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DeSmet - Math 152
Blitzer 5E
∫ 7.7
1. Imaginary Numbers:
We have thus far said that
- Complex Numbers
−1 is a non-real number.
That is because x 2 = −1 has no real number solution. No number on the real number line will
satisfy that equation.
(a) In the 1500s, some mathematicians began experimenting with what we now call imaginary
numbers.
They wanted this equation to have a solution. So they gave it one, i.
That is i 2 = −1 . Which also implies that
−1 = i .
Although this seems ________, the implications are huge. i appears in applications to
electronics and electrical engineering (AC current), engineering, physics, vibration analysis,
and much more (a reason being that a “complex number” can describe anything involving 2
dimensions).
2. Definition: i 2 = −1 or
−1 = i .
3. Roots of Negative Numbers:
Now we can take the square root of negative numbers:
Example 1: Simplify.
(a)
−64
(b)
−13
(c)
−50
A number of the form bi where b is a real number is called an imaginary number.
Section 7.7
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DeSmet - Math 152
Blitzer 5E
4. Complex Numbers:
A number of the form a + bi where both a and b are real numbers is called a complex number.
a is called the real part.
b is called the imaginary part.
Note:
a + 0i is a real number. 0 + bi is a purely imaginary numbers. Both are complex numbers.
5. Adding/Subtracting/Multiplying Complex Numbers:
A complex number is simplified when it is written in standard form: a + bi .
We can add, subtract, and multiply complex numbers just as we do polynomials, treating i like x.
The only exception is i 2 = −1 .
( a + bi ) + ( c + di ) = ( a + c ) + (b + d ) i
( a + bi ) − ( c + di ) = ( a − c ) + (b − d ) i
c ( a + bi ) = ( ac ) + ( bc ) i
( a + bi ) ( c + di ) → FOIL
6. Example 2: Simplify the following, write all answers in a + bi form.
( a ) ( 5 − 2i ) + ( 3 + 3i )
(b ) ( 2 + 6i ) − (12 − 4i )
(c)
7i ( 2 − 9i )
( d ) ( 5 + 4i )( 6 − 7i )
( e ) ( 2 − i )2
Section 7.7
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DeSmet - Math 152
Blitzer 5E
7. Also note, when multiplying square roots of negative numbers, you must take the root first (thus
involve i first), then multiply! The reason is a b = ab only if a and b are non-negative.
Example 3: Simplify
−4 −16 .
8. Division of Complex Numbers:
(a) What is wrong with the number
2+i
?
3− i
(b) a + bi and _______________ are called complex conjugates.
Multiply:
( a ) ( 3 − i )( 3 + i )
(b ) ( a + bi ) ( a − bi )
9. Example 4: Divide and simplify to the form a + bi form.
(a)
2+i
3− i
Section 7.7
(b )
3 − 6i
5i
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DeSmet - Math 152
Blitzer 5E
10. Powers of i:
A special property of i is how its powers repeat.
i=
i5 =
i9 =
i2 =
i6 =
i10 =
i3 =
i7 =
i11 =
i4 =
i8 =
i12 =
Example 5: Simplify each expression.
(a) i 37
(b) i 61
(c) i 44
(d) i 50
Section 7.7
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