How Modern Numbers Were Invented
Notation matters!
In this post, we’ll see how the modern number system elegantly solves the problem (discussed in this post ) of how to write numbers as big as we please and how to calculate using pen and paper. The modern way we write numbers and do arithmetic is one of the most important inventions in the history of ideas. Indian mathematicians around the 7th century AD saw that it was possible to write numbers using only ten symbols, a creative solution to a problem that had vexed mathematicians for thousands of years. We often think that mathematics is discovered, but how we chose to write down mathematical symbols was invented. There is no rule that says we must use ten symbols for numbers. Indian mathematicians invented that solution. When Arab mathematicians adopted the Indian numbers, they realized that they could easily write big numbers and that they could do arithmetic with pen and paper. That simple insight eventually led to an explosion of mathematical progress.
Modern numbers originated from the Indian number system
The Indian numerals, developed in the eight century A.D, were founded on two simple, but incredibly powerful ideas. First, you only need to group by ten using ten fundamental symbols, including the number zero. And second, you don’t need any symbols beyond these ten, since you can indicate other groupings such as the one hundred grouping or one thousand grouping by the position of each digit. So, you don’t need an L for 50 or a C for 100 as you would have in the Roman system. You also don’t need cuneiform symbols for one and ten as the Babylonians used for sexagesimals.
The Indian numbers, 1-9 and zero, originally looked like this.
Because these digits had to be hand copied into scholarly manuscripts, they would naturally change a little bit over time. Eventually, as they were copied over and over, they changed a bit after a few hundred years, with the changes indicated in red:
The horizontal line representing the number 1 eventually became vertical, probably because it saved space to write it that way. The other changes are very slight, and probably came about because it’s easier to make the symbol in one smooth motion without lifting the pen. So, the number for 2 likely came from drawing the top horizontal line from left to right, then moving the pen down diagonally to the left, and then making the bottom horizontal line. The modern three likely came about the same way: the three horizontal lines were drawn without lifting the pen. The numbers 4 and 5 are variations. Five came about by adding the curly cue to the bottom of the original 5, possibly as a flourish. The modern eight and nine seem to have developed from writing the number in one continuous circular motion.
During the Arab conquests, the Indian numerals were imported into Islamic science and mathematics and were used extensively by scholars. Arab mathematicians also developed the techniques that schoolchildren use today to do arithmetic by hand using paper and pen. But the Indian numbers never caught on with the general public in the Islamic world of the Middle Ages.
Europe eventually adopted the Indian-Arab number system
At the turn of the thirteenth century, the Indian number system was introduced into Italy by Leonardo of Pisa, also known as Fibonacci. Fibonacci was an Italian merchant and scholar who traveled extensively for business and for mathematical research. While visiting what is today Algeria, Fibonacci learned of the Indian numbering system and the arithmetic Arab mathematicians had developed. Fibonacci thought the Indian-Arab number system so superior to the Roman system and arithmetic with pen and paper so much better than the abacus that he resolved to replace the Roman numbers with the Indian-Arab numbers.
In 1202 AD, Fibonacci published a book, Liber Abaci, Latin for “The Book of Calculation,” although what it means more precisely is the book of calculation without the abacus. Fibonacci’s goal was to write a book that, while as logically rigorous as Euclid’s Elements, would also be an accessible and practical training manual for merchants and scholars. Fibonacci understood the value of marketing, even in mathematics, and so he filled the book with examples from the worlds of finance and trading.
Fibonacci began the first chapter with “The nine Indian figures are 9,8,7,6,5,4,3,2,1. With these 9 figures and sign 0, which the Arabs call zephyr, any number whatsoever is written, as is demonstrated below.” Fibonacci then explained how to do addition, subtraction, multiplication and division by hand, without the abacus, much as we learn to do it in elementary school today. The genius of the Indian system is that the positions of the numbers indicates whether they should be grouped by ones, tens, hundreds, thousands, or something bigger. Arithmetic now became possible without an abacus because we add, subtract, multiply, or divide numbers one position at a time. If we get a result that is too big or too small in the position we are focusing on, we carry the result over to the next position or borrow from the next position and then keep going. Because Arabic is read right to left, Arab arithmetic was done right to left, a custom we still maintain today.
For example, consider adding 45 and 76. 45 in the Indian-Arab notation means 4 tens plus 5 ones and 76 means 7 tens plus 6 ones. In the number 76, 6 is in the ones place and 7 is in the tens place. In the number 45, 5 is in the ones place and four is in the tens place. To add 45 and 76, we first add the 5 and 6 in the ones place. Moving right to left, we get 11, so we keep the 1 in the ones place and carry over ten, which means we carry over 1 ten into the tens place. In the second stage, we add the 1 ten carried over plus the 4 tens plus the 7 tens to get 12 tens. Then, we keep two tens from 12 in the tens place and carry over 1 hundred to the hundreds place. Overall, we have finished with 1 in the hundreds place, 2 in the tens place, and 1 in the ones place. Thus, 45 + 76 = 121. This simple calculation, that’s so easy using pen and paper, is very difficult to perform using Roman numbers, since there are no positions in the numbers indicating how counting should be grouped.
The Indian-Arab numbers introduced the concept of a ‘base’
Nowadays, we call this numbering system a base ten numbering system. The reason we use the term ‘base’ is that the Indian-Arab system is very flexible. It works with any grouping or ‘base,’ which is another word for number grouping. For example, in a base 2 system, there are two numbers, 0 and 1, and we indicate the value of the number by the positions of 0 or 1 in the number. In a base 2 number, reading from right to left, we have a ones place, a twos place, a fours place, an eights place, and so on. Each place is double the previous place. Base 2 numbers are also called binary numbers.
Base 2 or binary numbers
To illustrate how binary numbers work, let’s write 45 in base 2. It would be 101101 or 1 in the thirty-twos place plus 0 in the sixteens place plus 1 in the eights plus 1 in the fours place plus 0 in the twos place plus 1 one in the ones place, or 32 + 8 + 4 +1 = 45.
Arithmetic can be done in binary the same way as in base ten. Binary numbers are essential for computer arithmetic since numbers and calculations on computers are done in base 2. The beauty of Indian-Arab numbers is that we can use any base that is convenient. Computer scientists sometimes use base 16, which are called hexadecimal numbers. They work just like base 10 numbers, except they require 16 basic symbols, 0-9, and the letters a through f.
Indian-Arab numbers eventually replaced Roman numerals
Although Fibonacci’s book began to be used by Italian merchants almost immediately to train apprentices, it took 300 years for the Indian-Arab system to finally prevail over the Roman numbers. Many people resisted the new Indian-Arab numbers and the new arithmetic. Roman numbers had been used for over a thousand years without problems. Abacuses worked just fine. If it’s not broken, why fix it? Nevertheless, the advantages of the new numbers gradually won out and Roman numbers disappeared by and large by 1500 AD, although they are still used for limited purposes today, such as for numbering book chapters.
Decimals based on the Indian-Arab numbers replaced the sexagesimal fraction system
Despite his advocacy for a place-value numbering system using a base of ten, Fibonacci didn’t go to the final step to realize that fractions could also be written in base ten. In our modern numbering system, 0.345 is the same as , essentially equivalent to the sexagesimal system except that ten is used as the base rather than 60. Ironically, Fibonacci thought that if sexagesimal numbers weren’t broken, why fix them? Mathematicians from the time of the Greeks through the1500s in Europe still used them. The Egyptian astronomer Ptolemy as well as Copernicus were able to perform their calculations with acceptable accuracy using sexagesimals, so why change the system?
Just as the advantages of using the Indian base ten place-value system won out for numbers for everyday use, they also won out for fractions. The French mathematician Francoise Viete argued to use decimals exclusively by the latter part of the 16th century. A little later, the mathematician Simon Stevin wrote a manual for how to do arithmetic with decimals, just as Fibonacci had done for natural numbers. People learned to do decimal calculations from Stevins’ book. By the beginning of the 17th century the modern base ten numbering system was in place.
Some lessons from the history of numbers
Why did it take over 3000 years, from the time of the Babylonians to the end of the Middle Ages, for the modern numbering system to be developed? There are some important lessons here that we see over and over in the history of mathematics.
First, simple but incredibly powerful ideas, such as the base ten numbering system, were often overlooked by even the greatest of mathematicians. Archimedes missed it.
Second, even when these powerful concepts are discovered, they may not go anywhere on their own. Base ten numbers were developed in the eighth century in India but they didn’t instantly take over. Base ten numbers might have never taken over if the Arab conquests hadn’t brought the idea into the Arab world, where it moved further into Italy. But it took 800 years for that to happen!
Third, somebody needs to understand the significance of an idea, explain it, and push it forward. In this case, that somebody was Fibonacci. Today, to the extent that Fibonacci is known at all he’s generally recognized by mathematicians for a sequence of numbers he is famous for, the Fibonacci sequence. He’s also known among botanists and some gardeners since Fibonacci sequences show up in how plants form and develop. Fibonacci should be remembered for championing perhaps the most important innovation in numbers in history. But even Fibonacci didn’t see that fractions could also be included in a base ten numbering system, which took almost another 100 years to develop. This episode reminds us of another lesson we see over and over in the history of mathematics.
It’s very difficult to know in advance how important a mathematical discovery will turn out to be. Mathematicians usually work on a problem for its own sake. Sometimes the answer becomes vitally useful and sometimes it doesn’t. The proponents of Roman numbers in the late Middle Ages were wrong, but they weren’t terribly wrong. Base ten numbers were better, but they were not that much better than Roman numbers plus the abacus for most everyday purposes.
What the advocates of Roman numbers couldn’t see was that once numbers were severed from the abacus, new calculations that could never have been accomplished using Roman numerals and the abacus became possible. Fibonacci probably didn’t understand fully the profound significance of what he was advocating either. Base ten numbers catapulted mathematics dramatically forward in a startling way that most people at that time could not have foreseen.