Mathematicians have designated a special
number 'i' which is equal to the square root of minus 1. Then, it follows that i2 =
-1. To determine the square root of a negative number (-16 for example), take the
square root of the absolute value of the number (square root of 16 = 4) and then multiply it by
'i'. So, the square root of -16 is 4i.
When a number has the form a + bi (a real number plus an imaginary number) it is called a
"complex number". How do complex numbers "crop up" in mathematics? A good example would be the
roots of the quadratic equation x2 -6x + 25 = 0 where the 2 roots are 3 + 4i and
3 - 4i. Can we be sure these are the roots of the equation?
Addition and subtraction of complex numbers pretty much follow the rules of basic arithmetic and
so we won't discuss these. Multiplication starts getting a little tricky. Consider:
Were you wondering - is division more difficult than multiplication? Sure is. First we must define
a new term - conjugate, whereby the conjugate of a + bi = a-bi. (Example - the conjugate
of 3 + 4i is 3 - 4i). The main principle to remember in complex number division is that we
multiply the "top" and "bottom" of the fraction by the conjugate of the denominator. Time for
an example don't you think ?
Now we move on to even greater difficulty. Time to define another term - modulus,
whereby the modulus of a complex number a + bi equals the square root of
(a2 + b2). The modulus of a complex number is generally represented by
the letter 'r' and so:
Finally, the 2 square roots of a complex number are:
Find the square root of 12 + 16i.
Even though you have a calculator that can do these calculations for you, now you know the procedures for complex number arithmetic.
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