Well now, what would happen if we lost all the nines in our number
system? Have a go at writing the numbers out in this way and have a
look at the multiplications table.
Here are many ideas for you to investigate - all linked with the
Find all 3 digit numbers such that by adding the first digit, the
square of the second and the cube of the third you get the original
number, for example 1 + 3^2 + 5^3 = 135.
Published December 2007,March 2007,December 2011,February 2011.
But most of us seem perfectly familiar with what seems to be an easy to understand idea. Surely it is not difficult if we use it so much in our everyday language.. For example;
Can you think of other phrases you have used, or heard, that refer to "nothing" or "zero"?
Then, about 5000 years ago those small communities began to organise themselves and this was the beginning of civilisation as we know it today. Because people stayed in the same place and farmed they had more time to think and observe what was around them and one of the most important things they needed to know about was how to measure time and know where in the annual cycle they were. When
would it be best to plant crops or to expect rain in dry climates? They began to use the phases of the sun, moon and stars to help them keep track of time. For example, the Egyptians were able to tell when the Nile was due to flood by studying the dog star (Sirius). Time was central to the cycle of life so it was crucial to be able to 'control' the calendar. They needed symbols to count the days
and phases of the moon and to communicate with others.
How do you represent a tally? I suspect it is similar to my tallying:
Tallies are rather long-winded, especially for large numbers and ancient civilisations came up with more efficient ways of recording numbers. For example, the Egyptians used Hieroglyphs but there were other pictorial systems too.
Pictorial systems are all very well but they do not make calculation, even adding, easy. Systems were needed that would support calculation.
The 'Base' does not have to be 10, it could be 20 or we could use base 16 or even base 60... So, how did we end up where we are today?
Hieroglyphs were being used by the Egyptians in 3000 BC
Using the hieroglyphs we know that:
Here is one for you to try (answer at the end of the article):
Here are some hieroglyphs on King Narmer's War Club (about 3000 BC). Along the bottom, and outlined in red, the hieroglyphs tell how many bulls, goats and human prisoners the king took. Can you work it out?
Notice that the pictures or hieroglyphs can be written in any position - you know what they are worth by their shape.
Around 2600BC the Babylonians were using a system similar to the Egyptians but theirs was based on 10 and 60:
Later, and very importantly, they introduced what is known as a positional system. We use a positional system today and it means that the position of a symbol tells you about its value. This makes numbers easier to write and use to record large numbers as well as potentially making calculations easier. The written version of the numbers was called cuneiform:
They still used bases 10 and 60, but the positions of the symbols also had information about their value. So, what the Babylonians saw was based on numbers up to 60 formed by the tens and the units symbols:
These groups of symbols were then positioned according to whether they were units, sixties and so on... For example
Try this one (answer at the end):
Remember, the groups of symbols represent (from right to left) units, 60's, 3600s, and so on, rather than the units, 10's, 100's we use today, but very clever all the same!
The Romans only had a limited number of symbols I, V, X, C, L, D, M and from these they generated their numbers using what was very like a tally system with some shortcuts:
The numbers 1 to 10 (no zero notice) are:
I, II,III, IV,V, VI, VIII, IX, X
Then other numbers were created by using a combination of I's, V's and X's and the letters L (representing 50), C (standing for 100), D (500), M (1000).
So, for example 111 = CXI - could you have guessed that?
Notice the way 4 and 6 are written 4 is one less than 5 so it is written with the symbol for 5 (V) with a 1 (I) in front of it telling you that 4 is one before the 5. Because 6 is one more than 5, the I is put after (6 is one after 5). In the same way 9 is one before 10. So you could write 90 as XC (10 before 100).
Here are some more:
1999 MCMXCIX (1000 + 100 before 1000 + 10 before 100 + 1 before10).
Now try these:
They had two problems - firstly they did not have zero and secondly they did not have a system which allowed for "place value" in other words the position of a number told you nothing about its value (so in Roman numerals a C means 100 wherever you see it). Whereas, in our number system you know the value of a number by its column counted from the right (units, tens, hundreds, etc.)
The Romans, and Greeks, did not have a zero and their ability to calculate suffered as a result.
It was the trading of the Arabs that helped to spread the Indian system widely to Western Europe.
A useful number system took about 12000 years to arrive!
Firstly, it represents nothing - it enables us to write in symbols the fact that nothing is left, or there to start with.
Secondly it enables us to make an empty space in a number. For example in the number 650,081 the zeros tell us there are no hundreds and no thousands. Without the zeros our place value system would not work as we would not be able to distinguish 650,081 from 6,581.
But we also use zero to mean other things. Often we use it to create a reference point. For example:
When we talk about the height of land we mean its height above sea level. This means that someone decided that sea level would be at zero height.
The freezing point of water is zero when measuring temperature in centigrade but not in Fahrenheit. The freezing point of water is 32 degrees Fahrenheit. Zero Fahrenheit was the coldest temperature that the German-born scientist Gabriel Daniel Fahrenheit (after whom the scale is named) could reach with a mixture of ice and salt.
"Absolute zero" is -273.15 degrees centigrade. Absolute zero relates to the oscillation of molecules or atoms. In all materials a point is eventually reached at which all oscillations are the slowest they can be. This point is called 'Absolute zero'. You can never reach absolute zero.
'Nothing is real'
....or perhaps not!
King Narmer's Club:
400,000 bulls; 1,422,000 goats;120,000 human prisoners
36 x 3600 + 40 x 60 + 11 = 132,011
36, 47, 756, 1969
We can only count: