Complex Numbers Study Notes

Introduction to Complex Numbers

• Complex numbers are called "complex" not due to difficulty but because they include numbers that are not real.

• In particular, complex numbers help resolve situations where a square root of a negative number is required.

Definition of Complex Numbers

• A complex number can be expressed in the form: $z = a + bi$ where:

  • $a$ is the real part

  • $b$ is the imaginary part

  • $i$ is the imaginary unit, defined as the square root of -1

Imaginary Numbers

• When attempting to take the square root of a negative number, such as √(-1), calculators display an error since such calculations are not valid in the real number system.

• Thus, mathematicians introduced the symbol $i$ (imaginary unit) to represent √(-1).

• For example:

  • $ext{If } x^2 + 4 = 0$

  • $x^2 = -4$

  • $x = ext{±}√(-4) = ext{±}√(4) * i = ext{±}2i$

Types and Forms of Complex Numbers

• Complex numbers can be described in various forms, including polar forms.

Representing Complex Numbers

• Complex numbers can be represented as points in the Cartesian plane (xy-plane).

• Example: For a complex number $z_1 = 2 + 3i$

  • This represents the point (2, 3) on the xy-plane.

• Generally, complex numbers can be written as $z = a + bi$ where:

  • $a$ is the x-coordinate (real part)

  • $b$ is the y-coordinate (imaginary part)

  • It is essential to note that the real part always comes first in the representation.

Example Points of Complex Numbers

• Examples include:

  • $z_1 = 2 + 3i$ represented by point (2, 3)

  • $z_2 = 4 + 2i$ represented by point (4, 2)

  • $z_3 = 3 - 2i$ represented by point (3, -2)

  • $z_4 = -2 + i$ represented by point (-2, 1)

Coordinate System in Complex Numbers

• The x-axis represents the real axis, while the y-axis represents the imaginary axis.

• Complex numbers can also lie solely on the x-axis (purely real) or y-axis (purely imaginary)

  • Example:

  • For on the x-axis: $z_5 = 2 + 0i$ (just 2)

  • For on the y-axis: $z_6 = 0 + 3i$ (just 3i)

Operations with Complex Numbers

Addition of Complex Numbers

• To add two complex numbers, add the real parts together and the imaginary parts separately.

  • Example:

  • Let $z<em>1 = 2 + 3i$ and $z</em>2 = 4 + 2i$

    • $z<em>1 + z</em>2 = (2 + 4) + (3 + 2)i = 6 + 5i$

Subtraction of Complex Numbers

• Similar to addition, subtract the real parts and the imaginary parts.

  • Example:

  • Let $z<em>7 = 5 + 6i$ and $z</em>8 = 2 + 3i$

    • $z<em>7 - z</em>8 = (5 - 2) + (6 - 3)i = 3 + 3i$

Scalar Multiplication of Complex Numbers

• When multiplying a complex number by a real number (scalar), distribute the scalar to both real and imaginary parts.

  • Example:

  • Let $z_9 = 3 + 4i$ and multiply by scalar 5:

    • $5z_9 = 5(3 + 4i) = 15 + 20i$

Multiplication of Complex Numbers

• To multiply two complex numbers, apply the distributive property (FOIL method):

General Formula

If:

• $z<em>1 = a</em>1 + b<em>1i$ and $z</em>2 = a<em>2 + b</em>2i$
  Then:

• $z<em>1 * z</em>2 = (a<em>1a</em>2 - b<em>1b</em>2) + (a<em>1b</em>2 + a<em>2b</em>1)i$

• Example:

  • Let $z<em>1 = 1 + 2i$ and $z</em>2 = 3 + 4i$

    • $z<em>1 * z</em>2 = (1<em>3 - 2</em>4) + (1<em>4 + 2</em>3)i = -5 + 10i$

Understanding Imaginary and Real Parts

• Every real number can also be considered as a complex number with an imaginary part of zero.

  • For example:

  • The real number 5 can be expressed as $5 + 0i$

Visual Representation of Complex Numbers

• Complex numbers can be graphically represented on the Cartesian plane, where the x-coordinate indicates the real part and the y-coordinate the imaginary part.

• The length of a complex number (its modulus) can be calculated using the Pythagorean theorem:

  • $|z| = ext{√}(a^2 + b^2)$

  • Example: For $z = a + bi$:

  • If $z = 3 + 4i$

    • The modulus is $|z| = ext{√}(3^2 + 4^2) = ext{√}(9 + 16) = ext{√}(25) = 5$

Polar Form of Complex Numbers

• Complex numbers can also be represented in polar form, expressing the number in terms of magnitude and angle (argument).

• The argument (angle) can be calculated through tangent:
  $ext{arg}(z) = an^{-1}( rac{b}{a})$

• The polar form can be represented as:

  • $r( ext{cos}( heta) + i ext{sin}( heta))$

• where

  • $r$ is the modulus,

  • $heta$ is the argument

Conclusion

• Understanding complex numbers is crucial for solving equations involving negative square roots, thereby allowing for the resolution of polynomial equations that may result in missing roots if confined to real numbers.