Complex Numbers
Digital Lesson
Definition: Complex Number
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The letter i represents the numbers whose square is –1.
i =
Imaginary unit
If a is a positive real number, then the principal square root of �negative a is the imaginary number i .
= i
Examples:
= i
= 2i
= i
= 6i
The number a is the real part of a + bi, and b is the imaginary part.
A complex number is a number of the form a + bi, where a and�b are real numbers and i = .
i2 = –1
Examples of Complex Numbers
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Examples of complex numbers:
Real Part
Imaginary Part
a
bi
+
2
7i
+
20
3i
–
Real Numbers: a + 0i
Imaginary Numbers: 0 + bi
a + bi form
+ i
=
4 + 5i
=
+ i
=
Simplify using the product property of radicals.
Simplify:
= i
= i
= 3i
1.
= i
= 8i
2.
+
3.
Add or Subtract Complex Numbers
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To add or subtract complex numbers:
1. Write each complex number in the form a + bi.
2. Add or subtract the real parts of the complex numbers.
3. Add or subtract the imaginary parts of the complex numbers.
(a + bi ) + (c + di ) = (a + c) + (b + d )i
(a + bi ) – (c + di ) = (a – c) + (b – d )i
Adding Complex Numbers
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Add (10 + ) + (21 – )
= (10 + i ) + (21 – i )
i =
= 31
Group real and imaginary terms.
a + bi form
= (10 + 21) + (i – i )
Examples: Add (11 + 5i) + (8 – 2i )
= 19 + 3i
Group real and imaginary terms.
a + bi form
= (11 + 8) + (5i – 2i )
Subtracting Complex Numbers
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Examples: Subtract: (– 21 + 3i ) – (7 – 9i)
= (– 21 – 7) + [(3 – (– 9)]i
= (– 21 – 7) + (3i + 9i)
= –28 + 12i
Subtract: (11 + ) – (6 + )
= (11 + i ) – (6 + i )
= (11 – 6) + [ – ]i
= (11 – 6) + [ 4 – 3]i
= 5 + i
Group real and �imaginary terms.
Group real and imaginary terms.
a + bi form
a + bi form
Product of Complex Numbers
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The product of two complex numbers is defined as:
1. Use the FOIL method to find the product.
2. Replace i2 by – 1.
3. Write the answer in the form a + bi.
(a + bi)(c + di ) = (ac – bd ) + (ad + bc)i
Examples
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= 5i2
= 5 (–1)
= –5
2. 7i (11– 5i) = 77i – 35i2
= 35 + 77i
3. (2 + 3i)(6 – 7i ) = 12 – 14i + 18i – 21i2
= 12 + 4i – 21i2
= 12 + 4i – 21(–1)
= 12 + 4i + 21
= 33 + 4i
Examples:
1.
= i i
= 5i i
= 77i – 35 (– 1)
Product of Conjugates
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The complex numbers a + bi and a - bi are called �conjugates.
Example: (5 + 2i)(5 – 2i)
= (52 – 4i2)
= 25 – 4 (–1)
= 29
The product of conjugates is the real number a2 + b2.
(a + bi)(a – bi) = a2 – b2i2
= a2 – b2(– 1)
= a2 + b2
Dividing Complex Numbers
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Replace i2 by –1 and simplify.
Dividing Complex Numbers
A rational expression, containing one or more complex numbers, �is in simplest form when there are no imaginary numbers remaining in the denominator.
Multiply the expression by .
Write the answer in the form a + bi.
Example:
–1
–1
Example:� (5 +3i)/(2+i)
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Replace i2 by –1 and simplify.
Multiply the numerator and �denominator by the conjugate of 2 + i.
Write the answer in the form �a + bi.
In 2 + i, a = 2 and b = 1. �a2 + b2 = 22 + 12
Simplify:
–1