history of mathematics - National STEM Centre

62
The Arabs
Activity 5.6 Horner's method
overlooked by all the famous 'amicable number chasers' - to be
discovered in 1866 by an Italian schoolboy! Fortunately, there are other
approaches. The German mathematician Euler (1707-1783) found more
than 60 pairs, using methods he developed himself - methods that still
form the basis of present-day search techniques.
Algorithms for finding solutions to equations
The following two activities illustrate two ideas which originated in China and
which were used by Arab mathematicians and handed on to the West. The first is an
example of a numerical solution of higher-order equations by a Chinese method,
which culminated in the work of Chin Chiu Shao in 1247 entitled Chiu-chang suan-
shu, which appeared in its simpler versions in Arab mathematics about the same
time. Nowadays this method is known as 'Horner's method'.
In 1819, an English schoolteacher and mathematician, W G Horner,
published a numerical method of finding approximate values of the roots
of equations of the type f(x) = a0xh + alx"~ ] + ... + an_ { x + an =0. The
procedure that Horner re-discovered is identical to the computational
i scheme used by the Chinese over 500 years earlier.
Here is an example in which this method is used to solve x 2 + 252x - 5292 = 0.
First, find that there is a root between 19 and 20. Then make the substitution
y = x - 19 to obtain (y + 19) 2 + 252(y + 19) - 5292 = 0, or y 2 + 290y -143 = 0,
which you know has a solution between 0 and 1. Now say that
143
y = ——-— = 0.491, so that* = 19.491.
(1 + 290)
To find the cube root of 12 978 (that is, to solve the equation jc 3 -12 978 = 0), first
note that there is a solution between 20 and 30, so you substitute _y = x - 20 to
obtain the equation y 3 + 6Qy 2 + \200y- 4978 = 0, which has a solution between 0
and 10. Now check that this equation has a solution between 3 and 4, so write
z = .y-3 and obtain the equation z 3 + 69z 2 + 1587z-811 = 0. Then you can say
81 1
that z = ————— = 0.49, so that x = 23.49.
1+69+1587
Work in a group for this activity. Be prepared to make a short oral report on
question 3 at your next review.
1 Use Horner's method to find the square root of 71 824.
2 Use Horner's method to solve the equations
a x 2 -23x- 560 = 0
b x 6 + 6000 x = 191 246 976.