history of mathematics - National STEM Centre

64
The Arabs
Indian text. For over a thousand years, the Chinese used this method, refining and
extending it, and then passed it on to the Arabs who in turn gave it to the Europeans.
And the method is still used in modern numerical methods. Its Chinese antecedents,
however, are not well known in the West.
Arab algebra
Perhaps the best-known mathematician of the Arab era was al-Khwarizmi. His
work, Arithmetic, introduced the Hindu-Arabic system of numerals to the Arabs,
and thence to western Europe. Unfortunately, only translations survive.
The word al-jabr appears frequently in Arab mathematical texts that followed al-
Khwarizmi's influential Hisab al-jabr w'al-muqabala, written in the first half of the
9th century. Two meanings were associated with al-jabr. The more common one
was 'restoration', as applied to the operation of adding equal terms to both sides of
an equation, so as to remove negative quantities, or to 'restore' a quantity which is
subtracted from one side by adding it to the other. Thus an operation on the equation
2x + 5 = 8 - 3x which leads to 5x + 5 = 8 would be an illustration of al-jabr.
The less common meaning was multiplying both sides of an equation by a certain
number to eliminate fractions. Thus, if both sides of the equation %x + ^ = 3 + ^-x
were multiplied by 8 to give the new equation 18;c + 1 = 24 + 15;c, this too would be
an instance of al-jabr. The common meaning of 'al-muqabala' is the 'reduction' of
positive quantities in an equation by subtracting equal quantities from both sides.
So, for the second equation above, applying al-muqabala would give successively
1 = 24 + 15*
and 3x = 23 or x = ^.
Eventually the word al-jabr came to be used for algebra itself.
Notation
The Arabic word 'jadhir' meaning 'root' was introduced by al-Khwarizmi to denote
the unknown x in an equation. With the use of terms such as 'number' for constant,
'squares' for jc 2 , 'kab' (cube) for x , or combinations of these terms, the Arabs
were able to represent equations of different degrees. For example, you can translate
'one square cube and three square square and two squares and ten roots of the same
equal twenty' into modern notation as x 5 + 3x 4 + 2x 2 + lOx = 20.
Al-Khwarizmi distinguished six different types of equations which he then
proceeded to solve, providing both numerical and geometrical solutions.
The six different types of equation were
squares equal to roots
squares equal to numbers
roots equal to numbers
squares and roots equal to numbers
squares and numbers equal to roots
roots and numbers equal to squares
ax = bx
ax 2 =c
bx = c
ax 2 +bx = c
ax +c = bx
bx + c = ax 2