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Transcript
Objectives Complex Numbers Section 2.1 i = the square root of negative 1 • In the real number system, we can’t take the square root of negatives, therefore the complex number system was created. • Complex numbers are of the form, a + bi, where a = real part & bi = imaginary part. • If b = 0, a + bi = a; therefore a real number (thus reals are a subset of complex #). • If a = 0, a + bi = bi; therefore an imaginary # (imaginary # are a subset of complex #). Multiplying Complex Numbers • • • • Add & subtract complex numbers Multiply complex numbers Divide complex numbers Perform operations with square roots of negative numbers Adding & Subtracting Complex Numbers • Add real to real, add imaginary to imaginary (same for subtraction) • Example: (6+7i) + (3-2i) (6+3) + (7i-2i) = 9+5i • When subtracting, DON’T FORGET to distribute the negative sign! • (3+2i) – (5 – i) (3 – 5) + (2i – (-i)) = -2 + 3i EXAMPLE Find the product of (2 + 3i) • (3 – 6i) • Treat as a binomial times binomial. • BUT what is i times i? It’s -1!! Why?? • Let’s consider i raised to the following powers: i 2 = ( − 1) 2 = −1 i 3 = i 2 ⋅ i = (−1)i = −i i 4 = (i 2 ) 2 = (−1) 2 = 1 1 Dividing Complex Numbers • It is not standard to have a complex number in a denominator. To eliminate it, multiply by a wellchosen value of 1: ( conjugate/conjugate). EXAMPLE Divide. ( 3 – 8i ) ( 4 + 3i ) • The conjugate of a + bi = a – bi. • We use the following fact: (a + bi ) ⋅ (a − bi ) = a 2 − abi + abi − b 2i 2 = a 2 − b 2 ⋅ (−1) = a 2 + b 2 2