1.4 Complex Numbers Review radicals and rational exponents. We need to know how to add, subtract, multiply and divide complex numbers.

Complex Numbers The imaginary unit i is defined as

Example Basically, when you have the square root of a negatively signed number, instead of writing “not a real number”, we will immediately take out the square root of negative one, which is i. Then simplify the radical as usual.

Complex Numbers The set of all numbers in the form a+bi with real numbers a and b, and i, the imaginary unit, is called the set of complex numbers. The real number a is called the real part, and the real number b is called the imaginary part, of the complex number a+bi. Equality of Complex Numbers a+bi = c+di if and only if a = c and b = d

Adding and Subtracting Complex Numbers (a+bi) + (c+di) = (a+c) + (b+d)i (a+bi) - (c+di) = (a-c) + (b-d)i That is, we simplify by removing the parenthesis and combining like terms, as usual. Like terms have all the same rules as before, now we will also consider imaginary numbers “like”. Combine the real number parts, then combine the imaginary number parts.

Text Example Perform the indicated operation, writing the result in standard form. (-5 + 7i) - ( i) Remove parenthesis: i i Combine the real and imaginary parts: (-5+11) + (7+6)i = 6+13i

Example =

As you can see, we FOIL as usual. The only difference being that we have i 2, which we must simplify by replacing it with -1. FOIL Simplify Combine like terms

Conjugate of a Complex Number The complex conjugate of the number a+bi is a-bi, and the conjugate of a-bi is a+bi. Ex: What is the conjugate of 7 + 2i Ex: What is the conjugate of -2 - i

Example: simplify the rational expression Remember that i is the SQUARE ROOT of -1, so it cannot be left in the denominator. Just as before, if we have two terms in the denominator, we multiply the numerator and denominator by the ________________ to simplify.

Principal Square Root of a Negative Number For any positive real number b, the principal square root of the negative number -b is defined by

Example: simplify Remember: when dealing with radicals SOS. Simplify, perform the Operation(s), the Simplify again. Do p 128#38. (S)implify Perform the (O)perations (S)implify