This then begs the question, what is 0! be?

Unlike division by zero, this has an answer. The answer is 1. Mathematicians define it as 1 because the factorial of any number is that number multiplied by the factorial of the number before it. So for 1! be equal to 1, 0! 1 as being equal to 1.

In calculus, there are tons of these cases of two functions divided by each other that are of the form 0/0 or ?/? or 00 Or ?? . These are not real figures but limits, and there are techniques to solve this type of problem, which I will not detail here.

This brings me to other philosophical problems faced by mathematicians, which brings me to the subject of this episode.

Before doing that, just a quick reminder.

A positive number multiplied by another positive number is a positive number.

A negative number multiplied by a negative number is a positive number.

A square is a number multiplied by itself, and a square root is a number which, multiplied by itself, is the number in question.

Take for example the square root of 4.

2×2 equals 4, so 2 is the square root of 4.

However, -2 x -2 is also equal to 4, so -2 is also the square root of 4.

Thus, 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?

A positive number and a negative number when multiplied by itself will be positive. So what does it even mean when you’re trying to find the square root of a negative number?

This problem has existed for a very long time.

The first time we know someone ran into this problem was our pal Hero of Alexandria in the first century.

You may remember him as one of the first 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 found was the square root of 81-144 or the square root of -63.

The square root of -63 didn’t make sense to Hero and he just assumed he had made a mistake, so he just changed it to the square root of 144-81 and left it at that.

The next person we know who addressed the issue was another person who was mentioned several times on this podcast. The great Islamic mathematician Al-Khwarizmi.

Al-Khwarizmi’s solution to the problem was quite simple and, to be completely honest, made sense. He simply said that only positive numbers are squares, so the square root of a negative number is meaningless.

His solution was similar to the division by zero problem. Get rid of it.

However, the negative square root problem was not the same as the division 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 discovered 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 if they made absolutely no sense, was totally necessary to solve real problems.

It was very different from dividing by zero.

In 1637, the French philosopher and mathematician Renée Decartes coined the term “imaginary numbers”.

The next big breakthrough came in 1748 with one of the greatest mathematicians of all time, Leonhard Euler. He discovered a relationship between trigonometric functions and the exponential function.

The exponential function is the number e raised to a 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 lowercase 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 fundamental constants.

As mathematicians popped these imaginary numbers into the equations they solved, there was a big problem. It was more of a metaphysical problem than a mathematical problem.

The number “Ididn’t exist anywhere on the number line, but it clearly fit into the math, and the equations that used it worked. But what was it??

A big step towards clarifying this problem was made by the Danish mathematician Caspar Wessel in 1799. He expressed these imaginary numbers geometrically by thinking of the numbers as a two-axis plane.

The x-axis was the old regular number line. The y-axis was the imaginary numbers. So starting from 0 you would have 1i, 2i, 3i, 4i, etc. Similarly, you could go down and have -1i, -2i, -3i, etc.

You can then choose a point on this plane to create a number with a real part and an imaginary part. So you might have a number like 3+4i.

This type of numbers, which have been used by Cardano, are known as complex numbers, and the plane is known as the complex plane.

Wessel’s publication of the complex aircraft did not attract much attention, and it was rediscovered several times in the 19th century.

Thanks to 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 used complex analysis for their discoveries, and now complex analysis is an integral part of mathematics as a discipline.

The philosophical angst that early mathematicians suffered from imaginary numbers has disappeared, and they are considered as normal as real numbers.

All normal mathematical operations of addition, subtraction, multiplication and division can be used on them.

The term “imaginary number” is a term that is rarely encountered in mathematics today.

If you’re just going about your daily business, you’re probably not going to encounter many complex numbers. Even heavy-duty jobs like bookkeeping don’t need to use them.

However, they are important in the fields of science and engineering, and of course mathematics. Complex numbers are essential for any field studying waves, including anything related to radios, wifi, sound, fiber optics, GPS, and MRI machines.

Even though these numbers may 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.

Associate producers are Thor Thomsen and Peter Bennett.

Today’s review comes from listener Fiønn, at Podbean. They write,

I’m here with a big nerdy smile, having listened to your show for the first time. Your cheese episode was great. I love your pacing and your balance of facts and “life is like that” humor. It’s now marked to follow, and I aim to get in there and listen every morning. Thanks

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