Print this Page

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.

Alg2A 3.08 - Complex Conjugate

  • 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}.

Alg2A 3.08 - Complex Numbers

 

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.

Alg2A 3.08 - Squares of Negatives

 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

Alg2A 3.08 - Sqrt -1 Equals i

Alg2A 3.08 - Sqrt -1 Equals i Patter

Alg2A 3.08 - Rules for i

  • 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

Alg2A 3.08 - Rules for Remainders


  • 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

Alg2A 3.08 - Complex Number Ex

  • Adding & Subtracting with Complex Numbers

Alg2A 3.08 - Complex Numbers Add Subt

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

Alg2A 3.08 - FOIL i

  • FOIL with Imaginary Numbers

Alg2A 3.08 - FOIL ex

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

Alg2A 3.08 - FOIL CC

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

Alg2A 3.08 - Division


 

Alg2A 3.08 - Complex Division

 Quadratics with Imaginary Numbers
  • 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

Alg2A 3.08 - Quadratics with Imaginary

  • Quadratics with Imaginary Numbers Using the Quadratic Equation
    • Step 1: Set up the equation in standard form
    • Step 2: List the a, b, and c and substitute them into the equation
    • Step 3: Solve and simplify

  • Example

Alg2A 3.08 - Quadratics with Imaginary 2

Permanent link to this article: http://newvillagegirlsacademy.org/math/?page_id=1438