Essential Operations with Complex Numbers to Know for Algebra 2

Complex numbers combine real and imaginary parts, expressed as a + bi. Understanding their operations—like addition, multiplication, and division—helps solve equations and graph them on the complex plane, connecting to broader concepts in Algebra 2.

1. Definition of complex numbers (a + bi form)

   • A complex number is expressed in the form a + bi, where a and b are real numbers.
   • 'a' is called the real part, and 'b' is called the imaginary part.
   • 'i' represents the imaginary unit, defined as the square root of -1.

2. Graphing complex numbers on the complex plane

   • The complex plane is a two-dimensional plane where the x-axis represents the real part and the y-axis represents the imaginary part.
   • Each complex number corresponds to a point (a, b) on this plane.
   • The distance from the origin to the point represents the absolute value of the complex number.

3. Addition and subtraction of complex numbers

   • To add or subtract complex numbers, combine their real parts and their imaginary parts separately.
   • For example, (a + bi) + (c + di) = (a + c) + (b + d)i.
   • This operation is straightforward and follows the same rules as adding or subtracting polynomials.

4. Multiplication of complex numbers

   • Multiply complex numbers using the distributive property (FOIL method).
   • For example, (a + bi)(c + di) = ac + adi + bci + bdi², where i² = -1.
   • Simplify the result by combining like terms and substituting i² with -1.

5. Division of complex numbers

   • To divide complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator.
   • The complex conjugate of a + bi is a - bi.
   • This process eliminates the imaginary part in the denominator, allowing for simplification.

6. Finding the complex conjugate

   • The complex conjugate of a complex number a + bi is a - bi.
   • It reflects the complex number across the real axis in the complex plane.
   • Conjugates are useful in division and finding absolute values.

7. Absolute value (modulus) of complex numbers

   • The absolute value of a complex number a + bi is calculated as √(a² + b²).
   • It represents the distance from the origin to the point (a, b) in the complex plane.
   • The absolute value is always a non-negative real number.

8. Powers of i (i², i³, i⁴, etc.)

   • The powers of i cycle every four: i² = -1, i³ = -i, i⁴ = 1, and i⁵ = i.
   • This cyclical pattern can simplify calculations involving higher powers of i.
   • Understanding these powers is essential for simplifying complex expressions.

9. Simplifying complex expressions

   • Combine like terms and use the properties of i to simplify expressions.
   • Replace i² with -1 to reduce higher powers of i.
   • Ensure the final expression is in standard form (a + bi).

10. Solving quadratic equations with complex solutions

    • Quadratic equations can have complex solutions when the discriminant (b² - 4ac) is negative.
    • Use the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).
    • If the discriminant is negative, the solutions will be in the form of complex numbers.