Development of number

through the

history of mathematics

Zero – the place holder

Development of number through the history of mathematics

1Zero – the placeholder

Topic: Zero – the place holder

Resource content

Teaching

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

Activity sheet 1: Ishango Bone Supporting historical information (Activity sheet 1) Activity sheet 2: Ishango Bone (simplified) Activity sheet 3: Egyptian hieroglyphics Activity sheet 4: Egyptian numbers 1 Activity sheet 5: Egyptian numbers 2 Supporting historical information (Activity sheet 5) Activity sheet 6: Babylonian cuneiform writing (Plimpton 322) Supporting historical information (Activity sheet 6) Activity sheet 7: Babylonian numbers – Clay tablet obverse Activity sheet 8: Babylonian numbers – Clay tablet reverse

Resource description

A selection of artefacts demonstrating number systems prior to the use of a symbol forzero. Learners work in groups to examine, interpret and compare these numbersystems with the decimal system in current use and the other systems.

Teacher comment

The resource is set within context of the history of mathematics and how mathematics isdiscovered and/or invented. Considerable importance is attached to the use of ‘real’artefacts and historical accuracy – though interpretation is more open to debate.Options available to link to real artefacts.

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Mathematical goals

To help learners to:

develop an understanding of zero as a place holder gain a sense of the timescale involved in the development of a new

mathematical idea begin to recognise that numbers may be written in different bases utilising ideas

from the history of mathematics

Starting points:

This session requires no specific prior knowledge.

It is aimed at KS2 and 3 but can be used with younger learners by focussing on one-to-one matching, counting and tallying, or with older learners as a historical introduction tothe development of number systems.

The resource on multiplication could be used before this activity.

Materials required:

For each pair or small group of learners you will need:

materials to create posters Activity sheet 2: Ishango Bone (simplified) Activity sheet 4: Egyptian numbers 1 Activity sheet 7: Babylonian numbers – Clay tablet obverse Activity sheet 8: Babylonian numbers – Clay tablet reverse internet for some/all learners (optional) modelling clay (or equivalent) with sticks to make cuneiform numbers (optional)

Interactive whiteboard and projection resources

You may find it easier to project the Activity sheets, using a data projector, a visualiseror an overhead projector with a transparency. Alternatively you might want to use thePromethean ActivStudio and Smart Notebook IWB versions of the activities.

Wherever items for display are subject to copyright restrictions direct links are providedfor them.

Activity sheets 1 to 8

‘Supporting historical information’ Activity sheets 1, 3, 5 and 6

Time needed:

Between 1 and 3 hours depending on how much detail and background you choose touse.

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What I did:

Beginning the session

I show Activity sheet 1: the picture of the Ishango bone and say something like: “thiswas discovered in the 1950s and was then thought to be 20,000 years old. If this is thecase then it is probably the oldest existing mathematical artefact. However there issome question over its actual age at the moment” (see the Wikipedia discussion aboutthis).

Whole group discussion (1)How long ago did people first start to use numbers?

I give out Activity sheet 2 and ask the learners to discuss what the marks on the bonesmight mean and talk about any issues that arise, e.g. how might they be made moreefficient?

I gather ideas on the board and use the information from the page called Supportinghistorical information (Activity sheet 1) as appropriate and say: “The Egyptians were oneof the first civilisations, around 3000 BCE to start to write their numbers more efficiently.They used to carve hieroglyphics into the walls of their temples.”

I show Activity sheet 3 as an example.

I continue: “Let’s look at how they wrote their numbers using hieroglyphics. Thehieroglyphics used varied over the different eras of Ancient Egyptian history, but alwaysfollowed a pattern similar to that on Activity sheet 3.”

Working in groups (1)Deciphering hieroglyphics

I give out Activity sheet 4 and ask learners to work in pairs or threes and work out whatnumbers the hieroglyphics stand for (Activity sheet 5 can be used to help).

An online number translator can be used to convert our numbers to hieroglyphics.

Suggestion: Learners could make up some Egyptian numbers for other people todecode or could set up some questions to add, subtract or multiply by 10.

The ancient Egyptians did not mind whether the numbers were written left to right, rightto left or vertically, although they did tend to put them in order of highest value symbol tolowest value symbol.

Whole group discussion (2)Comparing hieroglyphics to the decimal system

I display Activity sheet 5. Note that some background information is available on thefollowing page Supporting historical information (Activity sheet 5).I then ask them to discuss the meaning of the individual symbols and how they arepositioned within the numbers.

I ask: “How it is the same as, and different to, our modern number system”?

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same: it is a decimal system different: we have individual digits from 0 - 9, the Egyptian use a form of tallying, and

juxtaposition different: the order matters in our system but not in the Egyptian system different: if there are no 10s (for example) the Egyptians don’t have any there whereas

we put a zero to mean ‘no 10s’

The next interesting development in number systems was due to the Babylonians.

I show Activity sheet 6. I tell them that this is a tablet (known as Plimpton 322) and isfrom approximately 1800 BCE. It shows tables of numbers written in cuneiform script,formed by pressing a specially shaped stick into a clay tablet to form each symbol.

I make use of the following page: Supporting historical information (Activity sheet 6) asappropriate.

Working in groups (2)Deciphering Babylonian cuneiform

I hand out Activity sheets 7 and 8. (N.B. These could be cut out and stuck together backto back to form the tablet.)

I ask learners to decode it, working in pairs or threes.

it is the 9 times table.

Whole group discussion (3)Comparing Babylonian cuneiform to the decimal system

I gather relevant information together on the board.

e.g. what 10 looks like only 2 symbols used how is it the same and different to our number system? how is it the same and different to the Egyptian number system?

I ask individuals to write a selection of numbers which lie between 1 and 59 on theboard.

I ask what happens at 60. I get them to look at 7 x 9 = 63 on the tablet.I discuss the two different uses of the symbol for 1 and the problems that might occur.For example, how would the Babylonians write exactly 60?

Working in groups (3)Working with Babylonian cuneiform numbers

I write some numbers on the board and ask everyone to write them down in cuneiforme.g. 17, 42, 59, 68, 75, 123, 140 and to check if they have the same as their neighbourswhen finished.

I encourage learners to set other numbers for their neighbour to encode/decode simplearithmetic questions in cuneiform involving addition, subtraction and multiplication by10.

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Reviewing and extending learningComparison of number systems (homing in on lack of zero as a place holder)

Finally, I hold a whole group discussion on how the Babylonian numbers compare to theEgyptian number system and to our number system. In what ways are they the sameand different?

Babylonians only use two symbols. the Babylonian system uses ‘place value’, i.e. the order of the symbols is important. they leave a space if there are no sixties, for example. they count in groups of 60 (called base 60).

Draw out issues relating to (lack of) zero as a place holder.

It is said that the Babylonians thought there were 360 days in a year so a full circle wassplit into 360 degrees, which worked very well with their number system. It is alsothought that the fact we have 60 minutes in an hour and 60 seconds in a minute stemsfrom the Babylonians (however we are not sure).

The Egyptians appeared to be ahead of the game here. They also had 360 days in ayear. One third of the year was for the flood; one third for planting crops and one thirdfor the harvest. But they knew there were 365 days and used the extra 5 days forfestivals.

At this point it would be possible to move onto looking at numbers written in differentbases if wished.

Which of methods of writing numbers we have looked at is best and why?

Reflection

Learners often struggle with the idea of place value. By seeing how numbers developedthey can appreciate the need for a zero as a place holder.

What learners might do next:

You could ask learners:

to create sets of cards to match numbers from different systems to investigate other ancient number systems e.g. Mayan, Roman, Chinese to investigate Egyptian mathematics e.g. working with unit factions, perfect

numbers, Egyptian multiplication problems from the Rhind Papyrus to investigate mathematics Babylonians did e.g. using tables of reciprocals to

work with fractions, using tables of squares and cubes to perform calculations(Plimpton 322)

Further ideas:

Other modules that use a similar approach are found at the History of MathematicsMathemapedia entry at the NCETM portal.

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Artefacts and resources:

Ishango BoneInformation used in this lesson was taken from Wikipedia, though it should be noted thatthere is some dispute about statements made (see the Discussion tab). However theRoyal Belgian Institute of Natural Sciences, where you can see the bone, provide a lotof the same information. You can also get the same information in French. See also anarticle by Simon Singh.

Egyptian mathematics sites for teachersMacTutor History of Mathematics has a section on Egyptian mathematics.

Egyptian mathematics sites for learners and teachersJo Edkin provides a page on Egyptian numbers. Multiplication is done slightly differentlyto what is shown here.Jo Edkin also offers a page on many number systems, including Egyptian.Papyrus paper with copyright clearance at Wikipedia.Mark Millimore’s website offers many things for teaching about the Egyptians includingan ancient Egyptian calculator, photos and other products.The Eyelid website also offers some Egyptian problems translating numbers intosymbols. They offer the symbols below (and you can use these “I have no objection topeople using the material on this site for Educational, non-profit purposes provided I'mcredited with a link back to this site”.)

The British Museum also provides more on Egypt including information about thepyramids.The Great Scott! Site provides a hieroglyph translator and also translates numbers intoEgyptian.

Plimpton 322Information in this lesson is taken from Plimpton 322.Plimpton on display at the University of Pennsylvania, November 2010 to January 2011.Maor, E. (2002). Trigonometric delights. Princeton University Press. This includes achapter on Plimpton 322.The Plimpton 322 website at Columbia University (where it is kept) contains three parts:Introduction and bibliography; Transcriptions of the tablet; and Interpretation andpossible construction.Robson, E. (2002). Words and Pictures: New Light on Plimpton 322

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Activity sheet 1Ishango bone

Image provided by the Royal Belgian Institute of Natural Sciences.

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Supporting historical information (Activity sheet 1)

The Ishango bone is now housed at the Royal Belgian Institute of Natural Sciences in Brussels.

Although it was found amongst tools for hunting, fishing and food production it looks like it has more significance than simply tallying. The marksare grouped together in blocks as follows:

Top row: 9 19 21 11Bottom row: 19 17 13 11

Both rows total 60.The top row could be evidence of working in base 10 i.e. 10-1 20-1 20+1 10+1The bottom row list primes between 10 and 20.

The obverse (back) is marked 7 5 5 10 8 4 6 3

and could be evidence of doubling strategies i.e. 3,6 4,8 5,10.

For more information see:George, G. J., 1992. The Crest of the Peacock: The Non-European Roots of Mathematics. Penguin Books. Ch. 2.Mathematics of the African DiasporaRoyal Belgian Institute of Natural SciencesWikipedia - Ishango bone

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Activity sheet 2Ishango bone (simplified)

Traced from image at: Wikipedia on the History of Computers in Education.

Development of number through the history of mathematics

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Activity sheet 3Egyptian hieroglyphics

This hieroglyphic is found in the Louvre museum in Paris.

Development of number through the history of mathematics

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Activity sheet 4Egyptian numbers 1

These numbers are written the Egyptian way below. Try to work out what whichnumber matches each Egyptian number.

1 2 3 8 1014 17 21 26 319

1045 10,200 300,050 2,000,000

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Activity sheet 5Egyptian numbers 2

Taken with permission, from the Eyelid website.

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Supporting historical information (Activity sheet 5)

Egyptian numbers were written in three different ways, hieroglyphics (pictorial), hieratic (symbolic) and demotic (popular) script. The mostcommonly known is the pictorial or hieroglyphics where some say the pictures some significance to their meaning for example a 10 isrepresented by a piece of rope in a horseshoe and 100 is a coil of rope. (see Rope Stretchers). Less well known is hieratic which is based onpowers of 10. It is thought that hieratic became more popular as a method of writing on expensive papyrus as it uses significantly less symbolsfor the same number.

With hieroglyphics, they didn't need zero as a place holder because each place is represented by a different symbol. Also there is no reasonwhy symbols should be written in any particular order, although they were usually placed from left to right in descending value order and ifnecessary form top to bottom. However the examples in the exercise are not always in value order in order to emphasise that the order doesn'tmatter.

For more information see:George, G. J., 1992. The Crest of the Peacock: The Non-European Roots of Mathematics. Penguin Books. Ch. 2.MacTutor History of Mathematics website entry for Egyptian numerals.Wikipedia entry on Egyptian numerals.

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Activity sheet 6Babylonian cuneiform writing (Plimpton 322)

Source: Plimpton 322 at Wikipedia.

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Supporting historical information (Activity sheet 6)

The Babylonians lived in Mesopotamia, the region between the Rivers Euphrates and Tigris, i.e. modern day Iraq. Mathematics developedthere from around 3500 BCE.

Babylonian numbers are base 60. They use a place value system. Originally there was no symbol for zero and so a space was left as a placeholder. Later on a small double triangle symbol was used as a place holder. Activity sheets 7 & 8 uses spaces.

There are about half a million clay tablets still in existence from the Babylonian era in museums around the world showing how the Babyloniansused tables of square, cubes and reciprocals to perform calculations as well as their familiarity with Pythagoras’ Theorem and completing thesquare.

The Plimpton tablet is described fully in Crest of the Peacock.

For more information see:George, G. J., 1992. The Crest of the Peacock: The Non-European Roots of Mathematics. Penguin Books. Ch4MacTutor History of Mathematics website entry for Babylonian numerals.Wikipedia entry on Babylonian numerals.Also see page 7 for more references to Plimpton 322.

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Activity sheet 7Babylonian numbers – Clay tablet obverse

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Activity sheet 8Babylonian numbers – Clay table reverse