Basic Concepts and Operations with Complex Numbers

Basic Concepts of Complex Numbers

Introduction to Complex Numbers

Complex numbers extend the real number system to include solutions to equations that have no real solutions, such as $x^2 = -1$. They are fundamental in algebra and have applications in engineering, physics, and mathematics.

• Complex Number: Any number of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

• Imaginary Unit ($i$): Defined by $i^2 = -1$. Thus, $i = \sqrt{-1}$.

• Real Part: In $a + bi$, $a$ is the real part.

• Imaginary Part: In $a + bi$, $b$ is the imaginary part.

• Equality: Two complex numbers $a + bi$ and $c + di$ are equal if and only if $a = c$ and $b = d$.

• Standard Form: A complex number is in standard form when written as $a + bi$.

Example: $3 + 4i$ is a complex number with real part 3 and imaginary part 4.

Square Roots of Negative Numbers

Square roots of negative numbers are expressed using the imaginary unit $i$.

• For any positive real number $a$, $\sqrt{-a} = i\sqrt{a}$.

• To write $\sqrt{-16}$ as a product of a real number and $i$: $\sqrt{-16} = 4i$.

Example: $\sqrt{-25} = 5i$

Operations on Complex Numbers

Adding and Subtracting Complex Numbers

To add or subtract complex numbers, combine like terms (real with real, imaginary with imaginary).

• Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$

• Subtraction: $(a + bi) - (c + di) = (a - c) + (b - d)i$

Example: $(3 - 4i) + (2 + 6i) = (3 + 2) + (-4 + 6)i = 5 + 2i$

Multiplying Complex Numbers

Multiply complex numbers using the distributive property (FOIL method), remembering that $i^2 = -1$.

• $(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$

Example: $(2 - 3i)(3 + 4i) = 2 \times 3 + 2 \times 4i - 3i \times 3 - 3i \times 4i = 6 + 8i - 9i - 12i^2 = 6 - i + 12 = 18 - i$

Additional info: The last step uses $i^2 = -1$, so $-12i^2 = 12$.

Dividing Complex Numbers

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.

• Conjugate: The conjugate of $a + bi$ is $a - bi$.

• Division Formula: $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$

• This results in a denominator that is a real number: $(c + di)(c - di) = c^2 + d^2$

Example: $\frac{3 + 2i}{5 - i} = \frac{(3 + 2i)(5 + i)}{(5 - i)(5 + i)} = \frac{15 + 3i + 10i + 2i^2}{25 + 5i - 5i - i^2} = \frac{15 + 13i - 2}{25 + 1} = \frac{13 + 13i}{26} = \frac{13}{26} + \frac{13}{26}i = \frac{1}{2} + \frac{1}{2}i$

Powers of the Imaginary Unit $i$

Powers of $i$ repeat in a cycle of four:

Power	Value
$i^1$	$i$
$i^2$	$-1$
$i^3$	$-i$
$i^4$	$1$
$i^5$	$i$
$i^6$	$-1$
$i^7$	$-i$
$i^8$	$1$

• For any integer $n$, $i^n$ can be simplified by dividing $n$ by 4 and using the remainder to determine the value.

Example: $i^{15} = i^{4 \times 3 + 3} = (i^4)^3 \times i^3 = 1^3 \times (-i) = -i$

Summary Table: Key Properties of Complex Numbers

Operation	Formula	Example
Addition	$(a + bi) + (c + di) = (a + c) + (b + d)i$	$(1 + 2i) + (3 + 4i) = 4 + 6i$
Subtraction	$(a + bi) - (c + di) = (a - c) + (b - d)i$	$(5 + 6i) - (2 + 3i) = 3 + 3i$
Multiplication	$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$	$(2 + 3i)(1 - 4i) = 14 - 5i$
Division	$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2}$	$\frac{1 + 2i}{3 - 4i} = \frac{-5 + 10i}{25} = -0.2 + 0.4i$

Additional Notes

• The set of real numbers is a subset of the set of complex numbers.

• If $b = 0$, the complex number $a + bi$ is a real number.

• If $a = 0$ and $b \neq 0$, the complex number is a pure imaginary number.

• Complex conjugates are used to rationalize denominators when dividing complex numbers.

Additional info: Complex numbers are essential in solving quadratic equations with negative discriminants and in representing oscillations and waves in physics and engineering.