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Name: ________________________ Date: ___________________ Period: ____________ Algebra 2: Section 4.8 Complex Numbers Notes In the past we said that you could not take the square root of a negative. Well we can!! The imaginary unit i represents the number whose square root is -1 i 1 and i2 = -1 Rewriting imaginary numbers For any positive real number k, k 1 k i k . Practice: 1) Evaluate 25 2) Evaluate 3) Evaluate 48 4) Evaluate 12 36 2 Complex Numbers Complex numbers are numbers that can be written in the form a + bi, where a and b are real numbers The real number a is the ________ part of the complex number. The real number bi is the ________ part of the complex number. The standard form of a complex number is a + bi. Practice: 5) Write in standard form: 18 2 36 3 6) Write in standard form: 18 2 25 10 Add and Subtract Complex Numbers To add and subtract complex numbers, add “like terms”: a bi c di a c b d i Practice: 7) 2 3i 6 7i 8) 5 9) 4 2i 2 7i 36 2 49 Multiply Complex Numbers To multiply complex numbers, use the distributive property, then combine “like terms” One thing to keep in mind: the product property of radicals does not apply when the radicand is a complex number, so evaluate the square root of the negative number first, then multiply the radicals. Correct: 25 4 25i 4i Incorrect: 25 4 5i 2i 10i 2 10 Practice: 100 81 10) 11) 2i 5 3i 12) 5 2i 1 3i 25 4 100 10 2 2 2 2 i i 2 2 2 2 13) Divide Complex Numbers The complex number a + bi has a _____________________ a – bi. When you multiply a complex number by its conjugate you get a _________ number. Example: 3 4i 3 4i To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. Practice: 14) 2i 4 3i 15) 6 5i 7i 16) 9 12i 3i 17) 9 12i 3i Using this new knowledge we can provide more specific solutions: 18) 5m 2 14 4 19) 2 x 2 18 72 To compute powers of i, use an exponent that is the remainder of dividing the original exponent by 4. This works because the powers of i repeat themselves every four factors of i, as can be seen from making a list of powers of i. 20) i 27 21) i 38 22) i 150 23) i 111