# HISTORY OF ALGEBRA

M AT H 1

WHAT IS ALGEBRA?

• Denotes various kinds of mathematical

ideas and techniques

• more or less directly associated with

formal manipulation of abstract symbols

and/or with finding the solutions of an

equation.

HISTORICAL OBJECTIVES

1. attempts to deal with problems devoted

to finding the values of one or more

unknown quantities.

2. the evolution of the notion of number

3. the gradual refinement of a symbolic

language

THE SEARCH OF “EQUATION”

• Egyptian Mathematics

Egyptian mathematical texts known to us dated

from about 1650 B.C.

• They attest for the ability to solve problems

equivalent to a linear equation in one unknown

• Later evidence, indicates the ability to solve

problems equivalent to a system of two

equations in two unknown quantities

THE SEARCH OF “EQUATION”

• Babylonian and Egyptian Mathematics

• Throughout this period there is no use of

symbols; problems are stated and solved

verbally, like in the following, typical example:

THE SEARCH OF “EQUATION”

• Method of calculating a quantity,

multiplied by 1 1/2 added 4 it has come to 10.

What is the quantity that says it?

Then you calculate the difference of this 10 to

this 4. Then 6 results.

Then you divide 1 by 1 1/2. Then 2/3 result.

Then you calculate 2/3 of this 6. Then 4 results.

Behold, it is 4, the quantity that said it.

What has been found by you is correct.

THE SEARCH OF “EQUATION”

• Babylonian Mathematics

• cuneiform texts preserved in clay tablets.

Babylonian arithmetic was based on a wellelaborated, positional sexagesimal system (base 60).

• BUT, no consistent use of zero.

• A great deal of Babylonian mathematics consists of

tables: multiplication and reciprocal tables, squares,

square and cube roots (though no cubes),

exponentials and others.

THE SEARCH OF “EQUATION”

• Babylonian Mathematics

• Beside tables, there are problem texts involving

the computation of an unknown number.

• These texts explain a procedure to be followed

in order to find the number.

• This is illustrated by a specific example, rather

than by abstractly describing its successive

steps.

• The starting point could be relations involving

specific numbers and the unknown, or its

square, or systems of such relations

THE SEARCH OF “EQUATION”

• Babylonian Mathematics

• Beside tables, there are problem texts involving

the computation of an unknown number.

• These texts explain a procedure to be followed

in order to find the number.

• This is illustrated by a specific example, rather

than by abstractly describing its successive

steps.

• The starting point could be relations involving

specific numbers and the unknown, or its

square, or systems of such relations

THE SEARCH OF “EQUATION”

• Greek Mathematics

• Proportion Theory, Elementary Arithmetic

• Greek mathematics was the discovery by the

Pythagoreans around 430 B.C. that certain ratios

among pairs of magnitudes do not correspond to

simple ratios among whole numbers.

• Proportions became a main tool of mathematics

in general.

THE SEARCH OF “EQUATION”

• Greek Mathematics

• The Greeks would state this in as strictly verbal

fashion.

• Even shorthand expressions, such as the much later

A:B :: R:S.

• The theory of proportions provided significant

mathematical results, yet it could not lead to

deriving.

• A main feature of Greek mathematics is that

comparisons or simultaneous

• manipulations can only be made among magnitudes

of the same kind

THE SEARCH OF “EQUATION”

• Diophanthus

• original methods for solving problems that in retrospect may be seen as linear or quadratic

• A problem whose solutions are all negative was called by him “absurd”.

Diophantus solved specific problems using ad-hoc

methods convenient for the problem at hand; he did not

• provide general methods suitable for some “standard” cases.

THE SEARCH OF “EQUATION”

• Diophanthus

• first to introduce some kind of...

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