It is easy to take our numbering system for granted. With zero and 1-9, we can write every possible number, we can count, perform math and buy and sell goods and services for intangible assets like money.

Numbers haven’t always existed – they are a human invention. In fact, there are some cultures where numbers still do not exist. Check out this story from Smithsonian Magazine:

When University of Miami associate professor Caleb Everett and other anthropologists worked with the indigenous Amazonian people known as the Pirahã, they realized the members of the tribe had no word used consistently to identify any quantity, not even one.

Intrigued, the researchers developed further tests for the Pirahã adults, who were all mentally and biologically healthy. The anthropologists lined up a row of batteries on a table and asked the Pirahã participants to place the same number in a parallel row on the other side. When one, two or three batteries were presented, the task was accomplished without any difficulty. But as soon as the initial line included four or more batteries, the Pirahã began to make mistakes. As the number of batteries in the line increased, so did their errors.

According to Charles Seife in his very interesting book Zero: The Biography of a Dangerous Idea, our ancient ancestors could not even count. The beginning of counting was distinguishing between one and many. That moved to one, two and many – then one, two, three and many and so on. Counting up to five and ten and then twenty was assisted by our fingers and toes.

The earliest evidence of using marks to assist with counting is from a wolf bone dating from around 30,000 years ago on which there are a series of notches carved. These notches were in groups of five and added up to a total of 55. Notches such as these were the beginning of numbers.

As the complexity of life increased, particularly the move to an agrarian society and the rise of specialization about 10,000 years ago, the use of just notches to assist in counting was not sufficient.

The first “numbers” were really just extensions of tally marks, like the early Egyptian and Greek numbering systems which arose in the 4th and 5th centuries B.C. Later Greek numbering systems used letters of the Greek alphabet to represent numbers.

Of course, then there are Roman Numerals – dating from about 1000 B.C., which originated from the ancient Roman Empire and which are used sporadically today. (Fascinating fact about IV vs. IIII on clocks and watch faces here.) The Roman Numerals first appeared around 900 to 800 B.C.E. The Roman Numeral system uses I, V, X, C, D, L and M.

Among the major problems with these systems is while they were fine for counting and keeping track of sheep or inventory, they weren’t good for math or writing large numbers. Interestingly, while the ancient Greeks had some outstanding mathematicians in terms of geometry (such as Euclid and Pythagoras), they didn’t know arithmetic because they lacked a numbering system that would have enabled them to calculate instead of just counting or recording. Likewise, it is quite difficult to add roman numerals such as as L + XC + LX (which adds to CC).

Some early numbering systems were binary base systems with numbers progressing as follows: one and two were their own discrete numbers, then three was “two and one” and four was “two and two” and five was “two and two and one” and so on. Others were base-five systems, supposedly because we have five fingers on each hand (and five toes on each foot). Base-10, 12, 20 and 60 systems all have been used as well.

Note that a “base” means that when a number is reached, whether 2, 5, 10, 12, or 60, a different mark is made representing all the previous numbers. At that point units continue to be added to the base until the next base is reached, and so on. Our numbering system is a base-10 system.

A major breakthrough in numbering systems occurred with the development of “positional notation.”  Prior numbering systems needed to have different symbols for larger numbers, such as the following in the Roman Numeral system: C for 100, D for 500, and M for 1,000. So, the number 2557 is written as MMDLVII in Roman Numerals.

A huge benefit of a positional system is that it can use the same symbols over and over and get to different values based on where the characters are located within the sequence.

The Babylonians, Aztecs and ancient Chinese all developed positional notation numeric systems independently. The first was developed by the Babylonians around 2000 BC and used a base-60 system and their numbers/system looked like this:

The Aztecs used a base-20 numbering system as did the Mayans. The Chinese used a base-10 system.

A base-10 numbering system with positional notation was developed by mathematicians in India between the 4th and 8th centuries. Around this time zero was also invented as a number. See IFOD on the number zero here. The development of zero was a huge breakthrough.

While pretty much any symbols can be used to represent one through nine, our current system of 1 – 9 was developed in the Arabic region of North Africa. Our base-10 system using 0 and 1-9 is known as the Hindu-Arabic System of Numbers and began being used in Europe in the 1200s and became the dominant numbering system in the Western World by the 1500s.

With the development of base-10 numbering systems with zero and positional notation, mathematics was able to leap forward. However, as noted by professor Leonard Mlodinow in his excellent book The Drunkard’s Walk: How Randomness Rules our Lives:

Even after the development of a usable number system, it would be many more centuries before people came to recognize addition, subtraction, multiplication, and division as the fundamental arithmetic operations . . . and it wasn’t until the 16th century that the Western world was truly poised to develop a theory of probability.

The = sign wasn’t invented until 1557 and the use of x for multiplication began being used in the 17th century. As for the division symbol, it’s interesting it its own right and you can read about it here.

The development of theories of probability over the centuries is a fascinating story, told well in Peter Bernstein’s Against the Gods: The Remarkable Story of Risk and probably the topic of a future IFOD.

A related and very highly clicked IFOD relates to whether mathematics is invented or discovered.