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Understanding Complex Numbers: Imaginary Unit, Addition, Subtraction, and Multiplication

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Dive into the fundamentals of complex numbers with an emphasis on the imaginary unit, defined as i = √-1. Learn how to identify real, imaginary, and pure imaginary numbers based on their components. Explore the processes of adding and subtracting complex numbers with practical examples, along with effective multiplication techniques using the distributive property and FOIL method. Additionally, grasp the significance of complex conjugates in division and discover how to handle square roots of negative numbers. Ideal for students and enthusiasts looking to master complex number operations.

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Understanding Complex Numbers: Imaginary Unit, Addition, Subtraction, and Multiplication

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Presentation Transcript

  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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