Development of number
through the
history of mathematics
Zero – the number
Development of number through the history of mathematics
Topic: Zero – the number
Resource content
Teaching
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Resource description
Teacher comment
Background to development of the history of mathematics resources
Mathematical goals
Starting points
Materials required
Time needed
What I did
Reflection
What learners might do next
Further ideas
Artefacts and resources
Activity sheets and supporting historical information
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Activity sheet 1: Counting sheep
Supporting historical information (Activity sheet 1)
Activity sheet 2: Numbers without zero
Supporting historical information (Activity sheet 2)
Activity sheet 3: Alexander the Great and his Empire
Supporting historical information (Activity sheet 3)
Activity sheet 4: Chinese numbers
Supporting historical information (Activity sheet 4)
Activity sheet 5: Chinese negative numbers
Activity sheet 6: Chinese calculations
Activity sheet 7: The first ever zero
Activity sheet 8: Brahmagupta’s rules for numbers
Supporting historical information (Activity sheet 8)
Activity sheet 9: Brahmagupta’s rules for dividing by zero
Resource description
Selection of artefacts demonstrating the emergence of zero as a place holder.
Learners work in groups to examine how zero came into being through working with
mathematical artefacts.
Teacher comment
The resource is set within the context of the history of mathematics and how
mathematics is discovered and/or invented. Considerable importance is attached to
use of ‘real’ artefacts and historical accuracy – though interpretation is more open to
debate. Options are available to link to real artefacts.
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Zero – The number
Development of number through the history of mathematics
Mathematical goals
To help/enable learners to:
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develop an understanding of the difference of zero as a number versus a
place holder
understand how zero extended the number line hence enabling negative
numbers to exist
gain a sense of the timescale involved in the development of a new
mathematical idea
Starting points:
This session requires no specific prior knowledge. It can be used to follow on from
the topic Zero – the place holder, or it can stand alone in its own right.
It is aimed at KS2 and KS3 but can be used with younger learners by using smaller
numbers or with older learners as a historical introduction to the development of
number systems (e.g. rational, real, complex).
Materials required:
For each pair or small group of learners you will need:
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materials for creating posters could be useful for group work
matchsticks, or equivalents (two colours are needed)
Activity sheet 2: Numbers without zero
Activity sheet 4: Chinese numbers
Activity sheet 5: Chinese negative numbers
Activity sheet 6: Chinese calculations
Activity sheet 8: Brahmagupta’s rules for numbers
Activity sheet 9: Brahmagupta’s rules for dividing by zero
internet for some/all learners (optional)
Interactive whiteboard and projection resources
You may find it easier to project the Activity sheets, using a data projector, a
visualiser or an overhead projector with a transparency. Alternatively you might want
to the use the Promethean ActivStudio and Smart Notebook IWB versions of the
activities.
Wherever items for display are subject to copyright restrictions direct links are
provided for them.
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Activity sheets 1 to 9
‘Supporting historical information’ Activity sheets 1, 2, 3, 4 and 8
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Zero – The number
Development of number through the history of mathematics
Time needed:
Between 1 and 2 hours depending on how much detail and background you choose
to use.
What I did:
Beginning the session
First, I set the scene, and ask some questions
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I ask what would it be like without the number zero?
in ancient times would people count their sheep if they had no sheep?
(Activity Sheet 1 contains sheep to count - if desired).
I write 305 on the board.
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elicit what each digit means.
how does the spoken number: “three hundred and five” compare with the
written number: 305?
(Note that the tens are not mentioned when spoken so the zero isn’t needed.)
Working in groups (1)
Writing numbers without zero (the placeholder)
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I ask the learners what they would do if the number zero didn’t exist.
I ask if they could still write the number: “three hundred and five”.
I either give out copies of Activity sheet 2 to cut out the numbers themselves or
create sets of digits 1-9 on card that has been laminated in advance.
I give the class a few minutes to decide how they would show the number 305
without a zero, and to work out how their method would work for some other
numbers. I encourage them to be inventive.
Whole group discussion (1)
Writing numbers without zero (the placeholder)
I gather examples from different groups of how they dealt with the problem.
I point out that in some ancient civilisations people simply left a space or used a
special symbol e.g. the Babylonians used a double wedge symbol
, the Mayan
used a shell and in Ancient Indian they use a dot so it would have been 3●5.
I start a discussion on whether the place holder is a number or not; drawing out the
idea that it was not. It was simply filling the space to show that nothing was there.
(See the lesson Zero – the place holder for an exploration of place value.)
Ancient civilisations such as Greece, Egypt and Babylon needed mathematics to help
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Zero – The number
Development of number through the history of mathematics
build temples and pyramids or to work out the size of people’s fields so that they
could be taxed according to how big their fields were. So in the West mathematics
was based on geometry because it developed as a way of working with lengths and
areas.
Meanwhile, in the East, countries such as India and China developed mathematics
by thinking about numbers without the influence of the Western thinkers until
Alexander the Great (356-323 BCE) marched to India from Babylon.
I show Activity sheet 3 – Alexander the Great and his Empire, and I explain how he
disseminated mathematical information – see the Supporting historical information for
Activity sheet 3.
I then look at how Chinese numbers work before going any further. The Ancient
Chinese used to carry a bundle of counting rods made from bamboo. We don’t have
bamboo so we’ll use matchsticks or equivalent.
Working in groups (2)
Understanding Chinese numbers
I show Activity sheet 4. I get the learners to use matchsticks to make a poster of
some Chinese numbers. I explain how you work out what the number is.
Whole group discussion (2)
Understanding Chinese numbers
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I ask how easy is it to make Chinese numbers.
I ask if they find any problems the Chinese might have had.
I ask how easy would it be to do arithmetic using Chinese rods e.g. add,
subtract, multiply by 10.
I ask them how they would use the rods.
The Chinese also had a system of negative numbers as early as 200BCE which they
used for recording commercial and tax calculations where the black cancelled out the
red. The amount sold was positive (because of receiving money) and the amount
spent in purchasing something was negative (because of paying out); so a money
balance was positive, and a deficit or owing money negative.
Working in groups (3)
How the Chinese used negative numbers
I hand out Activity sheets 5 and 6.I use them to explore how coloured counting rods
can be used to calculate with negative numbers.
One set of answers is:
7–3=4
6 – 10 = -4
8–5=3
5 – 7 = -2
-1 -8 = -9
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Zero – The number
Development of number through the history of mathematics
Whole group discussion (3)
How the Chinese used negative numbers
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I ask how does using coloured counting rods compare with our system of
writing negative numbers.
I ask them to discuss what is the same and what is different?
I ask why they think that Westerners did not like negative numbers, whilst the
Chinese were quite happy to work with them? (If necessary I remind them
that Westerners were using mathematics for working out sizes of buildings
and areas of fields.)
I ask what happens if you take the red and black cards for 8 and negative 8
and add them together.
I ask them what they would write down. What if zero didn’t exist as in Chinese
times?
The first time zero is known to be written down as a the result of a calculation i.e. as
a number, was by an Indian called Brahmagupta (598 – 670CE) who defined zero as
the result of taking a number away from itself.
The very first time a circle is known to have been written as the symbol for zero can
be see on a stone table at Gwalior Fort in India (circa 850CE).
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I show Activity sheet 7 or the images on All for nought
Working in groups (4)
Rules for negative numbers
I hand out Activity sheet 8 to see how Brahmagupta’s rules match rules used today.
Whole group discussion (4)
Rules for negative numbers
I review some examples and explanations of how they illustrate the rules.
Reviewing learning
What the learners have found out
I round off the discussion by asking how zero helps us to work with numbers e.g. as
a place holder; so that counting and arithmetic work correctly, as the answer to: x – x.
Reflection:
Learners often do not think about the meaning of the numbers they use.
Understanding of how and why zero was developed can help with understanding the
difference between the use of zero as a place holder and as a bona-fide number.
Most learners, when involved, become interested in finding out about the history of
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Zero – The number
Development of number through the history of mathematics
mathematics – and this has helped many to find out more.
What learners might do next:
I get them to look at Brahmagupta’s rules for dividing by zero (Activity sheet 9) and
ask them to discuss whether they are correct or not.
Further ideas:
Other modules that use a similar approach are found at the History of
Mathematics Mathemapedia entry at the NCETM portal.
Artefacts and resources:
Alexander the Great
For background on Alexander the Great see the BBC History of the World Episode
31: Coin with head of Alexander the Great which provides interactive images of the
coin, a link to both play and download the podcast as well as the transcript of the
podcast.
Also see Wikipedia entry on Alexander The Great.
Chinese mathematics sites for teachers
The MacTutor site at St Andrews offers an overview of Chinese mathematics for
teachers and information about Chinese numerals.
Victor Katz offers a reading list with books on Chinese mathematics in the section
headed Ancient Mathematics.
David Joyce offers a view of the history of Mathematics in China, though most links
do not work, but it does offer a timeline.
Li Yan and Du Shiran. (1988.) Translated by John N. Crossley and Anthony W.-C.
Lun. Chinese Mathematics. A Concise History. Clarendon Press.
Chinese mathematics sites for learners and teachers
Jo Edkin provides a page on Chinese numbers. Multiplication is done slightly
differently to what is shown here.
Jo Edkin also offers a page on many number systems, including Chinese. The site
also shows the formal way to write these numbers for financial reasons.
You can convert numbers into Chinese.
Chinese character for zero.
Toshuo.com is a personal blog that converts Chinese numbers into Arabic numbers,
but can be used to see the symbols for over 10,000.
Brahmagupta
For a biography of Brahmagupta see the MacTutor site at St Andrews MacTutor Brahmagupta or Wikipedia - Brahmagupta
Gwalior
Details of Gwalior fort can be obtained at Wikipedia - Gwalior Fort.
Background on the first zero is available at First zero.
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Zero – The number
Development of number through the history of mathematics
Activity sheet 1
Topic: Counting sheep
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Zero – The number
Development of number through the history of mathematics
Supporting historical information (Activity sheet 1)
It would appear that as long as people were counting physical objects such as sheep
there was no need for a number zero. Either you had some sheep and might want to
know how many, or you had none and therefore wouldn’t try counting them.
For more information see:
Boyer, C., 2010. A History of Mathematics. 3rd Edition. John Wiley & Sons. Ch.1.
Stewart, I., Taming the Infinite. Quercus Publishing Plc. Ch. 1.
George, G. J., 1992. The Crest of the Peacock: The Non-European Roots of
Mathematics. Penguin Books. Ch. 2.
Zaslavski, C., 1999. Africa Counts: Number and pattern in African cultures. 2nd
Revised edition. A Cappella Books.
Wikipedia - Ishango_bone
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Zero – The number
Development of number through the history of mathematics
Activity sheet 2
Topic: Numbers without zero
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Zero – The number
Development of number through the history of mathematics
Supporting historical information (Activity sheet 2)
In the year 2000 when we were all celebrating the millennium no-one knew whether
the new millennium started at the beginning or at the end of the year. This is because
there is no year zero. The timeline goes straight from 1 BCE to 1 CE. A discussion of
how we give the age of a baby would illustrate why this is a problem.
For more information see:
Seife, C., 2000. Zero The biography of a dangerous idea. New Edition. Souvenir
Press Ltd. Ch. 2. (Blind dates)
See 101 uses of a Quadratic Equation for taxes in Babylonian times
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Zero – The number
Development of number through the history of mathematics
Activity sheet 3
Topic: Alexander the Great and his Empire
Source: Battle of Issus 333BC mosaic detail
Source: Macedon empire
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Zero – The number
Development of number through the history of mathematics
Supporting historical information (Activity sheet 3)
Alexander the Great was born in Macedonia in Greece. On taking his place as king
he was able to secure his own kingdom and then make the Persian Empire his own.
He was the undefeated victor in a series of battles. Having subdued the Persians he
then extended his empire even further over to India in the East.
During his travels he took the Babylonian number system to India where unlike the
Westerners who abhorred zero as the void which could be filled by the devil, ancient
eastern civilisations understood the idea of null or nothing i.e. non-existence. This
was strongly linked with religious beliefs such as Hinduism.
(see Alexander The Great)
In Sanskrit (India) the word for empty was: sunya. The Arabic translation was: sifr
which later became the English word cipher.
(see Crest of the Peacock Ch 3)
It is reputed that whilst studying mathematics Alexander asked his teacher,
Menaechmus, for a shortcut to geometry. According to legend Menaechmus replied:
“O King, for travelling over the country there are royal roads and roads for common
citizens; but in geometry there is one road for all.”
(see A History of Mathematics, Carl Boyer, Ch5)
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Zero – The number
Development of number through the history of mathematics
Activity sheet 4
Topic: Chinese numbers
Hengs and Tsungs
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Hengs
Tsungs
Hengs were use for units, hundreds and tens of thousand etc.
Tsungs were used for tens, thousands, hundreds of thousands etc.
For example: 2934 would be
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9
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Use matchsticks to make the following numbers in the same way that the
Ancient Chinese did. Then check with the person next to you to see if you
have the same.
219
4635
31067
500
Make some more numbers of your own and challenge the person next to you
to decode them.
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Zero – The number
Development of number through the history of mathematics
Supporting historical information (Activity sheet 4)
Counting rods were usually made of bamboo in the Han period – approx 3 centuries
CE. They were approximately 2.5 mm diameter and 140 mm long. They would be
tied in a hexagonal bundle for ease of carrying.
By the sixth century CE they were shorter and rectangular and were sometimes
made of wood, cast iron, jade or ivory. (Crest of the peacock page 143)
The Chinese used the rods in a similar way to an abacus repositioning them to get
the sum.
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Hengs
Tsungs
For example: 15 - 7 would be
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5
8
The ten tsung would be converted to 2 lots of 5 hengs and then 7 units removed and
the remaining rods realigned to give the correct heng for 8.
(See The History of Negative Numbers)
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Zero – The number
Development of number through the history of mathematics
Activity sheet 5
Topic: Chinese negative numbers
Red rods stand for positive numbers
Black rods stand for negative numbers
231
5089
-407
-6720
Source: Wikipedia - Counting rods
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Zero – The number
Development of number through the history of mathematics
Activity sheet 6
Topic: Chinese calculations
Cut out the Chinese numbers and arrange them in to 6 sets of 3 so that in
each set the first 2 added together make the last one.
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Zero – The number
Development of number through the history of mathematics
Activity sheet 7
Topic: The first ever zero
Gwalior Fort
Source: Gwalior Fort
The number 270
Source: All for Nought
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History of Mathematics
Zero – The number
Development of number through the history of mathematics
Activity sheet 8
Topic: Brahmagupta’s rules for numbers
Work out what each rule means and give an example that shows what it
means. Make a poster to show how Brahmagupta’s rules compare with the
rules we use today.
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The sum of two positive quantities is positive
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The sum of two negative quantities is negative
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The sum of zero and a negative number is negative
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The sum of zero and a positive number is positive
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The sum of zero and zero is zero.
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The sum of a positive and a negative is their difference; or, if they are
equal, zero
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In subtraction, the less is to be taken from the greater, positive from
positive
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In subtraction, the less is to be taken from the greater, negative from
negative
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When the greater however, is subtracted from the less, the difference is
reversed
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When positive is to be subtracted from negative, and negative from
positive, they must be added together
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The product of a negative quantity and a positive quantity is negative
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The product of a negative quantity and a negative quantity is positive
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The product of two positive, is positive.
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Positive divided by positive or negative by negative is positive
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Positive divided by negative is negative. Negative divided by positive is
negative
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History of Mathematics
Zero – The number
Development of number through the history of mathematics
Supporting historical information (Activity sheet 8)
Brahmagupta (598 – 670 CE) was an Indian mathematician and astronomer. He
published many works in the fields of arithmetic, algebra, geometry, trigonometry and
astronomy. In particular, he wrote the Brahmasphutasiddhanta which contained work
on a wide range of aspects of science and maths. Although the most relevant work
here is Brahmasputha Siddhanta (The Opening of the Universe), written in 628 CE.
Regarding negative numbers he gave the following list of rules as on Activity sheet 8
in addition to defining zero:
When zero is added to a number or subtracted from a number, the number remains
unchanged; and a number multiplied by zero becomes zero.
He defined the arithmetical rules in terms of fortunes (positive numbers) and debts
(negative numbers):A debt minus zero is a debt.
A fortune minus zero is a fortune.
Zero minus zero is a zero.
A debt subtracted from zero is a fortune.
A fortune subtracted from zero is a debt.
The product of zero multiplied by a debt or fortune is zero.
The product of zero multiplied by zero is zero.
The product or quotient of two fortunes is one fortune.
The product or quotient of two debts is one fortune.
The product or quotient of a debt and a fortune is a debt.
The product or quotient of a fortune and a debt is a debt.
Brahmagupta then tried to extend arithmetic to include division by zero (see Activity
sheet 9)
Source: Brahmagupta
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History of Mathematics
Zero – The number
Development of number through the history of mathematics
Activity sheet 9
Topic: Brahmagupta’s rules for dividing by zero
This was the earliest attempt to define division by 0.
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A positive or negative number when divided by zero is a fraction with
the zero as denominator
Zero divided by a negative or positive number is either zero or is
expressed as a fraction with zero as numerator and the finite quantity
as denominator
Zero divided by zero is zero
What do Brahmagupta’s rules mean? Are they correct?
Make a poster to explain your answers.
Source: Wikipedia - Brāhmasphuṭasiddhānta
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History of Mathematics
Zero – The number