Addition and subtraction of complex numbers :
Suppose a, b, c, and d are real numbers. Then
Multiplying complex numbers :
Suppose a, b, c, and d are real numbers. Then,
Division of complex numbers :
Suppose a, b, c, and d are real numbers, with c + di ≠ 0. Then (a + bi)/(c + di)
= [(ac + bd) / (c2 + d2)] + [(bc − ad)(c2 + d2) i]
Example 1 :
Add the following complex numbers
(2 + 3i) and (3 - 4i)
Solution :
To add two complex numbers which is in the form a + ib and c + id, we use the following method.
a + ib + c + id = (a + c) + i(b + d)
That is, we have to combine the real part and imaginary part separately.
(2 + 3i) + (3 - 4i) = (2 + 3) + i(3 - 4)
= 5 - i
Example 2 :
Add the following complex numbers
(4 - 5i) and (-2 + 3i)
Solution :
(4 - 5i) + (-2 + 3i) = (4 - 2) + (-5i + 3i)
= 2 - 2i
Example 3 :
Add the following complex numbers
(-5 + 8i) + (9 - 11i)
Solution :
(-5 + 8i) + (9 - 11i) = (-5 + 9) + i(8 - 11)
= 4 - 3i
Example 4 :
Add the following complex numbers
(3 + 2i) and (-6 - 9i)
Solution :
(3 + 2i) + (-6 - 9i) = (3 - 6) + i(2 - 9)
= -3 - 7i
Example 5 :
Subtract
9 - 11i from 2 + 3i
Solution :
= (2 + 3i) - (9 - 11i)
= (2 - 9) + i(3 - 11)
= -7 - 8i
Example 6 :
Subtract
3 + 4i from 4 - 5i
Solution :
= (4 - 5i) - (3 + 4i)
= 4 - 5i - 3 - 4i
= (4 - 3) - 5i - 4i
= 1 - 9i
Example 7 :
Subtract
(-7 + 5i) from (-8 + 9i)
Solution :
= -8 + 9i - (-7 + 5i)
= -8 + 9i + 7 - 5i
= -8 + 7 + 9i - 5i
= -1 + 4i
Example 8 :
Subtract
(-11 - 13i) from (-8 - 9i)
Solution :
(-8 - 9i) - (-11 - 13i) = -8 - 9i + 11 + 13i
= (-8 + 11) + i(-9 + 13)
= 3 + 4i
Complete any two and write in standard form.
Example 9 :
-6 - (-2 - 2i) - (5 - 4i)
Solution :
= -6 - (-2 - 2i) - (5 - 4i)
Using distributive property, distributing negatives we get
= -6 + 2 + 2i - 5 + 4i
= -6 + 2 - 5 + 2i + 4i
= -9 + 6i
Example 10 :
-4(-7 + 8i)(-5 + 6i)
Solution :
= -4(-7 + 8i)(-5 + 6i)
Using distributive property, distributing negatives we get
= -4(35 - 42i - 40i + 48i2)
= -4(35 - 82i + 48(-1))
= -4(35 - 82i - 48)
= -4(- 82i - 13)
= 328i + 52
Example 11 :
(-3 + i)2
Solution :
= (-3 + i)2
Looks like the algebraic identity,
(a + b)2 = a2 + 2ab + b2
= (-3)2 + 2(-3) i + i2
= 9 - 6i - 1
= 8 - 6i
Example 12 :
(9 + 4i)/6i
Solution :
= (9 + 4i)/6i
Multiplying both numerator and denominator by the conjugate of the denominator, we get
= [(9 + 4i)/6i][-6i/-6i]
= (9 + 4i)(-6i) / -36i2
= (-54i - 24i2) / -36(-1)
= (-54i - 24(-1)) / 36
= (-54i + 24) / 36
= -3i/2 + 2/3
= (2/3) + (-3i/2)
Example 13 :
(-5 + 5i) - (4 - 2i) + (-8 - 7i)2
Solution :
= (-5 + 5i) - (4 - 2i) + (-8 - 7i)2
(a - b)2 = a2 - 2ab + b2
= -5 + 5i - 4 + 2i + (-8)2 - 2(-8)(7i) + (7i)2
= -9 + 7i + 64 + 112i + 49i2
= 55 + 119i + 49(-1)
= 55 + 119i - 49
= 6 + 119i
Example 14 :
(4 + i)/(8 - 7i)
Solution :
= (4 + i)/(8 - 7i)
Multiplying both numerator and denominator by the conjugate of the denominator, we get
= [(4 + i)/(8 - 7i)][(8 + 7i)/(8 + 7i)]
= (4 + i)(8 + 7i)/(8 - 7i)(8 + 7i)
= (32 + 28i + 8i + 7i2) / (82 - (7i)2)
= (32 + 36i - 7) / (64 + 49)
= (25 + 36i) / 113
= (25/113) + (36i/113)
Example 15 :
(-7i)(-2 + 7i)(2 + 6i)
Solution :
= (-7i)(-2 + 7i)(2 + 6i)
= (-7i)(-4 - 12i + 14i + 42i2)
= (-7i)(-4 + 2i - 42)
= (-7i)(-46 + 2i)
= 322i - 14i2
= 322i - 14(-1)
= 322i + 14
= 14 + 322i
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
©All rights reserved. onlinemath4all.com
Jul 02, 25 07:06 AM
Jul 01, 25 10:27 AM
Jul 01, 25 07:31 AM