“Where did it come from?” - the historical background of (selected) topics from the BC K-7 Mathematics IRP
IRP Topic
People or cultures connected with topic, and approximate date of development
NUMBER CONCEPTS
Counting
Anthropologists believe that almost all cultures had some awareness of number, although this may have been as
primitive as “one, two, many”. Animal bones, notched with tally marks (often grouped in fives for the fingers of
a hand) have been dated to about 40,000 BC. Egyptian objects from 3000 BC provide the first written evidence
of a number system, and by this time the count extended to the millions.
Number system in use today
The “Hindu-Arabic”, base ten, place-value, ciphered system that we use today is based on the Hindu system that
had developed by about 600 AD. The Arabs adopted this and spread it throughout their Empire. Various
Europeans promoted it from the 11th century onwards, but it was not commonly adopted until after the invention
of the printing press in the fifteenth century. At this time, it was still a whole-number system. (see also
Decimal Fractions)
Place Value
The earliest place-value system was the base 60 system of the Babylonians (clay tablets found from c.1600 BC).
Later civilisations had base 10 (Chinese, Hindu) and base 20 (Mayan) place-value systems.
Even & Odd.
Multiple, Factor, Composite, Prime
Zero
Fractions
Decimal fractions
Negative numbers
The Pythagoreans (500 BC) believed that “Number is the substance of all things” and they investigated many
properties of numbers (even & odd, prime & composite, figurate numbers), possibly using pebbles to illustrate
the patterns.
The Babylonians introduced zero as a place holder around 300 BC, and the Mayans also had such a symbol by
400 AD. Zero was first accepted as a number by Hindu mathematicians around 850 AD. This proved to be a
hard concept to grasp, and even some of the most prominent mathematicians of the sixteenth and seventeenth
centuries were unwilling to accept zero a the solution to an equation.
The Egyptians used unit fractions. Hindu and Chinese mathematicians used symbols similar to those in use
today. In China (100 BC) the numerator was called the “son” and the denominator “mother”.
The Babylonians included numbers smaller than one as part of their base 60 place-value system (“sexagesimal
fractions”), and this method of fractions continued to be used by scholars until the end of the Middle Ages.
Decimal fractions occur quite early in Chinese documents (300 AD), but Steven (1585) was one of the first
European mathematicians to promote decimal fractions.
Over 2000 years ago, the Chinese calculated with rods of two colours (red for positive and black for negative),
but other early civilisations did not consider the existence of negative numbers. Even by the 16th century,
prominent European mathematicians referred to such numbers as “fictitious” or “absurd”, and negative numbers
were not fully accepted until the middle of the 19th century.
NUMBER OPERATIONS
Algorithms for addition, subtraction,
multiplication & division
+, - x, ÷ signs
Calculation aids
Many different algorithms (recipes) for the arithmetic operations have been invented, and even now the methods
of calculation vary from country to country. At first, people used to calculate on various types of abacus, and it
was not the Middle Ages that written calculations became common. Those used today derive from those the
Arabs used on their “dust boards” around 800 AD.
The signs used for operations today were invented over a period of about three centuries. The + sign appeared
first in the work of the French mathematician Oresme (c.1350), probably as an abbreviation of the Latin “et”.
The German mathematician Widman used the signs + and – in 1480, but these did not come into common usage
until after the 1557 publication of Robert Record’s Whetstone of Witte (the first English text to use them).. The
English mathematics Oughtred introduced the cross for multiplication in 1628. The present division sign,
originally used for subtraction, was first used for division by the Swiss mathematics Rahm in 1659.
Fingers were the first aid for calculation. Simple abacuses were probably constructed in all ancient civilisation
but the earliest one known is from Ancient Greece (600 BC). Around 500 BC, the Chinese introduced the
counting board and rods. In 1614 Napier developed logarithm tables and “Napier’s Rods” for multiplication. A
simple calculating machine was invented in 1623. Babbage conceived of the principles of modern “Computers”
in 1830, but the technology to construct them did not exist for another century. ENIAC (Electronic Numerical
Integrator and Computer) was built in 1945. The first pocket calculators appeared in 1971, with microcomputers
following 4 years later.
VARIABLES & EQUATIONS
“Equals” sign
In 1557, Robert Record (Whetstone of Witte) wrote “to avoid the tedious repetition of these words – is equal to- I
will set … a pair of parallels, or gemow [twin] lines of one length, thus
, because no 2 things can be
more equal”.
Simple, one variable equations
All the early civilisations had methods for solving equations that would now be written as (e.g.) 2x+4=10.
Algebraic notation
From the Babylonians times onwards, most algebraic equations were written entirely in words. By the 15th
century, symbolic expressions began to appear, but the “x “ and “y” notation commonly used today was
invented by Descartes in 1637.
MEASUREMENT
Standard and non-standard units
Metric units/SI units
From the earliest civilisations, people have developed their own units, often based on body measurements.
However, as soon as people needed to trade, systems of measurements had to be standardised. These varied
from country to country, and by the 18th century, the need for a single, universally accepted standard was
acknowledged.
The metric system was developed in France in the 1790s. In 1960, the International Systems of Units (SI) was
adopted, based on upon the metre and kilogram of the metric system, together with other coherent sets of base
units that had been developed over the previous century to measure such quantities as time and temperature.
Temperature
The Fahrenheit scale was introduced in 1724 by the inventor of the first mercury thermometer. The Centigrade
scale (which divided the temperature range between the freezing and boiling points of water into 100 degrees)
was developed by Celsius in 1742, and is officially named after him. The Kelvin scale was invented in 1848,
using centigrade degrees but starting at “absolute zero”.
3D OBJECTS & 2D SHAPES
Naming shapes and objects
Angle measurement
The Greek derivations of the words used to describe polygons and polyhedra shows the fascination of
geometrical shapes for this ancient civilisation. Other words, such as quadrilateral, have Roman origins.
The division of a circle into 360 degrees was devised by the Babylonians, who used a base 60 number system.
TRANSFORMATIONS
Translations, rotations & reflections Islamic Art showed an intuitive understanding of transformations, but they were not studied formally until the
(Slides, Turns and Flips)
Coordinate geometry
mid-19th century.
The idea of using a grid to mark points has been around since the days of the Ancient Egyptian surveyors, but the
format of a pair of co-ordinate lines (the “x-axis” and “y-axis”) is often called the Cartesian coordinate system,
after the 17th century French mathematician Descartes. However, the system he developed did not include a
vertical axis or negative numbers – these modifications evolved gradually over the next 150 years.
PROBABILITY
Intuitive probability (“more likely”
etc), and experimental probability.
Theoretical probability
From earliest times, gamblers have developed an intuitive idea of probability based on observing the results of
many dice games etc. These ideas form the basis of the concept of “experimental probability”.
Theoretical probability is generally considered to have been started by a question on how to settle an unfinished
gambling game, proposed by the Chevalier de Méré to Pascal in 1654. Pascal worked on the problem with
Fermat, and they not only solved the Chevalier’s problem but many investigated other games of chance and laid
the basis for formal probability theory.
A general reference for mathematics is <http://en.wikipedia.org/wiki/List_of_mathematical_topics>, which includes a some information on the history
of mathematics. Other useful historical references are
Earliest known uses of mathematical symbols,< http://jeff560.tripod.com/mathsym.html>
Earliest known uses of some of the words of mathematics,< http://jeff560.tripod.com/mathword.html
MacTutor history of mathematics archive, <http://www-groups.dcs.st-and.ac.uk:80/~history/>.
An excellent printed text is the book “Math through the Ages: A Gentle History for Teachers and Others” by William Berlinghoff and Fernando Gouvea
If you would like more information about the history of mathematics, please contact me at <[email protected]>
Irene Percival, October 2005