Multiply and divide the complex numbers - Watch video (Algebra II)

Algebra II: Multiply and divide the complex numbers

This is a free lesson from our course in Algebra II


	In this lesson you'll learn how to multiply or divide the complex numbers. It is noted that the commutative, associative and distributive properties are used for working the product or multiplying the two complex numbers, together with i = -1. E.g. The two complex numbers z₁ = a₁ + ib₁ and z₂ = a₂ + ib₂ when multiplied yield: z₁z₂ = (a₁a₂ - b₁b₂) + i(a₁b₂ - a₂b₁). E.g. if z₁ = 4 + 7i and z₂ = 6 + 9i, then z₁z₂ = (4 + 7i)(6 + 9i) = -39 +78i. (More text below video...)

Other useful lessons:

Imaginary Unit

Plotting complex numbers on a 2D plane

Adding and Subtracting Complex Numbers

Classification of roots using the discriminant

Solving quadratic equation with imaginary roots

(Continued from above) Remember:
� the product of two complex numbers is a complex number.
� the product of a complex number and its conjugate is a real number, and is always positive.
Division is slightly trickier, because you want answer to have the form a + bi and not that of a ratio of such things (though a and b can be ratios).To get this you use the wonderful fact that any complex number multiplied by its complex conjugate (what you get by reversing the sign of its b) is a real number. You will follow basically the steps explained for rationalizing the denominator of a rational expression, as the fractions cannot have imaginary numbers in their denominator. Division is as follows:
z₁ z₂ = (a₁a₂ + b₁b₂)/ (a₂² + b₂²) + i(-a₁b₂ + b₂a₁) /(a₂² + b₂²)
For example, (3 + i)/(2 + 3i) = (9 � 7i)/13.

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