Imaginary and Complex Numbers 18 October 2010

Question: If I can take the, can I take the ? Not quite…. 

Answer??? But I can get close!

= Imaginary Number or

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

Example:

Your Turn:

*Powers of

*Powers of, your turn: Observations?

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: i ÷ 4 = 13 remainder 2 i 54 = i 2 = -1

Complex Numbers Real PartImaginary 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, i 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:

Examples, cont.

Your Turn:

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

Subtraction Example

Your Turn:

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

Multiplication Example 2

Your Turn:

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

Division Example 2

Your Turn:

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