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Avon High School Section: 1.4 ACE COLLEGE ALGEBRA II - NOTES Complex Numbers Mr. Record: Room ALC-129 Day 1 of 1 The Imaginary Unit i The Imaginary Unit i The imaginary unit i is defined as i 1, where i 2 1. Equality of Complex Numbers a bi c di if and only if a c and b d Complex Numbers a bi For the complex number, 4 6i a, the real part is -4 Example 1 a. (2 6i ) (12 i) b, the imaginary part is 6 Real Numbers a bi with b 0 Operations with Complex Numbers Perform the given operation and simplify. Conjugate of a Complex Number The complex conjugate of the number a bi is a bi and vice versa. The multiplication of two complex conjugates gives a real number. a bi a bi a 2 +b 2 a bi a bi a 2 +b 2 Imaginary Numbers a bi with b 0 b. (7 3i)(2 5i) Example 2 a. Using Complex Conjugates to Divide Complex Numbers Divide and express the result in standard form. 5 4i 4i b. 7 4i 2 5i Roots of Negative Numbers The square root of 4i and the square root of 4i both result in 16 : (4i)2 16i 2 16(1) 16 (4i)2 16i 2 16(1) 16 Consequently, in the complex number system 16 has two square roots, namely 4i and 4i . We call 4i the principal square root of 16 Principal Square of a Negative Number For any positive real number b, the principal square root of the negative number b is defined by b i b Example 4 a. 27 48 Operations Involving Square Roots of Negative Numbers Perform the indicated operations and write the result in standard form. b. 2 3 2 c. 14 12 2