history of mathematics - National STEM Centre

752
12 Summaries and exercises
Two texts which call for comment here represent versions of the same
exercise, and they are significant, not only because of the abstract interest
in numerical relations indicated by the enormous numbers involved, but
also because they concern the problem of irregular numbers.
Number 671 is written by a bungler who succeeded in making half a dozen
writing errors in as many lines ...
Jestin Number 50
se guru 7:l
sila 3 7
Jestin Number 671
(rev) se sila 3 7 ! guru 7
lu: 1 su ba-ti lu: 1 su ba-ti
lu-bi (obv) gurus
45,42,51 45,36,0 (written
on three lines)
se sila 3:3 su ?-tag 4 ?
Translation of Number 50
The grain (is) 1 silo.
7sila (7 liters)
each man received.
Its men:
45,42,51.
3 sila of grain (remaining)
A silo (guru) in this period contained 40,0 gur, each of which contained 8,0
sila. The correct answer seems to have been obtained by the following
process:
(1) 5,20,0,0 times 0; 8,34,17,8 = 45,42,51; 22,40
(2) 45,42,51 times 7 = 5,19,59,57
(3) 5,20,0,0-5,19,59,57 = 3
a Explain in your own words what problem the students have been asked to solve.
b Explain where the number 5,20,0,0 comes from.
c Explain step (1) in your own words.
d Verify that 45,42,51 times 7 equals 5,19,59,57.
e What mathematical error has the pupil apparently made in Jestin 671 ?
5 Give a geometric and an algebraic description of the Babylonian method of
solving an equation of the form x +ax = b.
3 An introduction to Euclid
Chapter summary
• how to use the Greek rules to carry out constructions for drawing a
perpendicular, bisecting an angle, bisecting a line segment, drawing a
perpendicular to a line segment (Activities 3.1 and 3.2)
• how to construct a square equal in area to any given polygon (Activity 3.3)
• the significance of the three classical problems of ancient Greece (Activity 3.4)
• some Euclidean proofs to work through (Activities 3.5 and 3.6)
• how Euclid's approach established an axiomatic basis for geometry (Activities
3.7 and 3.8).
Practice exercises
1 Given a line segment AB and a straight line /, use Euclid's rules to construct
points C and D on / such that CD = AB.