# 3.8 – Imaginary Numbers

## Objectives

• Define imaginary number.
• Use arithmetic operations to simplify expressions that include imaginary and complex numbers.
• Given the square root of a negative number, write the equivalent imaginary number.
• Find an equivalent expression for i raised to the nth power.
• Define complex numbers.

## Key Terms

• Complete the Square – A way to solve quadratic equations.
• It involves adding a number, $(\frac{b}{2})^2$, to both sides of an equation to make one side a perfect square trinomial.
• The equation can then be solved by taking the square root of both sides and simplifying.
• Complex Conjugate Theorem – If a + bi is a root, then a – bi is also a root.
• Complex Conjugates – Two complex numbers that have the form a + bi and a – bi, where a and b are any real numbers.
• In any complex number, the a term is called the real part and the bi term is called the imaginary part.

• Complex Numbers – The set of all numbers of the form a + bi, where a and b are any real numbers and i equals $\sqrt{-1}$.

## Notes

Imaginary Numbers
• Imaginary numbers are not real numbers, but they are the product of reals number and i.
• You cannot take the square root of a negative number, because any number squared will be a positive.
• A positive times a positive is a positive.
• A negative times a negative is a positive.
• So, the square root of a positive is real, but the square root of a negative is imaginary.

Simplifying with i
• Simplifying with i
• Rule:  $\sqrt{-1}=i$, because:
• $x^2+1=0$
• $x^2=-1$
• $\sqrt{x^2}=\sqrt{-1}$
• $x=\sqrt{-1}$ is not a real number, so x is replaced by i (imaginary)
• $i=\sqrt{-1}$
• Example: Simplify: $\sqrt{-20}$
• Step 1: Factor -1 • 4 • 5 (-1 = i, 4 is the perfect square, and 5 is the remaining factor)
• Step 2: Bring the outside of the square root:  $i \sqrt{4} \sqrt{5}$
• Step 3: Take the square root of 4, which is 2, and bring it out of the square root:  $2i \sqrt{5}$
• Step 4: The 5 is not a perfect square, so leave it inside the square root:  $2i \sqrt{5}$
• To solve the problem above, follow these steps
• Steps for solving negatives inside square roots
• Step 1: Factor -20 into -1 • any perfect squares • remaining factors
• Step 2: The square root of -1 is i, so pull the i to the outside
• Step 3: Take the square root of any factors that are prefect squares (4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on…)
• Step 4: Leave remaining factors inside the square root
Powers of i
• Powers of i
• $i = \sqrt{-1}$
• $i^2 = (\sqrt{-1})^2 = -1$
• So, $i^2 = -1$
• $i^3 = i^2 \cdot i = -1 \cdot i = -i$
• So, $i^3 = -i$
• $i^4 = (i^2)^2 = (-1)^2 = 1$
• So, $i^4 = 1$

• Rule: To find really large powers of i, divide by 4 and look at the chart for the remainder
• The remainder of any number divided by 4 will be 1, 2, or 3.
• Use the chart above to see the result of $i^1, i^2, or i^3$
• Remember: $i^4 = 1$ and 1 multiplied by any number is that number.

• Example:  $i^{14}$
• $14 \div 4 = 3$ remainder 2
• because 4(3) = 12, and 12 + remainder 2 = 14
• $(i^4)^3 \cdot i^2 = 1 \cdot -1 = -1$

• What about –i raised to a power?
• $(-i)^1=- \sqrt{-1}=-i$
• $(-i)^2=- \sqrt{-1} \cdot - \sqrt{-1}= -1$
• $(-i)^3=- \sqrt{-1} \cdot - \sqrt{-1} \cdot - \sqrt{-1}= -1 \cdot -i = i$
• $(-i)^4=- \sqrt{-1} \cdot - \sqrt{-1} \cdot - \sqrt{-1} \cdot - \sqrt{-1}= i \cdot -i = -i^2 = -(-1) = 1$
Complex Numbers
• Complex Number Examples

• Adding & Subtracting with Complex Numbers

• Multiplication (FOIL) with Complex Numbers
• How to Multiply Complex Numbers
• Use FOIL to multiply.
• Substitute -1 for $i^2$.
• Simplify

• FOIL with Imaginary Numbers

Complex Conjugates
• The complex conjugate of a number a + bi is a – bi.
• Ex. The complex conjugate of 3 – 2i is 3 + 2i.

• Division with Imaginary Numbers Requires Using Complex Conjugates
• Multiply the numerator and denominator by the complex conjugate of the denominator

• Quadratics with Imaginary Numbers by Completing the Square
• Step 1: Complete the Square
• Step 2: When you take the square root of both sides, factor out i for -1
• Step 3: Simplify

• Example