history of mathematics - National STEM Centre

14 Answers
b The required number is -2 as 2 + (-2) = 0 . The same
is true if the addition were carried out in the reverse
order. For this reason -2 is called the inverse of 2 under
addition since under addition 2 and -2 combine to give
the identity under addition. All numbers have an inverse
under addition; given any number, it is always possible to
find another which will combine with it under addition to
give 0.
c The number which is the inverse of 2 under
multiplication is j since 2x^ = ^x2 = 1. It is not
always possible to find an inverse under multiplication
for every number. Take 0 for example: there is no
number which when multiplied with 0 gives the identity
under multiplication.
d The statement is not true because 0 does not have an
inverse under multiplication. It would be true if it were
modified to the following.
If a non-zero number is chosen at random, then the
axioms require that it is always possible to find another
non-zero number which, when combined with the
original number using the operation x , gives 1 .
Activity 11.6, page 142
1 a The number 0 is an identity under f since it
satisfies f(a,0) = f(0,fl) = a.
The number 1 is an identity under g since for any a it
satisfies g(a,l) = g(l,a) = «•
For every number a there is another number, —a, which
combines with it under f so that f(a,—a) = f(-a,a) = 0.
For every non-zero number a there is another number, — ,
a
which combines with it under g so that
b It is true for f since the order in which we add
numbers together does not affect the outcome. For
example, 2 + 3 is the same as 3 + 2 . This is true in
general and as a result addition of numbers is said to be a
commutative operation.
The statement is true for g too since multiplication of
numbers is a commutative operation.
Activity 11.7, page 143
1 Complex numbers a + i b consist of a real part a and
an imaginary part b. The number b can be zero so that the
imaginary part can be zero. So all real numbers are at the
same time complex numbers, they are special complex
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numbers in the sense that the imaginary part is zero.
Zero, the identity under addition, is thus a perfectly valid
complex number which performs the same function in
complex numbers as it does to real; that is, when zero is
added to any complex number it leaves that complex
number unchanged. For example:
(2 + 3i) + (0 + Oi) = 2 + 3i . (0 + 0 i), or just plain 0, is the
identity under addition.
In a similar way (l + Oi), or 1, is the identity under
multiplication.
2 The inverse under addition of 2 + i is (-2 -i) since
The inverse under multiplication is a bit trickier to find.
You are trying to find a complex number which when
multiplied by 2 + i gives (1 + Oi) . Suppose that the
inverse is a + ib then
(2 + i)(a + ib) = 2a + ia + 2ib + i 2 b
= (2a-b) + (a + 2b)i
This should be (1 + Oi) and so you know that 2a-b = \
and a + 2b = 0 . You can solve these simultaneously to
find that a = ^,b = -\.
Activity 11.9, page 144
1 a To build up a step of length 1 2 to the left from the
basic building blocks: first take a step of length 1 to the
right, increase its length by multiplying by 1 2 and then
reverse the direction by incorporating a minus sign in
front which is effectively multiplying by - 1 . Thus
l-> 12-^-12.
b When tripled, a step of 2 to the left becomes a step of
6 to the left; reversing produces a step of 6 to the right;
then doubling produces a step of 1 2 to the right.
Using Clifford's notation, this is
&2(r3(-2)) = +12 , or in usual notation
The same outcome can be achieved by for example
taking the step of the 2 to the left, reversing it, then
tripling and then doubling. The order of the tripling and
doubling operations can be interchanged and they can
take place before or after the reversal. The reversal
operation commutes with the other operations and they
commute with each other, that is, the order in which they
are carried out does not affect the outcome.
2 a To do this you take the standard step and reverse it
and then multiply it by one-half, or vice versa.