Investigating How Ancient Humans Discovered Math

Added: 2026-06-01

[00:00:00]Nobody actually invented math. Not in the way we usually picture an invention where one clever person sits down one afternoon and comes up with the wheel and then everybody else just uses the wheel forever. Math did not happen like that. There was never a first mathematician who woke up one morning and figured out numbers and handed them to the rest of us. Instead, math got built up slowly over tens of thousands of years by an enormous number of people who mostly were not trying to do math at all. They were trying to count their sheep or keep track of who owed them grain or measure a field. so they could tax it or figure out how many days were left until the river flooded. And the strange thing, the thing this whole video is going to slowly walk through is that the single most powerful idea in the entire history of counting. The idea that runs every phone and every computer and every bank on the planet today was an idea for nothing. a number that means there is nothing here. And when that idea finally reached Europe, people were so suspicious of it that one of the richest cities in the world made it illegal. So to give you a sense of how far back this goes, we have to start long before anybody was writing anything down. Long before cities, long before farms, before there was a single word carved into a single wall anywhere on earth, we have to start with a bone.

[00:02:09]Somewhere in the mountains between South Africa and the small country now called Esatini in a place called the border cave.

[00:02:21]Somebody found a piece of a baboon's leg bone. And on that bone there were 29 cuts just 29 little notches carved deliberately into the bone in a row. It is called the labbo bone after the mountains it came from and it is somewhere around 44,000 years old. to put that in some kind of order that is older than farming, older than pottery, older than the last ice age ending, older than basically everything you think of when you think of ancient history. The pyramids are about 4 and a half,000 years old.

[00:03:14]This bone is roughly 10 times older than that. And someone all that time ago sat down and made 29 careful marks on it.

[00:03:28]Now, nobody knows for certain what those 29 marks were counting. That is the honest answer. And you are going to hear that honest answer a lot in this video because the further back you go, the less anyone can prove. But 29 is an interesting number to land on because it is very close to the number of days in a lunar month. The time it takes the moon to go through all its phases and come back to where it started. So, a lot of people who study this think the bone might be a way of keeping track of the moon, which would also make it a way of keeping track of time, which for people living off the land, hunting and gathering and following the seasons would have been an extremely useful thing to keep track of. It might also have been a way of counting something else entirely that we will simply never know about. But either way, the important part is this. Somebody looked at a pile of things, whether that was days or animals or anything else. And instead of just looking at the pile, they made one mark for each thing. One notch, one thing.

[00:05:00]One notch, one thing. And that, as simple as it sounds, is the beginning of the whole story. That is counting. There is a second bone and this one is even more famous and it is the one that really starts the arguments. It is called the Ishango bone and it was found in 1950 by a Belgian geologist named Jean de Hindelin who was poking around in what was then the Belgian Congo in central Africa near a river that eventually flows into the headarters of the Nile. He found this bone about the size of a pencil, dark brown, polished, smooth, with a sharp little piece of quartz stuck into one end of it, which makes a lot of people think it was used as a kind of engraving tool, like a handle you would hold to scratch marks into something. The bone itself is around 20,000 years old. Although the dating has been argued about for decades because there was a lot of volcanic activity in that area that messes with the usual ways of measuring age. And the reason the Ishango bone causes so many fights is the marks on it. There are 168 of them and they are not just scattered randomly. They are scratched into three separate columns running the length of the bone and within each column they are arranged in neat little groups. So in one column you get groups of 11, then 13, then 17, then 19, which add up to 60. And this is the part that gets people excited because 11, 13, 17, and 19 are all prime numbers.

[00:07:21]meaning numbers that cannot be divided cleanly by anything except themselves and one. They are in fact every single prime number between 10 and 20 in order.

[00:07:40]In another column you get groups that look like doubling. 3 and 6, four and 8, 10 and five. as if somebody was playing with the idea of taking a number and doubling it. And in the third column you get 11, 21, 19, and 9, which are 10 + 1, 20 + 1, 20 -1 and 10 -1 and which also happen to add up to 60. So you can see why some people look at that and say this is not a tally of sheep. This is somebody messing around with numbers for the sake of it 20,000 years ago in the middle of Africa. This might be the oldest piece of actual mathematics anybody has ever found.

[00:08:45]But and this is a big butt. Plenty of serious people think that is reading way too much into it. One historian of math, a man named Peter Rdman, points out that to even have the idea of a prime number, you first need the idea of division, of splitting a number into equal parts. And he argues that humans probably did not get to the idea of prime numbers until something like 500 years before the common era which is about 19,500 years after this bone was carved. So in his view, the fact that those numbers happen to be primes is just a coincidence, a pattern we are seeing because we already know about primes and we are looking for them. The bone might just be a counting stick where the groups mean something completely ordinary that we have lost. Nobody is going to settle this argument. The bone sits in a vault in a museum in Brussels and people will probably be arguing about it forever. But whichever side is right, it tells you something. It tells you that 20,000 years ago, people were not just counting. They were arranging their counts into patterns. And they cared about how those patterns looked.

[00:10:32]And to understand why even that even plain counting is a genuine invention and not just something the human brain does automatically the way it breathes.

[00:10:44]It helps to look at the rare groups of people alive today who never developed it because they show you what the starting line actually looks like. Deep in the Amazon rainforest, there is a group of hunter gatherers called the Pira. A few hundred people living in small villages along a branch of the river. And their language, as far as researchers can tell, has no exact words for numbers at all. Not even a word for one or a word for two. What they have instead are three loose, fuzzy words that mean roughly a small amount, a somewhat larger amount, and a lot, more like the way you might say a few or some, never a precise count. a linguist named Daniel Everett, who lived among them for years, and later his son Caleb Everett, who grew up there and went back to study it properly, ran some very simple tests. They would lay out a row of objects, a line of evenly spaced batteries, say, and ask a person to put down the same number of objects to match it. And as long as the line was only two or three objects long, people did it perfectly. But the moment the line got longer than about three, the matching started to fall apart because there was no mental tool to hold the exact count.

[00:12:31]And it was not for lack of trying. Over eight months, members of the group asked with real enthusiasm to be taught how to count to 10 in Portuguese, the language of the country around them. And after 8 months of daily lessons, they concluded together that they simply could not learn it and the classes stopped. Now, the reason this matters so much for our story is what it reveals about the raw equipment all humans are born with. Because it turns out that you too can only really see a few things at once without counting. If somebody flashes three dots in front of you, you know instantly that there are three without counting them one by one. And the same goes for one or two and just barely four. That instant knowing is something the brain does on its own and researchers call it subetizing.

[00:13:46]But past three or four, the magic stops.

[00:13:53]flash seven dots at someone too quickly to count and they will not know how many there are. They will guess. Human infants can tell the difference between small numbers of things and so can a surprising number of animals. But that built-in sense is rough and it runs out almost immediately. So everything past about three, every single number above three that you have ever used in your entire life is not natural equipment at all. It is technology. It is an invention passed down to you that lets your mind reach past the tiny handful of things it can grasp on its own and keep going into the hundreds, the thousands, the millions.

[00:14:52]Which means those notches on that baboon bone are not somebody recording a count. They are somebody building a tool to have a count in the first place. A way to hold a number that the human brain left to itself cannot hold. That is the actual invention.

[00:15:19]Not the writing down of the number, but the reaching past the limit of the mind to grasp it at all. Now before we leave the very deep past, there is one more piece of equipment we need to talk about. And it is a piece of equipment you are almost certainly resting your hands on or near right now, your fingers.

[00:15:46]Because the truth is that the first calculator any human ever owned came built into the body and you can still see the fingerprints of it so to speak all over the way we count today. Think about why we count in tens.

[00:16:07]There is no deep mathematical law that says numbers have to be grouped in tens.

[00:16:16]You could group them in any size you like. The reason almost every culture on earth landed on 10 is almost embarrassingly simple. We have 10 fingers. People counted on their fingers, ran out at 10, and started over. And that habit got baked so deep into how humans think about numbers that nearly the whole planet still does it.

[00:16:48]But not everybody stopped at 10. Some peoples counted on their fingers and their toes and got to 20. And you find counting systems built around 20 in places like Central America, which is going to matter a great deal later when we get to the Maya. And the French, weirdly enough, still carry a fossil of this in their language because the French word for 80 is more or less for 20s. And then there were the people who got really clever about it. And they are the ones who gave us the strangest counting habit we still have. The one hiding in every clock and every map.

[00:17:39]Because here is a question that should bother you more than it does. Why are there 60 seconds in a minute and 60 minutes in an hour? Why not 100? We count everything else in tens and hundreds. We have a 100 cents in a dollar. We measure distance in hundreds and thousands. But time and angles, the degrees in a circle, those we still chop up into 60s and 360s.

[00:18:18]And the answer is that we inherited that directly with almost no changes from people who lived more than 4,000 years ago in the place that is now Iraq. So to understand how that happened, we have to go to the part of the world where as far as anyone can tell, writing itself was invented and where it was invented for the most boring reason imaginable.

[00:18:55]Not poetry, not religion, not history, accounting. The place is a region called Sumer in southern Mesopotamia, the land between the Tigris and Euphrates rivers. And the time is roughly 8,000 years before the common era, which means around 10,000 years ago. This is right when people in that part of the world were figuring out farming, settling down in one place, growing grain, keeping animals, and for the first time owning more stuff than they could keep track of in their heads. And when you have a granary full of barley and a herd of goats and you are trading some of it and lending some of it and owing some of it, you run into a problem that hunter gatherers following a herd never really had. You need records. You need a way to say last month there were this many jars of oil and now there are this many and that person over there owes me that many sheep. And the solution that the people of the ancient near east came up with is one of the genuinely enormous inventions in human history. And it was of all things a handful of little clay shapes.

[00:20:33]Archaeologists have dug up thousands of them all over the Middle East. And the person who spent her whole career figuring out what they were is a researcher named Denise Schmant Besserat, who pieced together a story that goes like this. People made small tokens out of clay, each one shaped to mean a particular thing. A little cone might mean a small measure of grain. A sphere might mean a larger measure of grain. A little disc might mean an animal from the flock. Each token was a thing. And the number of tokens was the count. If you had five sheep, you had five sheep tokens.

[00:21:29]If you had 10 jars of oil, you had 10 oil tokens. It is the bone all over again. One mark for one thing, except now the marks are little clay objects.

[00:21:46]You can hold in your hand, sort them, line them up, move them around.

[00:21:54]hand them to somebody else. And this system stuck around almost unchanged for an unbelievably long time. We are talking thousands of years of people just using little clay tokens to keep their accounts. But then cities started to grow around 3,500 years before the common era. And with cities came workshops and craftsmen making all sorts of new goods. And so the number of token shapes exploded until there were something like 300 different kinds of token to represent all the different things a city could produce and trade. And that is where it gets interesting because once you are sending tokens back and forth as a record of a deal, you run into a new problem, how do you stop somebody from cheating? If I send you a sealed deal saying you owe me a certain number of jars of oil, what stops you from secretly throwing a couple of the tokens away before anyone checks? So, they came up with a fix. They started sealing the tokens for a deal inside a hollow ball of clay like a little clay envelope which scholars call a bula. The tokens went inside.

[00:23:37]The ball was sealed up and now nobody could mess with the count without obviously breaking the seal, which solved the cheating problem and created a brand new very annoying problem.

[00:23:52]Now you could not see what was inside without smashing the thing open. So to fix that, the people doing the accounting started pressing each token into the soft outside of the clay ball before sealing it up. So the outside of the envelope showed the shape of every token hidden inside. Five sheep tokens inside. Five sheep marks pressed on the outside. And now look at what they have accidentally created. They have a sealed clay ball and on the outside of it is a set of marks and those marks alone tell you everything you need to know. The marks on the outside say five sheep. And at some point, and this is genuinely one of the most important moments in the history of the human mind, somebody looked at that and realized something.

[00:24:56]If the marks on the outside already tell you the whole count, then you do not actually need the tokens inside at all.

[00:25:07]You do not need the ball to be hollow.

[00:25:10]You do not need the little clay objects rattling around in there. You can just take a flat piece of clay, press the marks into it, and that flat tablet of marks is the record. The tokens after thousands of years became unnecessary.

[00:25:33]The shape of the thing pressed into clay had replaced the thing itself.

[00:25:39]And that right there is the birth of writing. The very first writing in human history was not a story or a prayer or a law. It was a receipt. It was somebody noting down how much grain was in the warehouse. But there is an even deeper thing buried inside that moment. And it took people another long stretch of time to fully work it out. And it is the single biggest leap in this entire video. So it is worth slowing all the way down for. In the old token system, the count and the thing were stuck together. A sheep token was both the idea of one and the idea of a sheep all in one object. Five sheep was five sheep tokens. There was no way to talk about the number five by itself, separate from the sheep. The number was trapped inside the thing being counted. But on those early clay tablets, scribes eventually started doing something different.

[00:26:56]Instead of pressing five separate sheep marks, they would press a sign that meant five and next to it a sign that meant sheep. The five and the sheep became two different marks. And the moment you do that, the moment you write five sheep as a five and a sheep instead of as five sheepshaped marks, you have pulled the number loose from the thing.

[00:27:26]You have invented the idea of a pure number. A five that is just five. A five that could count sheep or jars or days or anything at all. That is abstraction.

[00:27:41]And it is the thing that makes everything afterward possible.

[00:27:46]Before that, a person had a count of sheep and a count of grain that lived in separate worlds. After that, they had numbers. Numbers that worked the same no matter what you pointed them at. The whole rest of mathematics is people doing more and more powerful things with that one idea. The idea of a number that does not belong to any particular pile of stuff. Now the people who took that idea and absolutely ran with it were the Babylonians who came along a bit later in that same part of the world in Mesopotamia and built one of the most sophisticated number systems the ancient world ever produced.

[00:28:44]And this is where we finally get the answer to that question about why an hour has 60 minutes. Because the Babylonians did not count in tens, they counted in 60s. Their whole number system was built around the number 60.

[00:29:09]what we call a base 60 or sexesimal system. And to be clear about how strange that is, it means that where we have separate symbols and a fresh column, every time we hit 10, they reset every time they hit 60. Now, 60 sounds like a wildly awkward number to build your entire civilization's math around.

[00:29:39]And people have argued for a long time about why on earth they picked it. The most popular explanation is that 60 is an incredibly friendly number when it comes to dividing things up. You can split 60 evenly into halves, thirds, quarters, fifths, 6ths, 10th, 12ths, 15th, 20ths and 30ths. It divides cleanly more ways than almost any small number you could pick, which is enormously useful when you are a merchant or a builder trying to split things into fair shares without ending up with ugly leftover fractions. There is also a neat idea that you can count to 60 on two hands if you use the thumb of one hand to point at the three little bones in each of the four fingers of the same hand which gives you 12. And then you use the five fingers of your other hand to count how many twelves you have gone through. And five 12elves is 60.

[00:31:04]Whether or not that is exactly how they did it, the point is that 60 stuck and it stuck so hard that thousands of years later you are still living inside it.

[00:31:17]Every time you look at the clock and see 60 seconds and 60 minutes, every time you measure an angle and the full circle comes out to 360°, which is 660s.

[00:31:37]You are using a number system that some Babylonian accountant settled on around 4,000 years ago.

[00:31:47]We never replaced it. We just kept using it. And the Babylonians did not stop at counting. They wrote thousands of these things down on clay tablets, pressing the marks in with a reed stylus while the clay was still wet. And then a lot of those tablets got baked hard, either on purpose or by accident.

[00:32:16]when a building burned down and clay that gets baked hard can sit in the ground for 4,000 years and come out perfectly readable. Which means that unlike a lot of ancient knowledge that rotted away on paper or papyrus, an enormous amount of Babylonian math survived, and some of it is genuinely startling. There is one tablet in particular and it is probably the most argued over object in the entire history of mathematics and it is called Plimpmpton 322.

[00:32:59]It is named somewhat unromantically after an American publisher named George Plimpmpton who bought it in the 1920s and it now lives at Colombia University in New York. It is a broken little slab with four columns and 15 rows of numbers written on it in that base 60 system.

[00:33:27]And it dates to roughly 3700 years ago.

[00:33:32]And by the way, the man who originally dug it up and sold it was an adventurer and antiquities dealer named Edgar Banks who spent years out in the deserts of the Middle East hunting for treasures and who is widely believed to be one of the real life people that the movie character Indiana Jones was loosely based on. So there is a faint thread connecting a whipcracking Hollywood archaeologist to a math homework tablet, which is the kind of thing that is true and that nobody tells you. What is actually on the tablet is what makes it famous. The rows of numbers are sets of three and they fit a pattern that any kid who has sat through a geometry class will recognize even if the name has long since faded.

[00:34:41]They are what we now call Pythagorean triples, sets of three whole numbers where you can build a right angle triangle, the kind with one perfect square corner. And where did two short sides squared and added together exactly equal the longest side squared? The simplest one is three, four, five. But the ones on this tablet are not the easy little ones. The first row, for example, uses the numbers 119, 120, and 169, which is not the sort of thing you stumble onto by accident.

[00:35:35]Somebody worked these out.

[00:35:38]And the part that should stop you in your tracks is the date. This tablet is about a thousand years older than Pythagoras, the Greek that this whole idea is named after. The rule that every school child on earth learns as the Pythagorean theorem was being written down in worked examples by Babylonian scribes a full millennium before the man it is named after was even born. Pythagoras did not discover it. He or his followers much later proved it in a general way which is a different and important thing that we will get to. But the raw fact of it, the pattern of those numbers, the Babylonians had it cold. A few years ago in 2017, a pair of researchers in Australia even argued that this tablet is not just a list of triples, but is actually a kind of trigonometry table, a tool for solving triangles, which would make it the oldest such tool by more than a thousand years.

[00:37:01]Not everyone agrees with that reading and the argument is still going, but even the cautious version of the story is remarkable enough. There is another Babylonian tablet, smaller and less famous, and it looks exactly like what it is, which is a piece of student homework. It is labeled again very unromantically Y B C 7,289 and on it somebody has drawn a square with its two diagonals crossing through the middle and along the side of the square they have written the number 30 and along the diagonal they have written a string of numbers in base 60. And when you translate that string of numbers out of base 60 and into the way we write numbers today, what you find is an estimate of a very particular value. It is the number you have to multiply the side of a square by to get the length of its diagonal, which is the square root of two. A number that starts 1.414 and then keeps going forever without ever settling into a repeating pattern.

[00:38:40]And the Babylonian students estimate scratched into clay thousands of years ago is correct to about six decimal places.

[00:38:52]It is so accurate that if you used it to build something today, the error would be smaller than the width of a hair.

[00:39:03]Somebody a long time ago sitting in a scribal school in the ancient Middle East calculated the square root of two well enough to embarrass a modern pocket calculator and then presumably handed it in and went home. We do not know their name. We never will. But the homework survived and they got it right.

[00:39:34]So while all of this was happening in Mesopotamia, a few hundred miles away down along another great river, the Egyptians were building their own version of math and theirs had a completely different flavor. Because the Egyptians were above almost everything else practical. Their math was the math of people who had to survey land that flooded every single year and then build some of the largest stone structures ever attempted by human beings. All without a single one of the tools we would think were necessary.

[00:40:23]Most of what we know about Egyptian math comes from a single astonishing document and it has a slightly silly name. It is called the rind papyrus and it is named not after the person who wrote it but after a Scottish lawyer and collector named Alexander Henry Rind who bought it in the town of Luxor in 1858.

[00:40:52]It now sits in the British Museum and it is a long roll of papyrus.

[00:40:59]the paperlike material the Egyptians made from reads covered in math problems.

[00:41:08]And there is a small detail here worth slowing down for. We actually know the name of the man who wrote it out because he signed it. His name was Amos, sometimes written amos, and he was a scribe. And right at the start, he tells us that he is not even the original author. That he's copying it from a much older document that was already ancient in his own time, going back another couple of centuries before him. So this is a copy made around 3650 years ago of an even older book that is now lost. a scribe carefully writing out the textbook of the generations before him and putting his name on it so that nearly 4,000 years later we know exactly who to thank. The papyrus is basically a collection of about 87 problems and it reads like a workbook. the kind of thing a young scribe in training would have studied to learn his trade. And the trade was serious because in ancient Egypt, the scribes were the people who ran the country's accounts, divided up bread and beer as wages, measured out grain, and calculated how much of a harvest the state was owed. So the problems are things like how do you split nine loaves of bread fairly among 10 men? Which sounds trivial until you realize how the Egyptians had to do it because the Egyptians had a very particular and frankly very stubborn way of dealing with fractions.

[00:43:18]And it is one of the strangest corners of this whole story. With one single exception, the Egyptians only used what we call unit fractions.

[00:43:32]Fractions where the top number is one.

[00:43:35]So they were perfectly happy with 1/2, 1/3, 1/4, 1/5, 1/10th and so on. anything that is one piece of something. But they had no direct way of writing a fraction like 3/4 or twoths.

[00:43:58]To them, a fraction like that had to be broken up and written as a sum of different unit fractions.

[00:44:08]So 3/4 for them was written as 1/2 + 1/4er and two fifths was written as 1/3 + 115th.

[00:44:23]And every single fraction that was not already a unit fraction had to be taken apart and rebuilt this way out of distinct unit pieces.

[00:44:37]never repeating the same one twice. A huge chunk of the Ryan Papyrus, something like a third of the entire document, is just a giant reference table for exactly this. It is a table showing how to take two divided by every odd number. 2 over 3, 2 over 5, 2 over 7, all the way up, and rewrite each one as a tidy sum of unit fractions, so that a working scribe would not have to figure it out from scratch every time.

[00:45:20]It is, in other words, a cheat sheet, a 4,000-year-old cheat sheet made so that the kid doing the accounts could look up the answer instead of grinding it out.

[00:45:34]And the way the Egyptians actually multiplied numbers together is its own small wonder because they did it without ever memorizing a times table. The thing every modern child is made to chant.

[00:45:50]Their whole method needed only two skills.

[00:45:54]The ability to double a number and the ability to add.

[00:46:00]Say you wanted to multiply 14 by 12. The Egyptian scribe would make two columns starting with one on the left and 12 on the right. And then just keep doubling both. So the next row was 2 and 24.

[00:46:21]Then 4 and 48.

[00:46:24]Then 8 and 96.

[00:46:29]stopping once the left column got close to 14. Then he would pick out the rows on the left that add up to 14, which are 8 and 4 and 2, and add up the matching numbers on the right, 96 and 40, 8 and 24. And that gives 168 which is exactly 14 * 12. And here is the part that should make you sit up because of where this video is going to end. Breaking a number down into a sum of doublings into ones and twos and fours and eights is precisely the way every computer on earth handles numbers today. The system we call binary where everything is built out of powers of two. The Egyptian scribe doubling his way down a column of papyrus 4,000 years ago was without any way of knowing it. Using the exact same logic that the machine you are listening on uses to do every calculation it has ever done. They got there first by thousands of years with a reed pen and a clever habit. And the same papyrus shows that the Egyptians could do far more than divide up bread. They could find the area of a circle, which is a genuinely hard thing to do because circles involve that troublesome number we call pi. the number that connects a circle's width to the distance around its edge. And pi is one of those numbers that goes on forever and never repeats.

[00:48:40]The Egyptians, of course, did not know any of that. What they had was a rule of thumb. To find the area of a circle, they said take the width of the circle, knock off a ninth of it and square what is left. And if you work out what value of pi that rule assumes, it comes out to 256 over 81, which as a decimal is about 3.16.

[00:49:20]The true value of pi is about 3.14.

[00:49:26]So the Egyptian estimate was wrong. But it was wrong by less than 1%. And they got there with nothing but a clever trick and no idea that pi was even a thing. They also worked out how to handle the slopes of their pyramids using a measurement they called the second which was a way of describing how steeply a pyramid side leaned in as it rose. essentially the runto- rise ratio of the face so that builders could keep the angle of a giant pyramid consistent all the way up from a base that might be hundreds of feet across. When you look at how precisely the great pyramids are built with sides that barely deviate from perfect over enormous distances, the sect is part of how they pulled that off. And tucked in among all the serious accounting problems, there is one problem in the rind papyrus that is not serious at all, and it has had a remarkable afterlife. It is a little number puzzle about houses and cats and mice and grain. The setup is that there are seven houses and in each house there are seven cats and each cat has caught seven mice and each mouse had eaten seven ears of grain and each ear of grain would have produced seven measures of crop. And the puzzle is to add it all up. And if that structure sounds weirdly familiar, it is because the exact same riddle sevens inside sevens inside sevens comes down to us thousands of years later as an old English nursery rhyme. The one that begins as I was going to St. wives.

[00:51:35]I met a man with seven wives, and each wife had seven sacks, and each sack had seven cats. The same joke, the same shape, the same delight in a number, multiplying out of control, passed hand to hand for 4,000 years. from an Egyptian scrib's workbook to a children's rhyme. Nobody planned that.

[00:52:05]It just kept getting copied because people liked it. So by this point in the story, you have these two great river civilizations, Babylon and Egypt. And between them, they can count. They can handle fractions. They can find areas and volumes.

[00:52:23]They know about the relationship between the sides of a right triangle.

[00:52:29]They can estimate pi and the square root of two to a frankly ridiculous degree of accuracy. They have by any reasonable measure a lot of math. But there is something they do not have and it is a thing so important that its arrival is usually treated as the moment math becomes something genuinely new. The Babylonians and the Egyptians knew an enormous number of things that were true. What they almost never did was ask why those things were true or prove that they had to be true. They had recipes.

[00:53:15]If you do this, you get that it works.

[00:53:20]So use it and it did work. But nobody seems to have sat down and demanded a reason. That demand, the demand for proof is the thing the Greeks brought and it changed what math even was. The man usually placed at the front of that change is named Tales from the city of Myitus on the coast of what is now Turkey and he lived around 2600 years ago. Now we have to be careful here because we have almost nothing written by the himself and most of what we know about him was written down centuries after he died. So a lot of it is half legend but the tradition is strong and consistent on one point. Tales was the first person we know of who tried to prove geometric facts using pure reasoning, stepping from one thing you already agree is true to the next instead of just measuring and trusting his eyes.

[00:54:40]There is a famous story and it may well be just a story that Tales visited Egypt and was asked or decided to measure the height of one of the great pyramids, which is not an easy thing to do when you cannot exactly climb up there with a tape measure. And the trick he supposedly used is beautiful in how simple it is. He waited until the time of day when his own shadow was exactly as long as he was tall, reasoning that at that same moment the pyramid shadow would also be exactly as long as the pyramid was tall. So he measured the shadow and he had the height. He never touched the pyramid. He just thought about it correctly. And there is a result still taught today that carries his name Talis theorem which says that if you take any triangle and one of its sides is the full width of a circle passing through all three of its corners.

[00:55:54]Then the corner opposite that side is always a perfect right angle every single time. No matter how you draw it, the point is not the theorem itself so much as the attitude behind it. The attitude that you can know something must be true for all triangles forever.

[00:56:22]Not because you measured a thousand of them and they all worked, but because you can reason your way to it and show that it could not possibly be otherwise.

[00:56:34]And then there is Pythagoras who is the most famous name in all of ancient math and also one of the most slippery because the man himself is buried under so many layers of legend that historians are not even sure how much of the math credited to him he actually did. He lived around 2500 years ago and he was not just a mathematician.

[00:57:05]He was the leader of what was by any honest description a secretive religious cult. A group of followers who lived by strange rules, believed that numbers were the hidden structure of the entire universe and were sworn to keep the group's discoveries secret.

[00:57:28]They had rules against eating beans.

[00:57:32]They believed souls came back in new bodies after death. And at the center of all of it was a belief about numbers so strong that it became almost a religion in itself. The belief that everything in existence, every length, every shape, every musical note could be expressed in terms of whole numbers and the ratios between them. They had good reasons to believe this. By the way, the Pythagorans discovered that musical harmony, the reason some notes sound pleasant together, comes down to simple ratios of whole numbers, that a string half as long sounds exactly an octave higher, and so on. So when they looked at the world and saw whole numbers hiding underneath the beauty of it, they were not being crazy. They had evidence.

[00:58:40]The universe really did seem to be built out of tidy ratios.

[00:58:46]And then one of their own discoveries blew the whole thing apart. And the way they reacted to it is one of the darkest and strangest stories in the history of mathematics.

[00:58:58]Because the Pythagorans working with that famous theorem about right triangles ran into a number that broke their entire world view. Take the simplest square you can imagine. a square whose sides are each one unit long and draw a line across it corner to corner the diagonal. How long is that diagonal? By their own beloved theorem, it has to be the square root of two, the same troublesome number the Babylonian student had estimated on that clay tablet.

[00:59:43]But the Pythagoreans were not interested in an estimate. They wanted to know the exact ratio of whole numbers that gave you the square root of two because their whole faith said such a ratio had to exist.

[01:00:05]And somebody in the group proved using exactly the kind of airtight reasoning the Greeks were so proud of that there is no such ratio. That the square root of two simply cannot be written as one whole number divided by another. No matter how big the numbers get, it is what we now call an irrational number. A number that escapes the net of whole number ratios entirely. And this was a catastrophe for them because it meant their central belief, the belief that all of reality could be captured in ratios of whole numbers was false. There were lengths, ordinary lengths, you could draw with a ruler that their numbers could not describe. The legend, and again it is only a legend pieced together from sources written centuries later and full of contradictions, is that the man who discovered this or who let the secret out was named Hippasses and that for the crime of revealing it, he was drowned at sea. Some versions say his fellow Pythagorans threw him overboard. Some say the gods themselves sank the ship as punishment for letting such a dangerous truth escape. The oldest sources do not even agree on whether the secret he spilled was about irrational numbers at all or about how to build a certain shape inside a sphere. So, the famous tale of a man murdered for discovering an inconvenient number is probably more story than fact. But the math underneath the legend is solid. Somebody in that group really did prove that the square root of two could not be a ratio of whole numbers. And it really did break the thing they believed most. The Greek tradition reached its peak with a man named Uklid who worked in the city of Alexandria in Egypt around 2300 years ago and who did something that in its own quiet way may be the most influential act of writing in the history of the subject. He took the enormous scattered pile of things the Greeks had figured out about geometry and numbers and he organized all of it into a single book called the elements.

[01:03:10]And the way he organized it is the thing that mattered. He started with a tiny handful of statements so simple that nobody could argue with them. things like you can draw a straight line between any two points and all right angles are equal to one another. He called these his starting points, the things he would simply assume. And then from those few simple seeds, using nothing but careful logic, step by step, he built up hundreds of much bigger and harder results, proving each one from the ones that came before it. So that the whole towering structure rested in the end on just those few obvious beginnings.

[01:04:12]Inside the elements is a clean airtight proof that the square root of two is irrational.

[01:04:21]The very thing the Pythagorans had broken themselves on. Now laid out so plainly that anyone willing to follow the steps has to agree the elements became the most successful textbook ever written. It was studied, copied, and taught more or less continuously for over 2,000 years. Which means that students were learning geometry out of Uklid's book until within living memory of people alive today. Almost nothing else humans have ever written has lasted like that. Now, while the Greeks were busy proving things in the Mediterranean, there was a fourth great tradition growing up on the far side of the world in China. And it deserves its place here because in some ways the Chinese got closer to the system we use today than anybody else in the ancient world did.

[01:05:36]They were not writing their calculations down on tablets or papyrus the way the others did. Instead, Chinese mathematicians did their arithmetic with a set of little sticks, small rods of bamboo that they laid out on a flat board or a mat ruled into a grid of squares. These were called counting rods, and they had been in use since at least 500 years before the common era.

[01:06:10]And the clever part is how they used them. Each square on the board stood for a place, units, then tens, then hundreds, marching from right to left.

[01:06:25]And the number of rods you laid in a square told you the digit. And the square it sat in told you its size.

[01:06:35]Which means the Chinese were using a true place value system. The same core idea as ours where the same handful of symbols mean different amounts depending on where they sit. more than a thousand years before that idea finally settled into Europe to keep a one in the unit's place from being confused with a one in the 10's place. They had a neat trick of laying the rods upright in one place and sideways in the next, alternating all the way along. So the eye could always tell the places apart, and they handled the empty place, too. When a column had nothing in it, the Chinese simply left that square on the board blank, an empty space where no rods were laid, and they had a word for it that meant of all things empty or void. The same idea the Indians would later call shunya. So they understood the hole in the number perfectly well. But notice the limit they ran into. The same one Babylon hit.

[01:08:04]The emptiness lived on the board in the gap between the rods. It was never turned into a written symbol. You could put in a row of digits on a page. A mark of its own. Leave a blank space on a board and you can see it. Try to write that blank down on a strip of bamboo and the blank just disappears.

[01:08:34]So, China like Babylon, like the Maya we are about to meet, had the empty place but never the written zero. What the Chinese did have and had earlier than anyone was the idea of a number less than nothing. Working with their rods, they used two colors. Red rods to stand for what they had and black rods to stand for what they owed. positive numbers and negative numbers laid out side by side on the board. There are great early mathematical text, the nine chapters on the mathematical art put together around 2,000 years ago. Lays out the rules for working with these.

[01:09:30]How to add and subtract a debt. How subtracting a positive from nothing gives you a negative and subtracting a negative from nothing gives you a positive. The exact rules a student learns today. And they used these raw layouts to solve whole systems of equations at once, several unknowns at the time by sliding the rods around the grid, which is genuinely close to a technique that would not be reinvented in Europe for many centuries.

[01:10:13]The Chinese were doing it with sticks on a mat while Rome was still standing. So now line them all up, these four great traditions, and look at what they had and what they lacked. The Babylonians with their base 60 and their square roots and their Pythagorean triples. The Egyptians with their fractions and their doubling and their pyramids.

[01:10:43]The Greeks with their proofs and their irrational numbers. The Chinese with their place value rods and their negative numbers. Between them they could count to the heavens measured the land proved the unprovable and reckoned with death. And yet not one of them, not Babylon, not Egypt, not Greece, not China, had ever taken the idea of nothing and turned it into a true number, a written symbol that sat in the row with the others that you could add and subtract and reason about all on its own. Several of them had a way of marking an empty place. None of them had a number called zero. And the long strange globe spanning story of how humanity finally got that number is the heart of this whole video. So settle in because it is going to take us from Iraq to Mexico to India to Baghdad and finally reluctantly to Europe. Now you have to understand why zero is so hard.

[01:12:19]Because at first it sounds like the easiest thing in the world. Zero just means nothing. And surely people have always understood the idea of having nothing. If you had three goats and they all wandered off, you have no goats. And a child understands that. So what was the big deal? The big deal is that there is an enormous difference between understanding nothing as an idea and treating nothing as a number.

[01:12:56]A thing you can write down. A thing that sits in a row with one and two and three. A thing you can add and subtract and terrifyingly divide by. For most of human history, people simply did not see why you would ever need a symbol for an amount you do not have.

[01:13:25]You would not write down a shopping list that said, "Buy zero apples." The whole point of counting was to count the things that were there. The first crack in that wall came once again from the Babylonians.

[01:13:42]And it came not because they wanted a zero, but because their number system forced one on them. Remember that they wrote numbers in columns where the position of a mark told you how big it was. The same way that in our system, the one in 19 means something completely different from the one in 91. And once you do that, you hit a problem. How do you show an empty column? How do you tell the difference between a number with something in the middle column and a number with nothing in the middle column? At first, the Babylonians just left a blank space. But a blank space is a terrible way to record anything because you can never be sure how many blanks are there. and a smudge could change the entire value. So eventually they invented a little symbol, a pair of slanted marks that they would drop into an empty column just to say nothing goes here. Keep counting. And that is a placeholder and it is genuinely useful.

[01:15:09]But notice what it is not. It is not a number. The Babylonians would never have written that symbol by itself as an answer. It was not something you could add or subtract. It was punctuation.

[01:15:26]It was a little sign that meant do not get confused. This column is empty. They had a mark for nothing. But they did not have the number zero, and they never once used it at the end of a number.

[01:15:43]which meant they still could not tell the difference between a number and that same number 60 times bigger except from context. They were halfway there and they stopped. And then on the complete opposite side of the planet, with absolutely no contact and no possible way of borrowing the idea, another people invented a zero entirely from scratch. And this one might be the most impressive of all. Over in Central America, the civilizations of Meso America, including the Mech and later, most famously the Maya, built their own number system, and theirs was based on 20. The fingers and toes count rather than 10 or 60. And to make their positional system work, and especially to run the spectacularly complicated calendars they were obsessed with, they needed a way to mark an empty place. So they invented a symbol for it, often drawn as a kind of shell shape, and they used it as a placeholder in their counts. The oldest zero anyone has found in the Americas is carved into a stone monument and it dates to about 2,000 years ago which makes it older than a lot of the zeros people usually credit.

[01:17:23]The Maya were in many ways more comfortable with the idea than the old world was. Building it right into the machinery of a calendar so precise it could track cycles of time stretching over thousands of years. And yet even they like the Babylonians mostly used it as a placeholder, a way to keep the columns straight rather than as a true number you would do arithmetic with. And because the Maya were cut off from the rest of the world, and because their civilization was later devastated, and their books burned by people who could not read them and did not care to, their beautiful, independent, zero, never spread anywhere. It was invented, used brilliantly and then lost, touching nothing outside its own world. It is in a way the loneliest invention in this entire story. So if the placeholder zero appeared in both Babylon and the Americas but went no further as a true number, where does the zero you actually use come from? The one in your phone, the one in your bank balance, the one that makes the whole modern world run.

[01:19:01]It comes from India. And the leap that happened in India is the one nobody else managed. They took the little placeholder for an empty column and they let it grow up into a full number. A citizen with the same rights as one and two and three. Something you could add, subtract, multiply, and reason about all on its own. The word they used for it tells you how they got there. In the ancient Indian language of Sanskrit, the word was shunya. And shunya meant empty or void.

[01:19:48]A word that came loaded with deep meaning in Indian philosophy and religion where the idea of emptiness and nothingness was something thinkers had spent centuries comfortably turning over in their minds.

[01:20:09]So while a Greek merchant might have found the idea of a number for nothing slightly absurd, an Indian scholar already lived in a world where nothingness was a respectable thing to think hard about.

[01:20:29]The culture had in a sense already prepared the ground and India's roots in math ran deep long before any of this which is part of why the breakthrough happened there. Going back many centuries before the common era, there was a body of writings called the Sulba Sutras and the word Sulba meant a cord or a rope used for measuring. So the name means something close to the rules of the chord which was simply their word for geometry. These were not abstract math books. They were instruction manuals for priests telling them exactly how to build the altars used in religious rituals.

[01:21:26]Because the belief was that an altar built to the wrong shape or the wrong size could ruin the whole ceremony. So the measurements had to be perfect. And to get them perfect, the writers worked out a remarkable amount of geometry.

[01:21:49]They state in their own words the same rule about right triangles that the rest of the world credits to Pythagoras, saying that the rope stretched along the diagonal of a rectangle makes an area equal to the two sides put together.

[01:22:12]They list off sets of whole numbers that form right angles, the same kind of triples on that Babylonian tablet. And they calculate the square root of two, the diagonal of a square correct to five decimal places. All so that a priest could lay out a fire altar that would please the gods. The deep habit of careful exact measurement was already centuries old in India before zero ever showed up. The oldest physical evidence we have of this Indian zero written as a small dot comes from an old manuscript written on 70 leaves of birch bark found by a farmer digging in a field in 1881 near a village called Bakshali in what is now Pakistan.

[01:23:12]It is full of practical math, the kind of thing traveling merchants would have used. And scattered all through it are little dots standing in for zero. For a long time, people thought this manuscript was only about a thousand years old. But a few years ago, some of the birch bark was carbon dated, and parts of it came back far older than anyone expected. possibly going back around 17 or 18,800 years, which would push the written history of zero back by centuries. The dating is argued about like everything else in this story, but the dots are there on the bark doing the job of zero very long ago. And the people writing this kind of math were not lightweights.

[01:24:07]Around the year 499, an Indian astronomer and mathematician named Aryabata, still only in his early 20s when he wrote his great work, was using a place value system to do serious astronomy, working out the movements of the planets, and calculating the value of pi as 3.1416.

[01:24:37]six which is correct to four decimal places and far better than the old Egyptian guess. He had a way of describing how each place in a number was 10 times the value of the place to its right which is the exact logic of the system you use today. So the machinery of place value was up and running in India in skilled hands doing real work. What was still missing was somebody to take the strange empty space at the heart of that machinery and give it a full set of rules to write down the law of nothing. But a dot in a merchant's notebook is still mostly a placeholder.

[01:25:35]The moment zero truly became a number.

[01:25:39]The moment somebody sat down and wrote out the rules for how this strange new thing behaves came in the year 628 with an Indian astronomer and mathematician named Brahma Gupta. He wrote a book whose title translates to something like the opening of the universe. And in it he did something nobody had ever done before. He treated zero as a number in its own right. And he wrote down the rules for working with it. He said that when you add zero to a number, the number does not change. And when you subtract zero, it does not change either. And when you multiply any number by zero, you get zero. which sounds obvious to you because you were drilled on it as a small child, but somebody had to be the first human being to write it down as a law. And that somebody was Brahagupta and he went further into territory that is genuinely difficult. He worked with negative numbers, numbers below zero, which were every bit as suspicious to most ancient people as zero itself.

[01:27:13]And he described them in a way that any merchant would instantly understand by talking about fortunes and debts. A positive number was a fortune, money you had. A negative number was a debt, money you owed. And so he explained, "A debt taken away from nothing becomes a fortune. And a fortune taken away from nothing becomes a debt." Which is exactly how a minus sign flips a number sign today. Written out in the language of being broke or being rich. He even reached for the one question about zero that is genuinely dangerous. The question of what happens when you divide a number by zero. And here for once he did not quite get it right because nobody could because dividing by zero turns out to be a question that breaks arithmetic itself in a way that would not be fully sorted out for another thousand years and more. But he asked it. He stared straight at the hardest part of the idea he had just created and he asked the question which is its own kind of greatness. The oldest zero anyone has found carved permanently into stone in India as opposed to written on bark or paper comes from a temple in a place called Gualor dated to the year 876 and what it records is wonderfully ordinary. It is an inscription about a garden noting that the garden measured a certain number of units one way and another number the other way and that a number of flower garlands were to be given to the temple each day. And in those measurements written plainly is a little circle our zero the round symbol you would recognize instantly.

[01:29:38]A piece of municipal recordkeeping about a flower garden accidentally preserving one of the most important symbols humanity ever came up with. So now the most powerful number system in the world exists sitting in India. The system we still use today. the one with the nine digits and the zero where the position of each digit tells you its size and a zero holds any empty place. And the obvious question is how on earth did it get from there to everywhere else? How did it get to you? And the answer to that runs straight through one of the great centers of learning the world has ever produced. In a city that for a few centuries was the place where the smartest people on the planet gathered to translate, argue, and build on everything that had come before. That city was Baghdad. In the 8th and 9th centuries, the rulers of the Islamic world, the khifs, set up an institution in Baghdad that later writers came to call the house of wisdom. And whether or not it was one single grand building or more of a loose movement of scholars and translators, the effect was the same. They went out and gathered up the written knowledge of every civilization they could reach, the geometry of the Greeks, including Uklid, the astronomy of the Persians, and importantly for us, the mathematics of India, including that brand new system of nine digits and a zero.

[01:31:43]Scholars in Baghdad translated all of it into Arabic, studied it, corrected it, and pushed it forward. And for a few hundred years, this was simply where the action was. While much of Europe had largely lost touch with the old Greek learning, it was being carefully kept alive and improved in Baghdad. And the single most important person for our purposes who worked there was a scholar named Muhammad Ibnam Musa al-Quarismi who lived roughly between the years 780 and 850.

[01:32:30]He came, as his name tells us, from a region called Quarism, which is out in Central Asia in what is now Pakistan.

[01:32:42]And he ended up in Baghdad doing math that would without exaggeration shape the language you use to talk about math to this very day. Because two of the most common words in the entire subject come directly from this one man and most people who use them every day have no idea. The first word comes from the title of his most famous book written around the year 830.

[01:33:15]A book about how to solve equations.

[01:33:18]How to find an unknown quantity when you only know certain things about it. The book is about the methods you use to do that. And two of those methods had names.

[01:33:34]One of them in Arabic was alab which meant something like restoring or completing.

[01:33:45]The idea of moving things around in an equation to balance it out, taking a piece from one side and restoring it to the other. And when his work was eventually carried into Europe and translated into Latin, that word algebra came along for the ride and it slowly turned into the word algebra. Every kid who has ever groaned about algebra homework is groaning about the word that is just a slightly worn down version of an Arabic term for balancing an equation coined in Baghdad 1,200 years ago. And the second word is even better because it comes from the man's own name. When Europeans translated his other great book, the one explaining that Indian system of numbers and how to calculate with it, they took his name Alarismi and they tried to write it in Latin and it came out as algorithmi and the stepbystep calculating procedures that his book taught, the methods for adding and subtracting and multiplying using those new numbers came to be called by a version of his latinized name. Which is why today the set of stepbystep instructions that runs a computer program, the thing that decides what shows up in your feed and how your phone finds a route home is called an algorithm. It is very literally named after a 9th century mathematician from Usuzbekistan.

[01:35:40]Every time anyone says the word algorithm, they are saying his name worn smooth by 1200 years of being passed from mouth to mouth across half the world. So Alqarismi takes the Indian number system with its zero and its nine digits and he writes the book that explains to the wider world how to actually use it. And from Baghdad it begins to spread. Carry it by traders and scholars along the roots that connected the Middle East to everywhere else. It moves through the Islamic world across North Africa and up into Muslim ruled Spain, getting closer and closer to the rest of Europe. And there it more or less stalls for a long time, bumping up against a continent that frankly was not interested because Europe already had a way of writing numbers. They had the Roman numerals, the system of letters where I is one and V is five and X is 10 and so on. The thing you still see on old clock faces and at the end of movie credits. And the Romans, mighty as they were, had never bothered with a zero at all because their system did not need one. It was not a positional system. The symbol for 10 was just its own letter X. So there was never an empty column to mark and therefore never any reason to invent a number for nothing. And Europe had been getting along with these numerals for over a thousand years. They were good enough for carving on monuments and writing down dates. The trouble is that Roman numerals are genuinely painfully bad at arithmetic.

[01:38:02]They are fine for writing a number down once it is finished. But try to actually add or multiply with them and you are quickly in misery. There is no neat way to line them up in columns and carry the one. So for any serious calculation, Europeans for centuries did not really calculate with the written numbers at all. They wrote the numbers in Roman numerals for the record. But to do the actual sums, they used a separate physical device, a counting board or an abacus, sliding beads or little markers along lines to do the arithmetic by hand. And then they wrote the answer down in Roman numerals afterward. The writing and the calculating were two completely separate jobs done with two completely separate tools. And this brings us at last to the man who finally dragged the new numbers into Europe. And his name was Leonardo of Pisa, though the world remembers him by a nickname he was given much later, Fibonacci.

[01:39:29]He was born around the year 1170 in the Italian city of Pisa, the one with the famous leaning tower. And the reason he matters is partly an accident of his father's job. His father was a merchant and a customs official. And he was posted to a trading port in North Africa in a town in what is now Algeria. And he brought his young son along and had him taught mathematics there. And in North Africa, the boy learned the Indian numbers, the Alquarismi system, the nine digits and the zero from the people who had been using them for centuries.

[01:40:17]He grew up traveled all around the Mediterranean trading and learning and he became convinced that this system was so much better than what Europe was using that it was almost embarrassing.

[01:40:29]And so in the year 122 he wrote a book to explain it to his fellow Europeans and he called it the liber abachi which means roughly the book of calculation and the opening of that book contains one of the great understated sentences in the history of the subject. He introduces the nine Indian figures the digits 1 through nine and then he writes that along with these nine figures and with this sign and here he writes a zero. You can write any number at all and he gives the zero a name a Latin word he borrowed and bent out of the Arabic. The Arabs had translated the Sanskrit Shunya, the void, into their own word for empty, which was cifir. And Fibonacci turned cifir into the Latin zephi.

[01:41:31]And that word zephirum traveled into Italian and got shortened in the dialect of Venice into zero. That is where the word comes from. It is shunya. The Sanskrit void passed into Arabic as cifher, passed into Latin as Sepharum, worn down by Italian tongues into zero.

[01:41:57]And that same Arabic word cifher, took a second path into European languages and became the word cipher, which is why to this day a cipher can mean both a zero and a secret code. The number for nothing and the idea of a hidden message share a single ancestor. Now the liberabachi is a big serious practical book full of problems about buying and selling and exchanging currencies and splitting profits. The real working math a merchant would actually need. But tucked away inside it as just one puzzle among hundreds is a little problem about rabbits that Fibonacci almost certainly thought was one of the least important things in the entire book and which is now by far the thing he is most famous for to the point where most people who know his name know nothing else about him. The puzzle goes like this. Suppose you put a single pair of rabbits in a field. And suppose that every month each grown-up pair produces a new pair. And that a new pair takes one month to grow up before they can start breeding themselves. How many pairs of rabbits will you have after a year? And when you work it through month by month, the number of pairs goes one, then one, then two, then three, then five, then 8, then 13, then 21, then 34, then 55, then 89, and then 144.

[01:43:55]And if you stare at that list for a second, you will spot the trick, which is that each number is just the two numbers before it added together. 1 and 1 make two. 1 and two make three. Two and three make five. Three and five make eight. And it keeps going like that forever. That is the Fibonacci sequence.

[01:44:24]And it turns out to show up in the strangest places in nature, in the spiral of a sunflower seeds, in the arrangement of leaves on a stem, in the curl of a pine cone. And it is tied to a famous ratio that artists and architects have chased for centuries.

[01:44:46]But here is the part that says everything about how history works.

[01:44:52]Fibonacci did not think this was a big deal. He wrote it down as a minor curiosity, a fun little puzzle, and then moved straight on to the next problem without making any fuss about it. And it sat there more or less ignored for over 600 years until a French mathematician in the 1800s named Edward Lucas got interested in it.

[01:45:22]studied it properly and stuck Fibonacci's name on it. So the one thing the man is now famous for is something he barely noticed he had written named after him by somebody else hundreds of years after he died. The actual gift he gave the world, the number system that the whole book was built to teach, is the thing almost nobody thinks to thank him for. Because you would think, wouldn't you, that once Europe saw this new system with its easy arithmetic and its powerful zero, everybody would have switched over immediately and thanked Fibonacci on the way. That is not what happened.

[01:46:20]What happened instead is one of the best examples in all of history of people stubbornly refusing a better tool because it made them uncomfortable.

[01:46:34]And the resistance was strongest in exactly the place you would least expect among the bankers and merchants of Italy. the very people who had the most to gain. In the year 1299, the city of Florence, one of the great commercial powers of Europe, passed a rule that actually banned its merchants from using the new numbers in their official account books. Ban them. The most useful calculating tool ever to reach Europe. made illegal by the people who did business for a living. And there were a couple of reasons and they are weirdly understandable once you hear them. The first was fraud. The new digits people complained were just too easy to fake and to alter. A zero could be turned into a six or a nine with a single stroke of the pen. A one could become a seven. A figure in a ledger could be secretly changed and nobody would be able to tell. The old Roman numerals written out in letters were clumsy, but they were much harder to tamper with after the fact, so the ban was in part an anti-fraud measure by people who did not trust how slippery the new symbols were. And the second reason was zero itself. People simply did not trust it. It was the strangest part of the whole package. A symbol that was not a number that meant nothing and yet that could be stuck onto the end of a number and suddenly make it 10 times bigger. That felt to a lot of people like a trick, like something that should not be allowed, a piece of slight of hand hiding in the accounts.

[01:48:46]Negative numbers got the same treatment, the same suspicion, the idea of a quantity less than nothing striking many people as plainly absurd.

[01:49:00]And a few decades later in 1348, the University of Padua ordered that the prices of its books for sale be written out in plain clear letters and not in these new figures at all. And so for a long stretch of time, Europe lived in a strange split. The merchants who wanted the power of the new arithmetic but were not allowed to use it openly are said to have sometimes kept two sets of books.

[01:49:37]One set written in the old Roman numerals was the official record. The one you would show to the authorities, all proper and legal, and the other set written in the New Indian numbers was the one they actually used to do the math, kept off to one side because it was simply better. The forbidden numbers were too useful to give up, so people just hid them. There was even a name for the two sides of this long argument. The people who clung to the old counting boards and the abacus were called the abbisonists.

[01:50:20]And the people who pushed the new written calculation, the methods that traced back to Alqarismi were called the algorists after his name again. And for a couple of centuries, the abbisonists and the algorists were in a low-key way at war over how Europe ought to count. The Algorists won. Of course, they were always going to win because they were right. because the mat was simply faster and stronger and no amount of suspicion or banning could hold that back forever. By the time printed books arrived and spread across Europe, the new numbers were unstoppable and the counting boards and the Roman numerals slowly faded into decoration.

[01:51:23]But it took a long time and a lot of grumbling and more than one outright ban for Europe to accept the gift it had been handed. The thing you find so obvious that you cannot even imagine numbers without it was not so long ago treated as a fraud risk and made illegal in one of the richest cities on earth.

[01:51:50]So that is the journey. And when you lay it all out end to end, the thing that should strike you is that there is no single inventor anywhere in it. No one moment where math gets switched on.

[01:52:07]There is a person in southern Africa cutting notches into a baboon bone 40,000 years ago.

[01:52:18]There is a farmer in the ancient near east pressing clay tokens into a bowl to keep an honest record of his grain.

[01:52:31]There is a Babylonian student getting the square root of two right to six decimal places and handing in the homework. There is a Greek standing in the shadow of a pyramid and refusing to believe a thing until he can prove it.

[01:52:53]There is somebody in a secret society drowning or maybe not drowning over a number that should not exist. There is an Indian astronomer in the year 628 writing down for the first time that a debt subtracted from nothing is a fortune. There is a scholar in Baghdad whose name became the word for the instructions that run every computer on the planet. And there is a merchant son from Pisa who learned to count in North Africa writing a book to convince a continent that did not want to listen.

[01:53:41]Each of them solving their own small usually very practical problem. None of them building the cathedral.

[01:53:51]All of them laying a single stone and the stones somehow ending up over 40,000 years in the shape of all of modern mathematics.

[01:54:04]And there is one last thread worth pulling because it ties the whole thing into a knot so neat it almost feels planned though of course it was not.

[01:54:19]Think back to the very beginning to people counting before tokens, before tablets, before any of it. Using the simplest tool there is, small stones, pebbles, moved one at a time to keep track of a count. One pebble, one sheep.

[01:54:40]The ancient Romans did this constantly, sliding little stones along the lines of their counting boards to do their sums.

[01:54:50]And the Latin word for one of those little stones, those little counting pebbles, was calculus, which means that when you pick up a calculator or when a student sweats through a class called calculus or anytime anybody anywhere calculates anything at all, the word in their mouth is the Roman word for a pebble.

[01:55:20]The most advanced computing device you own is name at the root. After a small stone that a person 40 centuries ago pushed across a board to count their goats, we never really left the pebble behind.

[01:55:39]We just got better and better at moving it. And the strangest piece of all, the one to fall asleep on, is the number for nothing. The idea that so many brilliant civilizations either never had or had and feared or had and banned. The void the Indians called Shuna, the empty mark the Babylonians dropped into an empty column. The lonely shell-shaped zero.

[01:56:14]The Maya carved and then lost. The symbol that bankers in Florence once outlawed as a fraud. That nothing. That little circle that means there is nothing here turned out to be the most powerful something humans ever came up with. Every computer that has ever runs on nothing and something on zero and one. Two symbols standing in for off and on and an unbroken line of zeros and ones is the language underneath every word and image and number on every screen in the world. The entire digital age. The whole humming electronic world you live inside is built at the very bottom on a number for nothing that took the human race tens of thousands of years to be brave enough to write down nobody invented math. We just kept counting one pebble at a time until the pile got big enough to hold up the