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Section 2-5 Complex Numbers Section 2-5 • • • • complex numbers and i operations with complex numbers (+,-, x) complex conjugates and division solving quadratic equations with complex solutions • plotting complex numbers • absolute value of complex numbers Complex Numbers • we learned back in Algebra 1 that the square root of a negative number is not a real number • there is a way to work with these numbers using the imaginary unit, i • we use this simple definition: i 1 • for example: 9 3i and 8 2i 2 Complex Numbers • all numbers we work with are part of the set of complex numbers • this set consists of all real numbers and all imaginary numbers (contain i) • all complex numbers can be written in the form a + bi • a is the real part, b is the imaginary part Operations With Complex Numbers • to add complex numbers, add their like parts (same for subtraction) (a bi) (c di) (a c) (b d )i (a bi) (c di) (a c) (b d )i • to multiply complex numbers, use FOIL 2 • use the fact that i 1 Division of Complex Numbers • if a + bi is a complex number, then its complex conjugate is a – bi • in order to simplify the division of two complex numbers, multiply the top and bottom of the fraction by the conjugate of the denominator • use FOIL on both top and bottom; the bottom will no longer contain i Solving Quad.’s • now, when you solve a quadratic equation for which the discriminant is negative, you can find its complex solutions • the solutions will be complex conjugates Plotting Complex Numbers • complex numbers cannot be plotted on a single number line because they have both a real and imaginary part • instead, we plot them on a complex plane which looks a lot like a coordinate plane we use for ordered pairs • the axes of this plane are the real axis and the imaginary axis Plotting Complex Numbers b -5 + 3i a 3 – 6i Absolute Value of Complex Numbers • remember that absolute value means distance from 0 on a number line • for complex numbers, it’s the distance from the origin • we use the distance formula to compute it a bi a b 2 2