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

Starting point is 00:00:00 Over the span of human history, there are certain ideas that humans have had a very difficult time accepting. Ideas that no one has any problem with today, and are even grasped by children, actually took centuries to take root. And perhaps this is no more true than with the concept of negative numbers. Learn more about negative numbers and how they went from being absurd to commonplace on this episode of Everything Everywhere Daily. Do you ever climb into bed ready to sleep, only to have your mind start racing the moment your head hits the pillow? Thoughts bouncing around, replaying the day or jumping ahead to tomorrow. That is exactly why Catherine Nicolai created Nothing Much Happens. Each episode is a gentle, cozy bedtime story where, well, nothing much

Starting point is 00:00:53 happens. No drama, no tension, nothing you need to follow closely. Just soft narration, calming repetition, and soothing sensory details designed to help your mind slow down and your body relax. It's not about entertainment, it's about rest. And millions of listeners around the world use it every night to quiet their thoughts and finally fall asleep. If you've ever struggled to shut your brain off at night, this might be exactly what you've been missing. You can listen to Nothing Much Happens wherever you get your podcasts. Episodes are every Monday and Thursday. I've previously done episodes on the number zero, infinity, and complex or imaginary numbers.

Starting point is 00:01:34 All of these mathematical concepts were difficult for people at first to grasp, because they aren't things that we deal with in everyday life. Mathematics had its origin in counting simple objects. If you had two sheep, you could count one, two sheep. You couldn't count zero sheep. You couldn't count an infinite number of sheep. And there certainly can't be a negative number of sheep. The first instance of negative numbers, which is mentioned in historical accounts, comes from the Greek mathematician Diophantus of Alexandria. In the third century, he wrote a book titled Arithmetic, which was a series of solutions to algebraic equations. One such equation which he encountered was the simple equation 4x plus 20 equals 4.

Starting point is 00:02:17 If you do the math and rearrange the terms, you'll find the solution of this equation is negative 4. Solving this equation isn't really controversial, but Diophantus just considered the result to be absurd because he couldn't see how you could have a negative amount of something. Much of this had to do with the fact that algebra, as a separate abstract discipline, didn't really exist yet at the time. It was intrinsically tied up with geometry, and in geometry, you can't have a negative length of something. The Diophantine view of negative numbers ended up becoming the predominant

Starting point is 00:02:49 view of negative numbers in European mathematics for centuries. However, around the same time that diophantus was working in Alexandria, Chinese mathematicians were developing their own system of mathematics. The Chinese mathematician Liu Wei in the third century wrote the very first rules for the addition and subtraction of negative numbers. Lou's system of counting wasn't so much a mathematical innovation as it was an accounting innovation. He created a system of positive red symbols and negative black symbols. The black and red would cancel each other out and were used to determine how much tax someone owed. The negative numbers Liu Wei worked with weren't as controversial in China as the negative

Starting point is 00:03:29 numbers that Diophantus dealt with in Alexandria. Historians have wondered why negative numbers were embraced in China, but not in Greece. One theory is that it may have had something to do with the Chinese worldview of accepting a duality. However, it probably might have had something to do with what the numbers were trying to measure. Diophantus was thinking in terms of physical things and geometric lines. Lu Wei was thinking in terms of taxes. While it may be difficult to imagine a negative number of sheep, it's very easy to understand owing someone a debt.

Starting point is 00:04:01 The system which Lu Wei documented was probably in place in China for centuries before he wrote it down. The next big advance in negative numbers came from the land which gave us the concept of zero. India. The great 7th century Indian mathematician, Brahma Gupta, was the one who really figured out how to work with negative numbers and developed many of the rules we have regarding negative numbers today. If you remember back to my episode on the number zero, it was Brahma Gupta, who was the same person who developed the mathematical concept of zero. If you've taken an algebra class, you've probably dealt with the ideas of Brahma Gupta, even if you didn't know it. It was Brahma Gupta that created a general solution of the quadratic equation and allowed for solutions

Starting point is 00:04:42 that were negative or zero. For the purpose of this episode, Brahma Gupta's explanation of negative numbers used terms that his audience would have understood. He described negative numbers as debts and positive numbers as fortunes. Here are the rules that he created for doing basic arithmetic with negative numbers and zeros. In his own words, quote, a debt minus zero is a debt, a fortune minus zero, is a fortune. Zero minus zero is zero. A debt subtracted from zero is a fortune, and a fortune subtracted from zero is a debt. The product of zero multiplied by a debt or fortune is zero. The product of zero multiplied by zero is zero. The product or quotient of two fortunes is one fortune. The product or quotient of two debts is one fortune. The product or quotient

Starting point is 00:05:32 of a debt and a fortune is a debt, and the product or quotient of a fortune and a debt is a debt. End quote. Change the words around and these are the same rules that we use today for doing math with negative numbers. The story of negative numbers then moves to the Islamic world and the great Muslim mathematician Al-Qarizmi, who I've mentioned on many episodes. Al-Qarizmi did not actually embrace negative numbers. Al-Qarizmi wrote the book Al-Jabar, from which algebra gets its name. And in it he covers many of the concepts in algebra, but avoids anything dealing with negative. numbers. He didn't come right out against negative numbers, but he just avoided using them. We know that Al-Qarizmi was familiar with the works of Indian mathematician, such as Brahmagupta,

Starting point is 00:06:18 but his work was also grounded in the geometry of the Greeks, which led him to dismiss negative results. About a century later, another Islamic mathematician, Abu al-Wafa, used negative numbers to represent debts. Abul Wafa and the 12th century mathematician Al-Samal are some of the only Islamic mathematicians to have used negative numbers. At this point, despite the concept having been independently discovered in China and India, and having been used to a limited extent during the Islamic Caliphate, the idea of negative numbers still hadn't caught on completely. In the 12th century back in India,

Starting point is 00:06:54 Bashkara II was solving quadratic equations and getting negative results, but he too rejected the negative values and the work earlier done by Brahma Gupta. It wasn't until the 15th century that negative numbers started to appear in the works of European mathematicians. There was a study of Islamic and Byzantine mathematical text where negative numbers were used as solutions to equations. But even then, there was still resistance to the idea. One of the first Europeans to take advantage of the idea of negative numbers was the Italian Luca Patroli, who's considered to be the father of modern accounting and used negative numbers in double-entry bookkeeping. Other European mathematicians in the 15th and 16th and 16th,

Starting point is 00:07:33 16th centuries took a similar approach as Islamic scholars. They recognized that negative numbers could solve equations, but they rejected the results. The 15th century French mathematician Nicholas Chiquet called them absurd numbers, as did the 16th century German Monkin mathematician Michael Stifle. However, some mathematicians didn't have any problem with negative numbers. Leonard Euler and Carl Friedrich Gauss in the 17th century used negative numbers on a regular basis and as a part of their mathematical theorems. However, other mathematicians from the same era, such as Francis Miseres, said that negative numbers, quote, darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple, end quote.

Starting point is 00:08:18 The universal adoption of negative numbers didn't really happen until the 19th century. So, why am I doing a whole episode on negative numbers? It's because the idea of negative numbers and the story behind them is something that I'm guessing most of you have never even bothered to think about. The concept of negative numbers are in no way controversial, and they're very easy to understand. Almost every grade school classroom will have a number line in the wall with negative numbers on the left side of zero. Children can grasp negative numbers almost immediately, and you don't need anything more than grade school math to do arithmetic with them. Yet as simple as they are to understand, it was a process that took

Starting point is 00:08:59 almost 2,000 years before this simple concept was universally accepted. And that is the real story of negative numbers. That sometimes, even the simplest of ideas can take centuries to catch on. The executive producer of Everything Everywhere Daily is Charles Daniel. The associate producers are Thor Thompson and Peter Bennett. Today's review comes from listener Jim Tom is God over on Apple Podcasts in the United States. They write, Done yet never done.

Starting point is 00:09:31 Finally join the completionist class. Club, still awaiting my membership benefits. Could you check on that? Thanks. After literally hundreds of podcasts, I still look forward to my daily dose. Even on topics I know plenty about, there is almost always something I didn't know. You have become a welcome edition of my daily routine and further cemented my trivia dominance. Thanks. Well, thank you, Jim Tom. I will bring your concern to the board of directors of the Completionist Club. They will establish an ad hoc committee to appoint a special investigator who will then be in charge of a blue ribbon panel who will look into the matter. We should have this taken care of before episode number 2000.

Starting point is 00:10:06 Remember, if you leave a review or send me a boostogram, you two can have it run in the show.

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

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