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 z_{1} = a_{1} + ib_{1}
and z_{2} = a_{2} + ib_{2}
when multiplied yield: z_{1}z_{2} = (a_{1}a_{2}
 b_{1}b_{2}) + i(a_{1}b_{2}
 a_{2}b_{1}).
E.g. if z_{1} = 4 + 7i and z_{2} = 6 + 9i, then z_{1}z_{2} = (4 + 7i)(6 + 9i) = 39 +78i.
(More text below video...)
(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: z_{1}z_{2} = (a_{1}a_{2} + b_{1}b_{2})/ (a_{2}^{2} + b_{2}^{2}) + i(a_{1}b_{2} + b_{2}a_{1}) /(a_{2}^{2} + b_{2}^{2})
For example,
(3 + i)/(2 + 3i) = (9 – 7i)/13.
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