Page 4-4
Simple operations with complex numbers
Complex numbers can be combined using the four fundamental operations
(+-*/). The results follow the rules of algebra with the caveat that
i
2
= -1. Operations with complex numbers are similar to those with real
numbers. For example, with the calculator in ALG mode and the CAS set to
Complex, we’ll attempt the following sum: (3+5i) + (6-3i):
Notice that the real parts (3+6) and imaginary parts (5-3) are combined
together and the result given as an ordered pair with real part 9 and
imaginary part 2. Try the following operations on your own:
(5-2i) - (3+4i) = (2,-6)
(3-i)(2-4i) = (2,-14)
(5-2i)/(3+4i) = (0.28,-1.04)
1/(3+4i) = (0.12, -0.16)
Notes:
The product of two numbers is represented by: (x
1
+iy
1
)(x
2
+iy
2
) = (x
1
x
2
- y
1
y
2
) +
i (x
1
y
2
+ x
2
y
1
).
The division of two complex numbers is accomplished by multiplying both
numerator and denominator by the complex conjugate of the denominator,
i.e.,
2
2
2
2
2112
2
2
2
2
2121
22
22
22
11
22
11
yx
yxyx
i
yx
yyxx
iyx
iyx
iyx
iyx
iyx
iyx
+
−
⋅+
+
+
=
−
−
⋅
+
+
=
+
+
Thus, the inverse function INV (activated with the Y key) is defined as
2222
11
yx
y
i
yx
x
iyx
iyx
iyxiyx
+
⋅+
+
=
−
−
⋅
+
=
+
Changing sign of a complex number
Changing the sign of a complex number can be accomplished by using the
\ key, e.g., -(5-3i) = -5 + 3i
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