Everything Everywhere Daily: History, Science, Geography & More - Imaginary Numbers

Starting point is 00:00:00 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. What if your perceptions about the past were wrong? throughline is a podcast that takes you back in time to uncover the parts of the story that may have gone unnoticed. It effectively turned day into night.

Starting point is 00:00:52 And how it shaped the world now. Time travel with us every week on the ThruLine podcast from NPR. There are certain things in mathematics that go beyond the realm of hard problems. There's any number of these type which are 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 of zero was created by ancient Indian mathematicians,

Starting point is 00:01:28 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 any 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, it turns out, was right. The answer is that you just can't divide by zero. It can't be done.

Starting point is 00:01:54 It doesn't even make sense. Take, for example, 6 divided by 0. For this to make sense, there must be some number that you can multiply by 0 to make 6. The problem is that there is no number you can multiply by 0 to make 6, because any number multiplied by 0 is 0. Likewise, 0 divided by 0 is also impossible, even though, anything times zero is zero. Avoiding dividing by zero always takes precedent. Another example of this is zero factorial. A factorial is just a number followed by an exclamation point. You calculate

Starting point is 00:02:26 it by just multiplying all the numbers together from the number one. So, for example, four factorial would be 1 times 2 times 3 times 4. This then brings up the question. What would the factorial of 0 be? Unlike dividing by 0, this has an answer, and the answer is 1. mathematicians define it as one because the factorial of any number is that number times the factorial of the number before it. Therefore, for one factorial to be equal to one, zero factorial also has to be equal to one. In calculus, there are tons of these cases of two functions divided by each other, which are of the form zero divided by zero or infinity invited by infinity or zero raised to the power of zero or infinity raised to the power of infinity. These are not actual numbers, but limits, and there are techniques to solve these type of problem, which I'm not going to get into detail here, but they do exist. And this leads me to other such philosophical problems encountered by mathematicians, which brings me to the subject of this episode.

Starting point is 00:03:27 But 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 also a positive number. A square is any 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 four. Two times two equals four, so two is the square root of four. However, negative two times negative two also equals four.

Starting point is 00:03:57 So negative two is also the square root of four. So the square root of a positive number will have two correct answers, a positive number and a negative number. And this then raises the interesting question, what happens if you take the square root of a negative number? Both positive and negative numbers when multiplied by themselves 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 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 the steam engine.

Starting point is 00:04:38 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 minus 44, or the square root of negative 63. The square root of negative 63 made no sense to hero, and he just assumed that he had made an error, so he switched it to the square root of 144 minus 81 and left it at that. The next person we know of who dealt with the problem was another great person, who has been mentioned in this podcast many times, the great Islamic mathematician Al-Qarizmi. Al-Qurizmi's solution to the problem was pretty simple,

Starting point is 00:05:14 and, to be totally honest, kind of made 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,

Starting point is 00:05:34 with the Italian mathematician Dirolamo Cardano. He was working on solving cubic equations, which are 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.

Starting point is 00:05:58 This was very much unlike dividing by zero. In 1637, the French philosopher and math, mathematician Rene Descartes 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. And the exponential function is just the number E raised to some variable. The relationship he discovered only works if you use the square root of a negative number.

Starting point is 00:06:29 And he also created a convention which is still used today. He used the lowercase letter I to represent the square root of negative 1. In fact, his famous equation, known as Euler's equation, can be simplified to 1 plus E raised to the power of I times pi equals 0. It is one of the most elegant equations in all of mathematics, and it unifies all of the fundamental constants, E, I, Pi, 1, and 0. While mathematicians had these imaginary numbers appear in equations that 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.

Starting point is 00:07:16 But what was it? A huge step towards clarifying this problem was made by the Danish mathematician Kaspar Wessel in 1799. He expressed these imaginary numbers geometrically by thinking of numbers as a plane with two axes. The x-axis, the horizontal one, was the regular old number line. The y-axis, the vertical one, was the imaginary numbers. So going up from zero, you would have one i, two-i, three-i, four-i, etc. And likewise, you could go down from there and have negative one-i, negative two-i, negative three-i, etc. You could then pick a point anywhere on that plane to create a number with a real part and an imaginary part.

Starting point is 00:07:59 So you could have a number like 3 plus 4i. These sort of numbers, which were actually used as far back as 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 actually 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 was developed in the 19th century. Most of the great mathematicians of the last 200 years have used some complex analysis for their discoveries, and now complex analysis is a core part of mathematics as a discipline.

Starting point is 00:08:38 The philosophical angst suffered by early mathematicians because of imaginary numbers is now gone, and they're considered as normal as real numbers. all the normal mathematical operations of addition, subtraction, multiplication, and division can be used with complex numbers. The term imaginary number is one that is seldom encountered in mathematics today. If you're going about your everyday life, you probably aren't going to encounter many complex numbers. Even number-heavy jobs like accounting doesn't need to use them. However, they are important in fields of science and engineering, and of course, mathematics. Complex numbers are critical for any field studying waves, which includes,

Starting point is 00:09:15 anything to do with radios, Wi-Fi, sound, fiber optics, GPS, and even MRI machines. Even though these numbers might be imaginary, they are in fact very real in their usage and in their practical applications. Everything Everywhere Daily is an Airwave Media podcast. The executive producer is Darcy Adams. The associate producers are Thor Thompson and Peter Bennett. Today's review comes from listener Fiona over at Podbean. They write, I'm here with a big cheesy smile listening to your show for the first time.

Starting point is 00:09:49 Your episode on cheese was brilliant. I love your pace and balance of facts versus life's like that humor. It's now marked for following and I aim to pop in and listen each morning. Thank you. Well, thank you, Fjone. I always enjoy hearing from people who just discovered the show. And I'm glad you like the show and I hope it becomes a regular part of your morning routine. Remember, if you leave me a review, a boostogram, or send me an email.

Starting point is 00:10:12 You two can have it read on the show.

Everything Everywhere Daily: History, Science, Geography & More - Imaginary Numbers

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