Imaginary and Complex Numbers

18 October 2010

Question:

If I can take the , can I take the ?

Not quite….

25 25

Answer???

But I can get close!

1512512525

= Imaginary Number or

????1

1 i

12 i

Simplifying with Imaginary Numbers

Step 1: Factor out -1 from the radicand (the number or expression underneath the radical sign)

Step 2: Substitute for

Step 3: If possible, simplify the radicand

i 1

Example: 8

Your Turn:

1. 2.

3. 36

2 12

*Powers of

4

3

2

1

i

i

i

i

4

3

2

)1(

)1(

)1(

1

i

1

1

i

i

*Powers of , your turn:

Observations?

i

Simplifying Powers of i

To simplify a power of i, divide the exponent by 4, and the remainder will tell you the appropriate power of i.

Example: i54

54 ÷ 4 = 13 remainder 2 i54 = i2 = -1

Complex Numbers

Real Part Imaginary Part

a + bi

where a and b are both real numbers, including 0.

Complex Number System

Complex Numbers: -5, -√3, 0, √5, ⅝, ⅓, 9, i, -5 + -4i

Imaginary Numbers:

-4i

2i√2

√-1

i

Real Numbers: -5, -√3, 0, √5, ⅝, ⅓, 9

Irrational Numbers

-√3

√5

Rational Numbers: -5, 0, ⅓, 9

Integers: -5, 0, 9

Whole Numbers: 0, 9

Natural Numbers: 9

The complex numbers are an algebraically closed set!

Writing Complex Numbers

Step 1: Simplify the radical expression Step 2: Rewrite in the form a + bi.

Examples:

i

i

36

63

619

619

69

Examples, cont.

327

327

347

347

1127

1127

127

i

i

i

i

Your Turn:

1. 2.

3. 4.

481 361

726 11136

Operations on Complex Numbers

Operations (adding, subtracting, multiplying, and dividing) on complex numbers are the same as operations on radicals!!! Remember: the imaginary number is really just a

radical with a negative radicand.

Addition and Subtraction of Complex Numbers

You can only add or subtract like terms. Translation: You must add or subtract the real

parts and the imaginary parts of a complex number separately.

Step 1: Distribute any negative/subtraction signs. Step 2: Group together like terms. Step 3: Add or subtract the like terms.

Addition Example

)4()42( ii

Subtraction Example

)88(6 i

Your Turn:

1.

2.

3.

4.

)5()97( ii )24()53( ii

)31()76( ii )(2)512( ii

Multiplication of Complex Numbers

Doesn’t require like terms! Translation: You can multiply real parts by

imaginary parts and imaginary parts by real parts! Multiply complex numbers like you would

multiply expression with radicals. Monomials: Group together like terms, then multiply. Binomials: FOIL!

Multiplication Example 1

)4)(5( ii

Multiplication Example 2

)53)(32( ii

Your Turn:

1.

2.

3.

4.

2)6( i)5)(2( ii

)72)(8( ii

2)49( i

Division of Complex Numbers

When dividing two complex numbers, use the same rules for rationalizing the denominator. Monomial: Multiply by the denominator over the

denominator. Binomial: Multiply by the conjugate of the

denominator. You must FOIL! The conjugate of a complex number is called

a complex conjugate.

Division Example 1

i

i

2

43

Division Example 2

i

i

32

3

Your Turn:

1. 2.

3. 4.

i

i32

i

1

i

i

26

42

i

i

2

*Your Turn:

5. 6.

7. 8.

)4)(32(

2

ii )2)(2(

1

ii

i

i

3

26

i

i

i 32

3

2

1