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Complex Numbers

Complex Numbers. 1.4. Imaginary Unit. i = √-1 i 2 = -1. Complex Number. a + b i Real Imaginary part part. a+b i. If b≠0 then a+bi is a complex number called an imaginary number If b=0 then a+bi is a real number If a=0 then bi is a pure imaginary number. Examples. -4 + 6i

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Complex Numbers

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  1. Complex Numbers 1.4

  2. Imaginary Unit • i= √-1 • i2= -1

  3. Complex Number a+bi RealImaginary partpart

  4. a+bi • If b≠0 then a+biis a complex number called an imaginary number • If b=0 then a+bi is a real number • If a=0 then bi is a pure imaginary number

  5. Examples • -4 + 6i • 2i = 0+2i • 3=3+0i

  6. Simplified and Standard form • Simplified and standard form • a+bi • If b contains a radical we write the i before the radical. • 7+3i 7+3√5 i

  7. Adding and Subtracting Complex Numbers Subtracting (a+bi)+(c+di)=(a+c)+(b+d)i (5-11i)+(7+4i) (5-2i)+(3+3i) (-5+i)-(-11-6i) (2+6i)-(12-i) Adding (a+bi)-(c+di)=(a+c)-(b+d)i

  8. Multiplying Complex Numbers • Use the distributive property and FOIL method • After completing the multiplication replace i 2 with -1 • 4i(3-5i) • (7-3i)(-2-5i)

  9. Complex Conjugates and division • For the complex conjugate a+bi, its complex conjugate is a-bi. The multiplication of complex conjugates results in a real number. • (a+bi)(a-bi)=a2+b2 • (a-bi)(a+bi)=a2=b2 • The goal of the division procedure is to obtain a real number in the denominator. • Example • 7+4i 2-5i

  10. Roots of negative number • The square root of 4i and-4i both result in -16. • In the complex number system, -16 has two square rolls, we use 4i as the principal square root. • Principal square root • =i • When performing operations with square roots of negative numbers, begin by expressing all square roots in terms of i.

  11. Operations involving square roots of negative numbers • 2

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