N is for Numbers

Blackadder: Right Baldrick, let’s try again shall we? This is called adding. If I have two beans, and then I add two more beans, what do I have?
Baldrick: Some beans.
Blackadder: Yes… and no. Now try again. One, two, three, four. So how many are there?
Baldrick: Three.
Blackadder: What?
Baldrick: …and that one.
Blackadder: Three and that one.  Let’s try again shall we? I have two beans, then I add two more beans. What does that make?
Baldrick: A very small casserole.

Blackadder episode, “Head”

Yes, dear readers, there will be math today. I know you can do it. I know you can run intellectual circles around Baldrick.

The definitive work on this topic is The History of Mathematics by Merzbach and Boyer, which is already in a Third Edition, even though not much has changed for the Egyptians and Sumerians, who used what we’d consider basic counting systems to construct giant pyramids. Mainly, Merzbach and Boyer have added a “Logic and Computing” and “Recent Trends” chapters at the end. Remember when Computer Science was about logic and not Belarussians creating algorithms to stuff your social media full of outrage porn? How quaint!

Anyway, I digress. Today, I want to describe how different cultures approached numbers–not specifically whether they were smart enough to figure out Fermat’s theorem or Poincare’s theory–but how we as humans figured out what Baldrick apparently couldn’t. Thinking about math is hard, but we’ll also see that there are harder and easier ways to do it.

Three questions I want to address:

1. What does math mean? How do numbers work?
2. How did different cultures approach using numbers?
3. Why do humans need math? Why did they invent and re-invent it, expanding it over time?

By the way, Tony Robinson, the brilliant actor who played Baldrick in the Blackadder series, has been knighted, in part because he hosted a two-decade series called Time Team all about archaeological digs. I don’t know if he learned to count to five, but he’s now an expert on archaeology. I wonder if he ever did a program on tally sticks.

Tally Sticks

Suppose you have language but no writing. You can hunt and gather, but your little band isn’t big enough to plant and harvest, and food and game are plentiful anyway. You wear a little jewelry, shoot arrows at rabbits, and sit around the campfire. You’re in Paleolithic Africa, which is relatively warm, so you don’t need to store things for winter. What is there to count?

Tools for one thing. You have a lot of different kinds of tools. Also, days. We know that by 10,000 ya, people developed calendars, even though we don’t have lots of examples of them. Keeping track of time, either to note the phases of the moon or the passage of the sun, could have helped them predict or plan for the seasons.

Whatever the reason, we know that early humans inventing counting. It sounds bland, but it’s a significant step forward in cognition.. They aren’t exactly sure when, since bones with notched markings have surfaced from 40,000 ya to 20,000 to 10,000–all times which crossed the evolution timeline from Homo erectus into Homo sapiens. The 40,000 year old one, from northern South Africa, is the Lebombo bone; the 20,000 year old one, from the Democratic Republic of the Congo, is the Ishango bone. These sticks have grouped notches lined up in columns on multiple sides of the bone. Individually a notched stick may not seem impressive, but collectively they show that humans wanted to keep track of their stuff and discovered a special way to do it.

Tax Records

Still, the idea of how many was not yet a number. Numbers had to be created, and the growing population of Sumeria in 7500 BCE was the first source. We saw (letter “C”) that population centers were forming around crops and settlements; Turkey built stone towers and carved pillars with what appears to be public tracking of the calendar. Growing cities needed to be fed and coordinated.

The earliest “mathematicians” were the scribes who took inventory. In particular, they would send someone out to the farms to take a count of the sheep, cows, or grain, and that person would draw pictures on clay balls with a stick, then cover those balls with a larger, mud ball with a hollow center for transport back to the central city. Back at the scribe’s desk, the ball would be cracked open and the contents written on clay tablets.

The development of this bullae process was a feature of a central government. An individual farmer might still use tally sticks or even their own little cones with pictures to track their harvest, livestock, or acreage. But the administration collected a share in taxes because it also shared resources among its population. It “taxed” the farmers on a portion of the crops in order to feed the workers–scribes, soldiers, and temple priests. Slaves and the poor, namely widows and orphans, were also fed from the central stash.

The state gets bigger and a veritable army of workers–bureaucrats–are created to do the accounting. You see this structure replicated in multiple civilizations: Sumeria, Egypt, Rome, Indus Valley, China, and Mesoamerica. Most often the expertise in writing and accounting was under the supervision of the priests, and the centralized grain stores were in the temples. Those tax records at first were written on the bullae, but pictures of sheep and grain were changed to symbols, simple wedged marks on the tablets. Record-keeping drove the development of numbers.

In Sumeria, the leap from marks on little clay balls to a Base 60 counting system with positional notation–more on that in a second–was not immediate. Before the numerals 1 through 9 were standardized, there were unique pictures for numbers of sheep, numbers of grain and so on. Instead of a one, they had one sheep; one cow; one wheat; one bean; and so forth. Imagine learning how to add but only according to WHAT you were adding. It took centuries for them to “invent” the idea of a number separate from the THING being counted.

There’s an early Sumerian joke: May my yield be small so I can go home early. Kind of like saying, I hope my paycheck isn’t too big so that I don’t have to pay too much in tax. Hah ha! Even so, maybe farmers often preferred those small yields, too, just to avoid the headache of counting all those different ways.

Sumerian Tables and the Babylonian Theorem

Eventually, the Sumerians developed simple numerals, that abstract idea that one sheep and one cow could both be symbolized by 1. They created a way to show positional notation, which means they could put another mark in front front of the numerals to distinguish 1 from 11, 6 from 56, etc. They did this up to the number 60, then started over. Their number 61 was similar to our number 101. That’s what it means to use Base 60.

Why 60? As I commented when discussing the Calendar, the Sumerians graduated from crop yields and taxes to astronomy, geometry, and calendars. Sixty was used for the degrees in a circle, as well as minutes and seconds because 60 is a handy number, given that is divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15 and so on.

Base 60 helped the Sumerians (and Babylonians) get super good at fractions. They got so good at them, that they could do quadratic equations, cubics, and all manner of geometry. They especially excelled at numbers that we would consider complicated (if you’re math-proficient, think rational, irrational, and negative numbers). Square roots were no problem. In this example for the square root of two, which is 1.4142135624 and a bunch more, they would calculate it using components of base 60/. Here, the semicolon stands in as the cuneiform “decimal point.”

√2= 1;24,51,10 = 1 + 24/(60) + 51/(60²) + 10/(60³) ~= 0.0000084 off.

The way the Sumerians routinely handled complicated calculations like these was with standardized tables of numbers. Scribes would need to learn how to use these tables of fractions, just like the logarithm tables from my Algebra II class. Before calculators (yes I’m that old), our math books had logarithms, sines, and other tables in the back, just like the Sumerian scribes.

The Babylonians, who came a little after the Sumerians, also had a special table for something more familiar to us. This bit of excavated clay, known as Plimpton 322, has tables of triplets for the dimensions of a right triangle: A² + B² = C²

We call this formula the Pythagorean theorem, but think about it. The Sumerians, Chinese, Indians, Mesoamericans, and Egyptians all discovered the mathematical properties and ratios of this triangle independently. The Babylonian scribes and engineers used these tablets as far back as 1800 BCE, more than a thousand years before the Greek guy Pythagoras was born. It could just as well be called the Babylonian theorem.

Find Heap Among the Fractions

The Egyptians also had extensive proficiency in geometry and mathematics. Sumerians were building 90-meter ziggurats; the Great Pyramid of Giza is 150 meters tall. These ancient engineers needed the math as much for building robust structures as for tracking the grain storage in the temple.

The Egyptians also used tables and also had expertise with fractions, even though their numbering system was the old familiar Base 10. They loved the fractions one over any number (1/n)– like 1/3, 1/5, 1/57–and would convert all their problems so that they ended up with a combination of 1/n fractions. If they had a fraction m/n, meaning 4/3, 2/5, or 56/57, they change it. Instead of thinking of 3/5 as a fraction as we do today, they preferred to see it as the combination of 1/3 + 1/5 + 1/15, which just makes my head hurt. It was easier for them to them to figure out a number based on this combination of 1/n, rather than multiplying 1/n times some number.

Finding an unknown x was also something these Mediterranean cultures would do, although they didn’t do it with algebra as we know it. Suppose they wanted to find x, which they called “heap.” For example:

Find the value of heap if heap and 1/7 of heap is 19

First of all, the handy symbols for plus, minus, divide, and so on, weren’t invented until the Late Middle Ages, so all of their problems were words only. We’d solve that as x + 1/7x=19. 8/7x=19 and x=(19*7/8)=16 and 5/8 (16.625). It’s not a particularly round number.

But the Egyptians didn’t have solving equations that way in their toolbox. Instead, they used something called the “rule of false.” They would substitute in a number, like 7, which they knew would not work, then ratio up or down until they got the right answer. If you stuck in 7, then the “heap” side 7 + 7/7 would be 8, which is much less than 19. If you try 14, then the “heap” side 14 + 14/7 = 16 is still too small, so the answer must be even bigger than 14. Using proportions up and down to get 16.625 would seem cumbersome to us, but remember the Egyptians could handle a fraction like 16 + 1/2 + 1/8 faster than we can text LOL.

Complex Notation

If you’re thinking that it’s a good thing that the Greeks and Romans made things so simpler, think again. It’s always amazed me that the Greeks get so much credit for inventing math and geometry, when (a) they didn’t invent all of it and (b) they had a bizarre approach to writing numerals. Basically, they didn’t have unique numbers. The Greek alphabet — α β θ λ π — was used for their numbering system. In order to write a number, say 5742, they would put a mark at the front showing it was a number, then use Greek letters: ͵Εψμβ.

Think that’s wacky? The Romans, at least used fewer letters to represent their numbering system. Roman numerals are built around multiples of five, logically, since we have five fingers and toes. 1, 5, 10, 50, 100 became 1, V, X, L, C etc. I know it’s a party trick or a Jeopardy question to translate Arabic numerals into Roman numerals. But stop to consider what happens when you have CLII times XXIII, or 152 times 23. Are you ready? This is how they multiplied:

Don’t even ask me how they divided.

Or is it the Gou Gu Theorem?

The approaches to math that Sumeria and Egypt took were replicated my other ancient civilizations. Others also solved advanced equations and created complex formulas, most with Base 10 although the Mayans used Base 20. During the Shang dynasty, the Chinese developed a sophisticated lunisolar calendar whose astronomy relied sophisticated calculations; they needed good mathematicians. Like the Babylonians and Egyptians, Chinese math students had word problems, although these had that unique Chinese flavor which seems more eloquent than the “heap” problems. Consider this example from a Chinese math book in a section on the right triangle problems.

The Olmecs and Mayans, on the other side of the world, having had no contact with the Babylonians, Greeks, or Chinese, nevertheless, also created their own numbering system. By 31 BCE, the Olmecs had calculated the circumference of the earth, knew how to apply the right triangle theorem, and even had a zero–typically represented by a seed or a conch shell. In developing the notion of zero, they were ahead of Western civilization by a considerable amount of time.

Solving for Nothing

The Europeans stumbled across zero in the late Middle Ages, supposedly acquiring it when Leonardo Fibonacci went to Algeria. North Africa in the late 12th century was Islamic and the Muslims were light years ahead of Europe by then in science and mathematics. Fibonacci shared the Hindu-Arabic numbering system, and Europe was never the same.

The Muslims got the numbering system from India. The Hindus may, in turn, have been influenced by the Chinese, neighbors, trading partners, and partners in the spread of Buddhism. The physical Shang numerals, Brahmi numerals, and Indian numerals all seemed to imprint upon the West Arabic numbering system. Ironically, the numerals used today in the Middle East follow the East Arabic system, while we use what is called West Arabic. Perhaps ours should be called West Arabic-Hindu-Shang numerals.

We’ve come a long way since the ancients looked up fractions in tables and calculated using heap or even tally sticks. Much of what we’ve learned mathematically is pretty recent, aided by computing power. whose capabilities are to those Sumerian tablets of right angle triplets what those triplets were to the tally sticks. Leaps and bounds.

The largest prime number ever discovered, as of April 2025, is 2^136,279,841−1. This number has 41,024,320 digits in it (the previous record only had some 25 million digits). The number is also designated as M136279841, a Mersenne prime, because there is an entirely sub-class of primes and because not only do mathematicians need to find a number, but they need to describe using it several other numbers. And they needed thousands of people with dozens of computers to get there. Imagine if they had to use Babylonian tables!

Heap of trouble is what I’d call it.