Math 98
10.6 Complex Numbers
The square root produces a number.
When you multiply the number times itself you’ll get the number under the square root.
The square root of a NEGATIVE NUMBER will produce an IMAGINARY NUMBER.
We break up the square root into two square roots: the square root on the right will have “-1” under it,
this is the “imaginary” number we call “i”.
KEY IDEA: √−1 = 𝑖
Since a square root produces a number that times itself MUST give us the number under the square
root, we find that the only way this can work is if we say that i2 = -1.
KEY IDEA: 𝑖 2 = −1
ARITHMETIC
If we add two REAL NUMBERS (numbers that do NOT have an i), we get a REAL NUMBER.
If we add two IMAGINARY NUMBERS (numbers with an “i” or the square root of a negative), we get an
IMAGINARY NUMBER.
If we add a REAL NUMBER and an IMAGINARY NUMBER, the answer will have an “i” in it. So the result
will be considered an imaginary number
KEY IDEA:
𝑎 + 𝑏𝑖 is the STANDARD FORM for an IMAGINARY NUMBER.
“a” is the REAL TERM and “Bi” is the IMAGINARY TERM.
ARITHMETIC:
ADDITION: To add two imaginary numbers, combine like terms (REALS and REALS, IMAGINARY and
IMAGINARY)
SUBTRACTION: To subtract two imaginary numbers, distribute the negative through the parentheses on
the right. THEN combine like terms as you did with addition.
DISTRIBUTIVE MULTIPLICATION: Distribute what’s outside the parentheses to what is inside.
REPLACE ALL 𝑖 2 𝑤𝑖𝑡ℎ − 1
FOIL: Foil all terms… REPLACE ALL 𝑖 2 𝑤𝑖𝑡ℎ − 1. Combine like terms.
TWO COMPLEX NUMBERS ARE EQUAL when the REAL TERM on the LEFT equals the REAL TERM on the
RIGHT and the IMAGINARY TERM on the LEFT equals the IMAGINARY TERM on the RIGHT