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.
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.
Previously I’ve done episodes on the number zero, infinity, and complex or imaginary numbers.
All of these mathematical concepts were difficult for people to grasp at first because they aren’t things that we deal with in everyday life.
Mathematics had its origins 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 amount of sheep. 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 Arithemtica, which was a series of solutions to algebraic equations.
One such equation which he encountered was a simple equation: 4x+20=4.
If you do the math and rearrange the terms, you’ll find the solution to this equation is -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 at the time. It was intrinsically tied up with geometry, and in geometry, you can’t have a negative length of something.
The Diophantus view of negative numbers ended up becoming the predominant view of negative numbers in European mathematics for centuries.
Around the same time as Diophantus was working in Alexandria, Chinese mathematicians were developing their own system of mathematics.
The Chinese mathematician Liu Hui in the third century, wrote the first very rules for the addition and subtraction of negative numbers.
Liu’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 Hui worked with weren’t as controversial in China as the negative 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 might 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.
Liu Hui was thinking in terms of taxes. While it might be difficult to imagine a negative number of sheep, it is very easy to understand owing someone a debt.
The system which Liu Hui 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 Brahmagupta 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 zero, it was Brahmagupta 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 Brahmagupta, even if you didn’t know it. It was Brahmagupta that created a general solution to the quadratic equation and allowed for solutions that were negative or zero.
For the purposes of this episode, Brahmagupta’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 he created for doing basic arithmetic with negative numbers and zeros, in his own words:
A debt minus zero is a debt.
A fortune minus zero is a fortune.
Zero minus zero is a zero.
A debt subtracted from zero is a fortune.
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 of a debt and a fortune is a debt.
The product or quotient of a fortune and a debt is a debt.
Change the words around, and these are the same rules that we use today for doing math with negatives.
The story of negative numbers then moves to the Islamic world and the great Muslim mathematician Al – Khwarizmi, who I’ve mentioned on many episodes.
Al – Khwarizmi did not actually embrace negative numbers. Al – Khwarizmi wrote the book Al-jabar, from which algebra gets its name. In it, he covers many of the concepts in algebra but avoids anything dealing with negative numbers.
He didn’t come out against them but just avoided using them. We know that Al – Khwarizmi was familiar with the works of Indian mathematicians such as Brahmagupta, but his work was also grounded in the geometry of the Greeks, which led him to dismiss negative results.
A century later, another Islamic mathematician Abul -Wafa, used negative numbers again to represent debts. Abul -Wafa and the 12th-century mathematician Al – Samawal 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, Bh?skara II was solving quadratic equations and getting negative results, but he, too, rejected the negative values and the earlier work done by Brahmagupta.
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 mathematics texts where negative numbers were used as solutions to equations.
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 Pacioli, who was considered to be the father of modern accounting and used negative numbers in double-entry bookkeeping.
Other European mathematicians in the 15th and 16th centuries took a similar approach as Islamic scholars. They recognized that negative numbers could solve equations, but they rejected the results.
15th-century French mathematician Nicolas Chuquet called them absurd numbers, as did the 16th-century German monk Michael Stifel.
However, some mathematicians didn’t have any problem with negative numbers.
Leonhard Euler and Carl Friedrich Gauss, in the 17th century, used negative numbers on a regular basis as a part of their mathematical theorems.
Still, other mathematicians from the same era, such as Francis Maseres said that negative numbers “darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple.”
The universal adoption of negative numbers really didn’t happen until the 19th century.
So, why am I doing a whole episode on negative numbers?
It is because the idea of negative numbers, and the story behind them, is something that I’m guessing most of you have never even thought about.
The concept of negative numbers are in no way controversial, and they are very easy to understand.
Almost every grade school classroom will have a number line on the wall with negative numbers on the left side of the zero. Children can grasp negative numbers almost immediately and you don’t 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 almost 2000 years before this simple concept was universally accepted.
That is the real story of negative numbers. Sometimes even the simplest of concepts can take centuries to catch on.
The Executive Producer of Everything Everywhere Daily is Charles Daniel.
The associate producers are Thor Thomsen and Peter Bennett.
Today’s review comes of listener JimTom is God over on Apple Podcasts in the United States. They write:
Done Yet Never Done
Finally joined the Completionist Club. Still awaiting my membership benefits, could you check on that? Thanks. After literally 100’s 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 addition to my daily routine and further cemented my trivia dominance. Thanks.
Thanks, JimTom! 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 appoint a blue ribbon panel to look into the matter. We should have this taken care of before episode 2000.
Remember, if you leave a review or send me a boostagram, you too can have it read on the show,