Graphics Reference
In-Depth Information
−
4
−
3
−
2
−
1
0
1
2
4
3
Fig. 2.2.
Rotating numbers through 180
◦
reverses their sign.
So the letter i represents an anticlockwise rotation of 90
◦
. Therefore i2 is
equivalent to lifting 2 out of the number line, rotating it 90
◦
and leaving it
hanging in limbo. But if we take this '
imaginary
' number and subject it to a
further 90
◦
rotation, i.e. ii2, it becomes
−
2. There
for
e, we can write ii2 =
−
2,
1.Butifthisisso,i=
√
−
which means that ii =
1!
This gives rise to two types of number: real numbers and complex num-
bers. Real numbers are the everyday numbers we use for counting and so on,
whereas complex numbers have a mixture of real and imaginary components,
and help resolve a wide range of mathematical problems.
Figure 2.3 shows how complex numbers are represented: the horizontal
number line represents the
real component
, and the vertical number line rep-
resents the
imaginary component
.
For example, the complex number
P
(1 + i2) in Figure 2.3 can be rotated
90
◦
to
Q
by multiplying it by i. However, we must remember that ii =
−
−
1:
i(1 + i2) = i1 + ii2
=i1
−
2
=
−
2+i1
2 + i1) can be rotated another 90
◦
to
R
by multiplying it by i:
Q
(
−
i(
−
2 + i1) = i(
2) + ii1
=
−
i2
−
1
=
−
−
1
−
i2
i2) in turn, can be rotated 90
◦
to
S
by multiplying it by i:
R
(
−
1
−
i(
−
1
−
i2) = i(
−
1)
−
ii2
=
i1 + 2
=2
−
−
i1
