When people began needing to count items, they used sticks or stones with one-to-one correspondence.
For example, they would collect 1 stick for every sheep in their flock. When they wanted to see if they still had all of their sheep, they would ‘count’, matching one stick for one sheep. If at the end of the ‘count’ they had sticks left over, they had lost some sheep. If at the end of the ‘count’ they had sheep left over, they had gained some sheep.
Over time, people realised that the amount of sheep was the same as the amount of sticks, and so names and symbols could be assigned to each amount, and numbers were born. This is a fundamental and important concept in counting – that a number always represents the same amount.
The earliest number systems were additive – Roman Numerals are one example. Roman Numerals are additive because we need to add or subtract values of the symbols to get the value of the number. XCVIII is 100 minus 10 (90) plus 5 plus 1 plus 1 plus 1 (98). Additive systems worked while people only needed to represent numbers of small values.
By about 2000 years ago, civilisations were beginning to travel long distances by sea, and to study the skies. This meant they needed a more efficient way to represent large numbers.
Multiplicative number systems were the solution because multiplication makes numbers large much faster than addition (because multiplication is not just a quick way to add!!). Multiplication by a number larger than 1 makes it a number of times larger.
For example, if you multiplied me by 2, don’t imagine there are 2 of me, imagine I am 2 times larger. And dividing me by 2, don’t imagine I have been cut into 2 parts (in half) but that I am 2 times smaller or half the size.
Which way would you have more money – by repeatedly adding $10 ($10, $20, $30), or repeatedly multiplying by 10 ($10, $100, $1000)?
Ancient civilisations understood multiplication, and many created multiplicative place value systems.
The Hindus (who were so-named because they lived in the Indus Valley) knew this, so they based their number system on multiplication. And what number did they decide to multiply by? They decided to multiply by 10, simply because they had 10 fingers. Other civilisations multiplied by 20 (fingers and toes) and by 5 (fingers on one hand).
But why is the base 10 number system the one that the whole world uses today? The base 10 number system soon spread to the Arabic part of the world – and thus came to be called the Hindu-Arabic number system. And there it stopped for about 700 years. During this 700 years, Indian and Arabic civilisations made great progress in both mathematics and science, made possible because they were using a multiplicative number system. While Europe, who were still using the additive number system involving Roman Numerals, were experiencing the Dark Ages, making no progress in either mathematics or science.
Finally around the 1200s, the Hindu-Arabic number system spread to Europe. Following time spent in Muslim Algiers, Italian Mathematician Leonardo Fibonacci recognised the superiority of the Hindu-Arabic Place Value number system and introduced it to Europe through the publication in 1202CE of his Liber Abaci (Book of Calculation) – and the dark ages ended and the renaissance began!
From then it was widely used in European mathematics, and with the invention of the printing press in Europe around 1440CE, it replaced Roman Numerals in general use. As the Chinese were also using a Base 10 number system, transition to the Hindu Arabic Place Value number system was natural.
European and Arabic people then began their colonisation of the world, taking with them their cultures, their religions and their number system.
And so the whole world now uses the same base 10 multiplicative number system. This is fabulous because no matter what language we speak, we can all share the same mathematical understandings and apply these understandings to science.
Initially, the Hindu Arabic Place Value number system was used only for whole numbers. The value of the lowest place was ones. In this way, all other values could be calculated from the ones place.
Fractions were recorded using different notations – including numerator, vinculum and denominator.
Extending the Place Value system to include fractions, began with Jewish, European and Persian Mathematicians who recognised that using different systems for recording whole numbers and fractions made calculation difficult, and that not all values can be created accurately with fractions (as a ratio between 2 numbers). So they simply divided 1 by 10 to get a tenth, and continued dividing by 10 to get hundredths, thousandths etc.
When the base 10 Place Value number system was extended to include fractions, it soon became less obvious which place was the ones place when it wasn’t the lowest value.
For example 13 as 1 ten and 3 ones, and 13 as 1 one and 3 tenths, looked the same. Initially different people used different symbols to identify the ones place.
In his booklet De Thiende (‘the art of tenths’), first published in Dutch in 1585, Flemish Mathematician Simon Stevin used numbered circles.
Others used a bar over the ones place to identify it. European countries finally settled on a comma (,) however English speaking countries used the comma to separate groups of numbers so Britain used a mid dot (∙) and the United States used period (dot) on the base line. Australia adopted the British mid dot.
We call the symbol a decimal point, because the people at the time said ‘point’ rather than ‘dot’. If it was created today, we would call it a dot! It is a decimal point, because the entire number system is decimal – the prefix ‘dec’ means 10 – we are multiplying and dividing by 10. The computer keyboard, being invented in the US, has a dot on the base line, but no mid dot, so we in Australia just use a dot on the base line on computers.
Without our awesome number system, none of the inventions and technology that we take for granted today would be possible!
Place Value ‘Activities’ abound on the Internet. But nothing beats teacher understanding and quality pedagogy!
Further reading and viewing: