Imaginary Number Rules

Imaginary numbers expand our understanding of mathematics by introducing the imaginary unit, i, which represents the square root of -1. These concepts are essential in Honors Algebra II, connecting real and complex numbers for various applications in science and engineering.

1. The definition of i: i = √(-1)

   • i is the imaginary unit, representing the square root of -1.
   • It allows for the extension of real numbers to complex numbers.
   • Imaginary numbers are used in various fields, including engineering and physics.

2. Powers of i: i², i³, i⁴, etc.

   • i² = -1, which is the foundational property of i.
   • The powers of i cycle every four: i, -1, -i, 1.
   • Knowing the cycle helps simplify higher powers of i quickly.

3. Simplifying square roots of negative numbers

   • To simplify √(-a), rewrite it as i√a.
   • This process allows for the expression of complex numbers in standard form.
   • Understanding this is crucial for solving equations involving square roots of negative numbers.

4. Adding and subtracting complex numbers

   • Complex numbers are in the form a + bi, where a and b are real numbers.
   • To add or subtract, combine the real parts and the imaginary parts separately.
   • The result is another complex number in standard form.

5. Multiplying complex numbers

   • Use the distributive property (FOIL) to multiply (a + bi)(c + di).
   • Remember that i² = -1 when simplifying the result.
   • The product will also be in the form a + bi.

6. Dividing complex numbers

   • To divide (a + bi) by (c + di), multiply the numerator and denominator by the conjugate of the denominator.
   • This eliminates the imaginary part in the denominator.
   • Simplify the result to express it in standard form.

7. The complex conjugate and its uses

   • The complex conjugate of a + bi is a - bi.
   • It is used to simplify division of complex numbers and to find the modulus.
   • Conjugates have applications in solving polynomial equations and in signal processing.

8. Absolute value (modulus) of a complex number

   • The modulus of a complex number a + bi is given by |a + bi| = √(a² + b²).
   • It represents the distance from the origin to the point (a, b) in the complex plane.
   • Understanding modulus is essential for working with polar forms of complex numbers.

9. Graphing complex numbers on the complex plane

   • The complex plane has a horizontal axis (real part) and a vertical axis (imaginary part).
   • Each complex number corresponds to a point in this plane.
   • Graphing helps visualize operations and relationships between complex numbers.

10. Polar form of complex numbers

• A complex number can be expressed in polar form as r(cos θ + i sin θ), where r is the modulus and θ is the argument.
• This form is useful for multiplication and division of complex numbers.
• Euler's formula connects polar form to exponential form: re^(iθ).