Complex Numbers Or I’ve got my “ i ” on you.

Real Numbers Imaginary Numbers Rational Numbers Irrational Numbers COMPLEX NUMBERS

Standard Form of a Complex Number a + b i REAL PART IMAGINARY PART

i 2 = -1 i 2 = -1 is the basis of everything you will ever do with complex numbers. i 2 = -1 is the basis of everything you will ever do with complex numbers. Simplest form of a complex number never allows a power of i greater than the 1 st power to be present, so ……… Simplest form of a complex number never allows a power of i greater than the 1 st power to be present, so ………

Simplifying Powers of i Simplification Simplest Form i None needed i i2i2 By definition- 1 i3i3 i 2 x i = -1 x i =- i i4i4 ( i 2 ) 2 = ( -1) 2 = 1 i5i5 ( i 2 ) 2 x i = ( -1) 2 x ii i6i6 ( i 2 ) 3 = ( -1) 3 = - 1 i7i7 ( i 2 ) 3 x i = ( -1) 3 x i- i i8i8 ( i 2 ) 4 = ( -1) 4 = 1

Simplification Examples i 42 = Divide exponent by 2 (42 ÷ 2 = 21 R 0) Quotient is exponent; Remainder is extra power of i Quotient is exponent; Remainder is extra power of i Write as power of i ( i 2 ) 21 Simplify = (-1) 21 = - 1

Simplification Examples i 27 = Divide exponent by 2 (27 ÷ 2 = 13 R 1) Quotient is exponent; Remainder is extra power of i Write as power of i ( i 2 ) 13  i Simplify = (-1) 13  i = - 1  i = - i

Adding/Subtracting Complex Numbers Adding and subtracting complex numbers is just like any adding/subtracting you have ever done with variables. Simply combine like terms. (6 + 8 i ) + (2 – 12 i ) = 8 – 4 i (7 + 4 i ) – ( i ) = i – 10 – 9 i = -3 – 5 i

Multiplying Complex Numbers This will be FOIL method with a slight twist at the end. An i 2 will ALWAYS show up. You will have to adjust for this. (4 + 9i)(2 + 3i) = i + 18 i + 27 i 2 = i – 27 = i (7 – 3 i )(6 + 8 i ) = i – 18 i – 24 i 2 = i + 24 = i

Binomial Squares and Complex Numbers You can still do the five-step shortcut, or you can continue to do FOIL. You will still have to adjust for the i 2 that will show up. (7 + 3 i ) 2 = i + 9 i 2 = i – 9 = i (8 – 9 i ) 2 = 64 – 144 i + 81 i 2 = 64 – 144i – 81= -17 – 144 i

D2S and Complex Numbers Situations that in the real numbers would have been differences of two squares (D2S) demonstrate in the complex numbers what are known as conjugates. (3 + 4 i )(3 – 4 i ) = (3) 2 – (4 i ) 2 = 9 – 16 i 2 = = 25 When conjugates are used, there will be no i in the answer.

Things Not Allowed in a Denominator Negative sign Radical Fractional Exponent Complex Number Each one of these must be adjusted out of the problem.

Clearing Complex Numbers from the Denominator If there is a pure imaginary number in the bottom, multiply by i with the opposite sign. Example: If denominator contains 3 i, multiply both sides by - i. – –Why?This also takes care of a negative in the denominator. 3 i  ( - i ) = -3 i 2 = 3 Example: If denominator contains -2 i, multiply both sides by i. -2 i  i = -2 i 2 = (-2)  (-1) = 2

Clearing the Denominator (continued) If the denominator is of the form a + b i, then multiply both sides of the fraction by the conjugate. If denominator contains (7 + 3 i ), multiply both sides by (7 – 3 i ). (7 + 3 i )(7 – 3 i ) = 49 – 9 i 2 = = 58 If denominator contains (8 – i ), multiply both sides by (8 + i ). (8 – i )(8 + i ) = 64 – i 2 = = 65

Simplifying Square Roots of Negative Numbers √ – 9 does not exist in the reals because there is no number that can be squared to give a negative answer. Therefore, you must use i 2 to replace the negative. √ – 9 = √ 9 i 2 = 3 i √ – 20 = √ 20 i 2 = √ 4  5  i 2 = 2 i√ 5

Multiplying Square Roots of Negative Numbers Any time multiplication of square roots involves the square root of a negative number, you MUST replace the negative with i 2 before doing any computation. √ 6  √ – 3 = √ 6  √ 3 i 2 = √ 18 i 2 = √ 9  2  i 2 = 3 i√ 2 √ – 2  √ – 8 = √ 2 i 2  √ 8 i 2 = √ 16 i 4 = 4 i 2 = – 4

Solving Equations in the Complex Numbers x = 0 Remember this equation that we used to show why a sum of two squares never factors in the reals? x 2 = - 4  √x 2 = √-4 x =  √-4=  √ 4 i 2 =  2 i Complex solutions always come in conjugate pairs.