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Find complex number solutions by applying rules for graphing and performing complex number operations. • • • • • 𝑖 = −1 So, 𝑖 2 = −1 𝑖 3 = 𝑖 2 ∙ 𝑖 = −𝑖 𝑖 4 = 𝑖 2 ∙ 𝑖 2 = −1 ∙ −1 = 1 Keep this in mind for simplifying imaginary numbers. • Imaginary numbers exist wherever you have the square root of a negative number. • −𝑎 = −1 ∙ 𝑎 = −1 ∙ 𝑎 = 𝑖 𝑎 • −18 • −1 ∙ 2 ∙ 3 ∙ 3 • 3𝑖 2 • You try: −12 • 2𝑖 3 • So is it true that −64 = − 64? • No! • Any number of the form a + bi where a and b are real numbers and b≠0 • If b = 0 then the number is a real number. (ex: 6+0i = 6) • If a = 0 and b ≠ 0 then the number is a pure imaginary number. (ex: 0 + 3i = 3i) • Imaginary numbers and real numbers make up the set of complex numbers. • The x-axis represents the real axis • The y-axis represents the imaginary axis • A complex number is of the form a+bi • a is the real part and b is the imaginary part • Ex: graph the number 3 − 2𝑖 • This is represented by the point (a, b) or (3, -2) • The absolute value of a complex number is its distance from the origin. • 𝑎 + 𝑏𝑖 = 𝑎2 + 𝑏 2 • Think Pythagorean Theorem • Find the graph and absolute value of each: • -5 + 3i • Graph the point (-5,3) and the absolute value is 34 • 6i • Graph the point (0,6) and the absolute value is 6 • To add and subtract, combine like terms. • If you are subtracting, distribute the negative first • Ex: (5 – 3i) – (-2+4i) = 7-7i • To multiply, you distribute. • Use FOIL (First Outer Inner Last) if both numbers have two terms. • Ex: 4 + 3𝑖 −1– 2𝑖 = −4 − 8𝑖 − 3𝑖 − 6𝑖 2 • Remember that 𝑖 2 = −1 so we have 2 − 11𝑖 • Odds p.253 #9-25 • Complex Conjugates are the number pairs 𝑎 + 𝑏𝑖 and 𝑎 − 𝑏𝑖 • The product of complex conjugates is always a real number 𝑎2 + 𝑏 2 • To divide complex numbers, you must multiply by the conjugate of the denominator. • 9+12𝑖 Ex: 3𝑖 • 4 − 3𝑖 • 2+3𝑖 You try: 1−4𝑖 10 11 •− + 𝑖 17 17 needs to be multiplied by −3𝑖 −3𝑖 • Factor 2𝑥 2 + 32 • Factor out the GCF: 2(𝑥 2 + 16) • We know how to factor a difference of squares, but now we have sum of squares. This means the factors are the conjugates (x + 4i) and (x – 4i) • So we have 2(𝑥 + 4𝑖)(𝑥 − 4𝑖) • Find the solutions to 2𝑥 2 − 3𝑥 + 5 = 0 • Easiest to use quadratic formula. • 𝑥= −(−3)± (−3)2 −4(2)(5) 2(2) • 𝑥= 3± 9−40 4 • 𝑥= 3± −31 4 3 4 • 𝑥= ± 31 𝑖 4 • Odds p.253 #27-53