A.6a Complex Numbers Get busy living, or get busy dying. -Red Complex Numbers Complex numbers are numbers of the form a bi where a is called the real part and b is the imaginary part i 1 i 2 1 i i 3 i4 1 i5 i i19 ? Complex Numbers To add complex numbers, add the real parts and add the imaginary parts together (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, follow the usual rules for multiplying two binomials. DOUBLE DISTRIBUTION!!! (a bi) (c di) a(c di) bi(c di) i 2 1 Complex Numbers Given the following complex numbers, perform the indicated operations. w 5 3i z 1 4i 1) w z 2) w z 3) w z Complex Numbers If z a bi is a complex number, then its conjugate, denoted by z , is defined as z a bi a bi zz a2 b2 Complex Numbers Let’s find the conjugate of the following complex numbers and then multiply by the conjugate. 1) z 3 4i 2) z 1 8i 3) z 2i Complex Numbers We can apply multiplying conjugates to quotients of complex numbers. Let’s write the following in standard form! 2 3i 4 3i Complex Numbers Given the following complex numbers, perform the indicated operations. w 2 3i z 5 2i 1) w z 2) w z 3) w z w 4) z Solving Equations Let’s recall solving the following equation. x2 4 0 If we extend our number system to include complex numbers, we can solvethe previous equation. How cool is that! If N is a positive real number, we define the principal square root of –N, denoted by N , as N Ni x 4 N In Complex Number System Evaluate the square root of the following negative numbers in the complex number system. 1) 36 2) 8 3) 32 Equations in Complex Number System Solve the following equation in the complex number system. 2x 3 3x 2 6x 9 0 Equations in Complex Number System Solve the following equation in the complex number system. x 3 6x 2 12x 0 Equations in Complex Number System Solve the following equation in the complex number system. 2x 6 2x 4 24 x 2 0 Equations in Complex Number System Solve the following equation in the complex number system. 10x 2 6x 1 0 Equations in Complex Number System 2x 3 3x 2 6x 9 0 x 3 , 3i 2 x 3 6x 2 12x 0 x 0, 3 3i 2x 6 2x 4 24 x 2 0 x 0, 0, 3, 2i 3 1 x i 10 10 10x 2 6x 1 0 Now that we have extended our number system to include complex numbers, we should get as many solutions as the degree of the polynomial we are trying to solve! Character of Solutions For the equation ax 2 bx c 0 b 2 4ac 0 the equation has two unequal real solutions 2 b 4ac 0 the equation has a repeated real solution the equation has two complex solutions that b 4ac 0 are not real and are conjugates of each other 2 Homework: p.1008 #9 – 31 odd, 39 – 45 odd, 24, 28 A.6a Complex Numbers Get busy living, or get busy dying. -Red

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# A.6a Complex Numbers Get busy living, or get busy dying. -Red