Today, we use numbers confidently in our everyday lives without giving much thought to their history. Is it necessary to know at least a little about where they came from? Probably not. However, such knowledge gives us a deeper cultural understanding of the long journey humanity has taken in order to understand the world better. We will not go into great detail, but we will briefly mention some of the important milestones in the development of numbers.
The need for numerical symbols arose from the need to count and measure. At first, people used stones, bones or tally marks to count the animals they had hunted.
It took centuries to move from marks and tallies, which referred to very specific objects, to numerical symbols representing abstract ideas. Understanding this difference led to a qualitative change in human thinking and contributed to the development of the human brain and consciousness. This happened as primitive hunting societies gradually adopted a settled way of life, left their caves, built permanent homes and began developing trade.
The ten digits we use today are known as Hindu-Arabic numerals. They were introduced into Europe during the twelfth century by the Italian mathematician Leonardo of Pisa (also known as Fibonacci), who had studied in North Africa. The numerals we use today are adaptations of the original Hindu symbols. The symbol for zero was invented in India. Before that, a pebble or simply a dot was often used to represent the absence of quantity.
Thousands of years ago there were no symbols for numbers such as two or three. Instead, people used fingers, stones, sticks or even eyes to represent quantities. There were neither clocks nor calendars to measure time accurately. The Sun and the Moon helped people distinguish between different parts of the day and the passing of time.
Most early civilisations did not even have words for numbers larger than two or three. Instead, they used expressions such as "a flock of sheep", "a heap of grain" or "many people". There was little need for an advanced numerical system until people formed villages and cities and developed systems of trade and barter, which eventually created the need for money and accounting.
Since paper and pencils did not yet exist, other methods were developed to record numbers. The Babylonians pressed wedge-shaped marks into soft clay using a stylus. The Egyptians painted numbers on pottery and carved them into stone. Various civilisations invented numerical systems based on symbols rather than the digits we use today.
Anthropologists tell us that in Sumer, around 4000 BCE, people used small clay tokens to represent numbers—an important improvement over tally marks carved into wood or bone. An even greater development was that tokens could not only be added but also removed, giving birth to arithmetic, one of the greatest achievements in human history.
These Sumerian tokens made it possible to assess wealth, calculate profit and loss and, perhaps most importantly, collect taxes and keep permanent records. Many historians believe that this accounting system eventually gave rise to the world's earliest form of writing.
Some more primitive societies, such as the Warlpiri people of Central Australia, traditionally had little need for extensive number systems. Why, then, did the Sumerians, living on the other side of the world, develop mathematics? The answer is simple. They lived in organised cities. Grain had to be stored, goods distributed and taxes collected. All these activities required arithmetic.
The Egyptians loved everything on a grand scale—great buildings, enormous statues and mighty armies. They developed symbols not only for everyday numbers but also for very large numbers such as one thousand, ten thousand and one million.
One of their greatest achievements was transforming the idea of "one" from simply counting objects into measuring length. They introduced the cubit, defined as the distance from a man's elbow to the tip of his middle finger plus the width of his palm. Using this standard unit, the Egyptians built monumental structures such as the pyramids with astonishing accuracy.
The Chinese developed one of the oldest numeral systems, using counting rods placed on a table to perform calculations.
From around 450 BCE, the Greeks used several different numeral systems. The most common one assigned the first ten letters of the Greek alphabet to represent the numbers from one to ten. To distinguish numbers from ordinary letters, they often added a small mark beside them.
Around 520 BCE, Pythagoras founded his famous philosophical and mathematical school in Greece. Fascinated by whole numbers, he noticed that harmonious musical sounds were based on simple numerical ratios. Believing that the universe itself was built upon numbers, he attempted to express all geometrical relationships using whole numbers. His studies eventually led to discoveries that changed mathematics forever.
The famous Pythagorean Theorem is traditionally attributed to him. However, ancient Indian texts such as the Sulba Sutras (around 800 BCE) and the Shatapatha Brahmana show that this relationship was already known in India centuries before Pythagoras lived.
Later, during the third century BCE, the great Greek scientist Archimedes explored unimaginably large numbers. He even attempted to calculate how many grains of sand would be needed to fill the entire universe. Some of his investigations proved remarkably useful. For example, his work on spheres and cylinders later contributed to methods used in cartography for representing the curved surface of the Earth on flat maps.
When the Romans conquered Greece, they were far more interested in power than in abstract mathematics. Archimedes was killed during the Roman conquest of Syracuse in 212 BCE, an event often regarded as a symbolic loss for mathematics. Roman numerals were unsuitable for complex calculations, so calculations were usually performed on counting boards, the predecessors of the abacus.
Although Roman numerals spread throughout Europe and remained the dominant numeral system for many centuries, no Roman mathematician achieved the fame of the great Greek scholars. The Romans were more interested in recording military victories, taxes and administration than in developing mathematical theory.
Roman numerals are still used today, although some forms have changed over time. For example, the Romans often wrote four as IIII instead of IV. Today Roman numerals are commonly used for book chapters, outlines, clock faces and the numbering of monarchs and sporting events. We still encounter Roman numerals such as I, II, III, IV, V, VI, VII, VIII, IX and X in many aspects of everyday life.
Later in this book we will illustrate several different numeral systems in tables.
Finger numerals were used by the ancient Greeks, Romans, medieval Europeans and many Asian cultures. Even today, children around the world often learn to count using their fingers.
Odd and even numbers
In everyday life, we often use the concepts of odd and even numbers when grouping, sorting or dividing objects into equal parts. Even numbers are divisible by two without leaving a remainder, while odd numbers leave a remainder of one when divided by two. Although this seems like a simple idea, it has many practical applications in everyday life.
The number π (pi)
Most people will never need to use the number π in their daily lives. Nevertheless, it is one of the most famous numbers in mathematics and often fascinates people. Knowing a little about it may even help you impress someone with your mathematical knowledge.
The earliest evidence of the use of π comes from ancient Egypt and Babylon, showing that it has been known for thousands of years.
The number π is the mathematical constant representing the ratio of a circle's circumference to its diameter. It is famous because its decimal expansion never ends and never repeats.
Its approximate value is 3.14 (to two decimal places).
Rounded to one hundred decimal places, π is:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679...
In the picture below, we can see bones that are between 20,000 and 30,000 years old. The marks carved into them represent one of the earliest known methods of counting used by prehistoric hunting societies. These tally marks provide clear evidence that humans were already counting tens of thousands of years ago and represent one of the defining moments in the development of human civilisation.
(Elena S. Lyubenova)
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