Learn & Review: Complex Numbers 1(Definition, Addition, Subtr

Understanding Complex Numbers

This summary outlines the fundamental concepts of complex numbers, their representations, and basic algebraic operations.

1. Definition of Complex Numbers

• A complex number is any number that can be expressed in the form a + ib, where:

  • 'a' and 'b' are real numbers (positive or negative).
  • 'i' is the imaginary unit, defined as the square root of -1 ($\sqrt{-1}$).

• Historical Context: Before the 16th century, finding the square root of negative numbers was considered impossible. The introduction of 'i' in the 16th century allowed for the calculation of square roots of negative numbers.

  For example, $\sqrt{-4}$ can be solved as $\sqrt{4 \times -1} = \sqrt{4} \times \sqrt{-1} = 2i$.

• Examples of Complex Numbers:

  • $2 + 4i$
  • $-1 + 3i$
  • $5$ (can be written as $5 + 0i$)
  • $3i$ (can be written as $0 + 3i$)

2. Representations of Complex Numbers

Complex numbers can be represented in three main forms:

• Rectangular (Cartesian) Form:

  • This is the standard form: a + ib.
  • It can also be represented as an ordered pair: (a, b).
  • Example: $z = 2 + 3i$ is in rectangular form.

• Polar (Circular) Form:

  • Represented as: z = r(cos θ + i sin θ)
  • 'r' is the magnitude or modulus of the complex number.
  • 'θ' is the amplitude or angle of the complex number.

• Exponential (Euler's) Form:

  • Represented as: z = re^(iθ)
  • This form is closely related to the polar form.

3. Algebra of Complex Numbers

3.1. Addition of Complex Numbers

• To add two complex numbers, $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$:

  • Add the real parts together: $(a_1 + a_2)$.
  • Add the imaginary parts together: $(b_1 + b_2)$.
  • The sum is: $(a_1 + a_2) + i(b_1 + b_2)$.

  Example: $z_1 = -3 + 2i$ $z_2 = -1 - i$ $z_1 + z_2 = (-3 + (-1)) + (2 + (-1))i = -4 + 1i = -4 + i$

3.2. Subtraction of Complex Numbers

• To subtract two complex numbers, $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$:

  • Subtract the real parts: $(a_1 - a_2)$.
  • Subtract the imaginary parts: $(b_1 - b_2)$.
  • The difference is: $(a_1 - a_2) + i(b_1 - b_2)$.

  Example: $z_1 = -3 + 2i$ $z_2 = -1 - i$ $z_1 - z_2 = (-3 - (-1)) + (2 - (-1))i = (-3 + 1) + (2 + 1)i = -2 + 3i$

3.3. Multiplication of Complex Numbers

• Key Properties of 'i':

  • $i = \sqrt{-1}$
  • $i^2 = -1$
  • $i^3 = -i$
  • $i^4 = 1$

• To multiply complex numbers, distribute terms similar to multiplying binomials, and use the properties of 'i'.

  Example: Find $z^2$ where $z = 4 + 3i$. $z^2 = (4 + 3i)^2 = (4 + 3i)(4 + 3i)$ $= 4(4) + 4(3i) + 3i(4) + 3i(3i)$ $= 16 + 12i + 12i + 9i^2$ $= 16 + 24i + 9(-1)$ $= 16 + 24i - 9$ $= 7 + 24i$

  Example: Find $z_1 \times z_2 \times z_3$ where $z_1 = -3 + 2i$, $z_2 = 4 + i$, $z_3 = 5 - 10i$. First, calculate $z_1 \times z_2$: $(-3 + 2i)(4 + i) = -12 - 3i + 8i + 2i^2 = -12 + 5i - 2 = -14 + 5i$ Then, multiply the result by $z_3$: $(-14 + 5i)(5 - 10i) = -70 + 140i + 25i - 50i^2 = -70 + 165i + 50 = -20 + 165i$

3.4. Division of Complex Numbers

• Conjugate of a Complex Number:

  • The conjugate of a complex number $z = a + ib$ is denoted as $z_{bar}$ and is equal to $a - ib$.
  • To find the conjugate, simply change the sign of the imaginary part.
  • Example: If $z = 3 - 4i$, then $z_{bar} = 3 + 4i$.

• Rationalization: To divide complex numbers, it's improper to leave a complex number in the denominator. You must rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

  Example: Divide $\frac{3 + 2i}{2 + 3i}$. Multiply numerator and denominator by the conjugate of $(2 + 3i)$, which is $(2 - 3i)$: $\frac{(3 + 2i)(2 - 3i)}{(2 + 3i)(2 - 3i)}$ Numerator: $(3 + 2i)(2 - 3i) = 6 - 9i + 4i - 6i^2 = 6 - 5i + 6 = 12 - 5i$ Denominator: $(2 + 3i)(2 - 3i) = 4 - 6i + 6i - 9i^2 = 4 + 9 = 13$ Result: $\frac{12 - 5i}{13} = \frac{12}{13} - \frac{5}{13}i$

  Example: Express $\frac{(1 + i)^3}{2 + i}$ in the form $a + ib$. First, expand $(1 + i)^3$ using binomial expansion: $(1 + i)^3 = 1^3 + 3(1^2)(i) + 3(1)(i^2) + i^3 = 1 + 3i + 3(-1) + (-i) = 1 + 3i - 3 - i = -2 + 2i$ Now the expression is $\frac{-2 + 2i}{2 + i}$. Rationalize by multiplying by the conjugate of $(2 + i)$, which is $(2 - i)$: $\frac{(-2 + 2i)(2 - i)}{(2 + i)(2 - i)}$ Numerator: $(-2 + 2i)(2 - i) = -4 + 2i + 4i - 2i^2 = -4 + 6i + 2 = -2 + 6i$ Denominator: $(2 + i)(2 - i) = 4 - 2i + 2i - i^2 = 4 + 1 = 5$ Result: $\frac{-2 + 6i}{5} = -\frac{2}{5} + \frac{6}{5}i$