history of mathematics - National STEM Centre

Activity 2.10, page 22
1 0;30 2 +0;30 = 0;15 + 0;30 = 0;45
30 2 -30 = 15,0-30 = 14,30
2 jc 2 +jt = 0;45
3 The steps are
(jc + 0;30) 2 =1
(* + 0;30) = l
x = 0;30
4 To solve x2 - bx = c , take the coefficient of x, that is
b (ignoring the sign), halve it, \b, and square it to get
jb 2 . Add it to c to get jb 2 + c which is the square of
^b 2 +c . Add the \b to get jb + ^b 2 +c, which is
the result.
5 It is the algorithm of al-Khwarizmi in the Equations
and inequalities unit in Book 3.
6 The steps are
7 The ancient method is very similar to the modern one,
except that the idea of completing the square is kept
hidden, and the whole appears very much like a recipe,
rather like using the quadratic equation formula when
you don't know where it has come from. The major
difference is that the negative root is ignored.
8 Ilx 2 +7jc = 6;15 so (ILt) 2 +77x = 1,8; 45. Let
y = 1 Ijc, so y 2 + ly = 1,8; 45. The coefficient is 7, so
half is 3; 30. 3; 30x3; 30 = 12; 15. Add 12; 15 to 1,8; 45
to give 1,21. The square root is 9. As 9 - 3; 30 = 5; 30,
y = 5; 30 and x = 0; 30.
Activity 2.11, page 23
1 a x 2 +ox = b
b It is already in the list as type 1.
2 l(;c-.y) = 3;30 so (i(x-y)f = 12; 15, and
(^(x-y)) 2 +xy = 1,12; 15. Therefore
}(x-y)) +xy =8; 30 so
14 Answers
3 ±(x + y) = 3;l5 so (-^(x + y)) = 10; 33, 45, and
( 2 (x + y}} 2 ~ W = 3 '> 3' 45 • Therefore
and
x = 5
Activity 2.12, page 25
1 The only alternative is to use the Greek construction
methods which were not, so far as is known, available to
the Babylonians.
2 The radius is 3 1 ; 1 5 as on the tablet.
Activity 2.13, page 26
l-h
222
(/-3)+9 2 =/
90 - 61 = 0
90 = 61
1 = 15
Add 3 2 to 9 2 to find 61. Then divide by 2, then by 3 to
find /.
2
wall
15
15 2 = 3,45 and 9 2 =1,21. Then 2, 24 = (wall) 2 . To find
wall, take the square root of 2,24.
Activity 3.1, page 31
2 ABDC is a rhombus, and the diagonals of a rhombus
cut at right angles.
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