The development of number in Ancient civilizations
In today’s world, we use a base-10 place value number system, with one symbol for each of the 10
numbers.
Place value means that we can use the same symbol in different places in a number and it will have
different value (units, tens, hundreds) and base-10 means that each of these places is ten times greater
than the one before it.
But it was not always that way. The Ancient Egyptians used a base-10 system too, but did not have a
place value system, whereas the Babylonians used a place value system but it was not base 10! The
Indian mathematicians took the best bits of these two number systems and also introduced the numerals
we use today, including zero.
This section gives a brief outline of how these civilizations laid the foundations for the numbers we use
today, and gives some ideas for exploring them.
Egyptians
As mentioned above, the Egyptians did not have a place value system. This means they had to invent lots
of symbols for different sizes of numbers. Here are a few early ones they used:
So it didn’t really matter where they put the symbols in the number as they had a fixed value.
Here’s how they might have written 136:
What are the advantages and disadvantages of this system?
Apart from coming up with base-10, Egyptians also developed notation for writing fractions. However,
they only used unit fractions (of the form 1/n) and would write them like this:
The
symbol is called rho, and denotes a (unit) fraction.
So, how did the Egyptians then write non-unit fractions? They would write them as sums of unit
fractions.
How did they write 2/5? You might have thought they would just write 1/5 + 1/5 but for some reason
they didn’t – they had strict rules about how fractions like this were written. They created a table of 2/n
fractions on something now called the Rhind Papyrus.
In fact, they wrote 2/5 as 1/3 + 1/15 like this:
Is this the only way of writing 2/5 using unit fractions?
How do you think they decided which unit fractions to use?
How might the Egyptians have written 2/7?
Can you find a pattern that will help you write other fractions of the form 2/n? Or 3/n?
… and finally
Egyptian mathematics developed an excellent method for multiplication, sometimes called Dyadic
multiplication.
Here is their method for calculating the product 19 x 23:
1 x 23 = 23
2 x 23 = 46
4 x 23 = 92
8 x 23 = 184
16 x 23 = 368
So we have 23 + 46 + 368 = 437. Can you see how this works? Will it always work?
Notes on Egyptian mathematics
You could write 2/5 in lots of different ways, but the first unit fraction (1/3) is the largest one that is
smaller than 2/5, so 2/5 = 1/3 + 1/15 seems quite sensible. We could draw it like this:
This picture suggests a method for finding other 2/n fractions:
So we can write 2/7 = 1/4 + 1/28. This suggests a pattern; following it on we have 2/9 = 1/5 + 1/45 and so
on…
As it happens, the Egyptians did not use this pattern to write 2/n fractions. They had a range of criteria
for writing them. For example, the Rhind Papyrus gives 2/9 = 1/6 + 1/18 and 2/15 = 1/10 + 1/30 which
suggests the use of the formula 2/3n = 1/2n + 1/6n
… and finally the method given uses the fact that all numbers can be expressed using powers of 2. Here,
19 = 1 + 2 + 16 so we just use those calculations. This method will work for any number.
Babylonian number system
The Babylonians invented a place value number system, which meant they didn’t
have to invent lots of symbols for different sized numbers. In fact, they only used
two symbols for one and ten.
This is possibly because they wrote on clay tablets with a piece of reed, so were
limited in how detailed their numerals could be.
Although they used a place value system like we do today, they used a different base. We use base 10, so
a blob in the tens place is worth ten times more than a blob in the units place; here’s 136.
100
10


units

The Babylonians used a base 60 system so each next column would be worth 60 times more than the one
before. So, using blobs, they might have written 136 like this:
3600
60

units


You can see why they invented an extra symbol for 10 units - you could have up to 59 blobs in one place!
So they would write 136 like this:
What are the advantages and disadvantages of the Babylonian number system?
Can you think of where we use something related to the Babylonian number system today?
Why do you think they went for base 60?
If you were to invent a number system from scratch (try it!), what system would you use and why?
Indian mathematics
Indian mathematics has been going on for as long as there has been mathematics. Here are a few of the
mathematical ideas that they brought to the world in the first millennium AD.
Indian mathematicians are credited with the genius idea of expressing each number using one symbol
each, as we do today. The development of these symbols happened over more than 1,000 years into
something resembling our current symbols 0-9. Here is an image of Devanagari numerals from around
1000AD:
Note that zero appears in the above numerals. The birth of zero can be traced to one of the most famous
Indian mathematicians Brahmagupta (c. 700AD). He came up with the following rules for working with
zero:
Which ones are correct? Which ones are wrong? Can you explain why?
1.
2.
3.
4.
5.
Zero plus or minus zero is zero.
Anything (including zero) multiplied by zero is zero.
Zero dividing by anything is zero
Anything dividing by zero is a fraction with zero as the denominator
Zero divided by zero is zero.
Once zero came along, negative numbers weren’t far behind; Brahmagupta also explored rules for
working with negative numbers.
Can you explain what the diagrams below (produced by my year 7 students) represent?
Could you develop your own notation for working with positive and negative numbers?
What are the rules for multiplying and dividing with negative numbers?
Can you convince someone why they are true?
… and finally
Here is a picture from an Ancient Indian
mathematical text.
What do you think it represents?
Notes on Indian mathematics
Brahmagupta’s rules for dividing by zero are not quite right. Rule 4 doesn’t really say anything and Rule 5
is not correct either. Indian mathematicians in this millennium never really dealt with dividing by zero,
and eventually settled on 1/0 = infinity.
To check Brahmagupta’s rules, we can check division by using the equivalent multiplication calculation.
For example,
Rule 3: 0÷1 = 0 is equivalent to 0×1 = 0
Rule 4: 1÷0 = ? is equivalent to ?×0 = 1
Rule 5: 0÷0 = ? is equivalent to ?×0 = 0
Clearly Rule 3 is fine. Rule 4 has no solutions, so we must conclude that we can’t divide something by 0.
Rule 5 is also wrong but for different reasons… there are too many (infinite) answers to the multiplication
sum!
Regarding multiplication with negatives, here’s a quite convincing demonstration of why two negatives
must multiply to make a positive!
Consider the product 19 x 28 (= 532).
Using the box method in a slightly unusual but perfectly valid way we have:
x
30
-2
20
600
-40
So… the bottom right box is the answer to -1 x -2.
What must go in there?
Well, the other numbers add up to 530, and we know that the
answer to 19 x 28 is 532, so it must be +2.
-1
-30
???
We conclude that -1 x -2 = +2.
… and finally this picture gives a visual method
for multiplying two 2-digit numbers together:
So we get 72 + 180 + 80 + 200 = 532.
What would a similar diagram look like for
multiplying two 3-digit numbers together? How
about three numbers? How could you use this
for expanding quadratics?
Islamic mathematics
The period from the 8th to 13th century was known as the Islamic ‘golden age’. Islamic mathematicians
working at the ‘House of Wisdom’ in Baghdad translated the important Greek and Egyptian mathematics
that had gone before, and added many important results of their own.
One of the first Islamic mathematicians Al-Khwarizmi (around 800AD) is credited with the invention of
algebra. Although his algebra did not look as we know it today (he used words instead of symbols) he
developed methods for solving any linear and quadratic equation. In doing so, he brought mathematics
from the practical to the abstract.
Two of the methods he invented for solving linear equations were known as al-muqabala (reducing) and
al-jabr (restoring) from which the word algebra is derived.
Al-muqabala is the process of removing the
same thing from both sides of an equation.
Consider the equation 4x = 2x + 6. We could
visualize this using a diagram like this,
where red squares represent x and the
green blobs are units.
Then is it clear to see that we can just remove
two squares from each side to leave us with
this much simpler problem. It is clear from
here that a red square (x) is worth 3 blobs
(units).
Solving an equation with negative signs is slightly more difficult. Consider the equation 4x – 2 = 2x + 6.
Now we need the method of al-jabr (restoring) to eliminate the -2 on the left hand side. By adding two
unit to both sides of the equation we now get 4x = 2x + 8 and we can continue in the previous example.
Here is a transcript of Al-Khwarizmi’s method for solving quadratics.
Divide the roots into halves. In this problem five is the result, which you multiply by itself to give twentyfive. Add it to thirty-nine and you have sixty-four. You extract the root which is eight and deduct it from
half the root which is five and three is left which is the root of the square you are looking for which is nine.
Can you work out how this method works?
Could you write down the steps of the method using the algebra we use today?
Could you use it to solve other quadratics?