Arithmetic on a + bi.
Addition: z₁ + z₂ = (a + c) + (b + d)i Subtraction: z₁ − z₂ = (a − c) + (b − d)i Multiplication: z₁ × z₂ = (ac − bd) + (ad + bc)i Division: z₁ ÷ z₂ = [(ac + bd) / (c² + d²)] + [(bc − ad) / (c² + d²)]i
Given two complex numbers z₁ = a + bi and z₂ = c + di, the four basic arithmetic operations are defined as follows:
z₁ × z₂ = (ac − bd) + (ad + bc)i = (3×1 − 2×(−4)) + (3×(−4) + 2×1)i = (3 + 8) + (−12 + 2)i = 11 + (−10)i
Result: z₁ × z₂ = 11 − 10i
The product of z₁ = 3 + 2i and z₂ = 1 − 4i is 11 − 10i. The real part (11) comes from multiplying the real parts and subtracting the product of the imaginary parts (since i² = −1 flips the sign). The imaginary part (−10) is the sum of the cross-products of real and imaginary components. The result is a valid complex number in standard a + bi form.
Complex numbers extend the real number system by introducing the imaginary unit i, where i² = −1. Every complex number has the form a + bi, where:
When b = 0, the number is purely real. When a = 0, it is purely imaginary.
Complex numbers can be plotted on the Argand plane (complex plane), with the horizontal axis representing the real part and the vertical axis representing the imaginary part. This geometric interpretation is powerful in physics and engineering.
The modulus (or absolute value) of z = a + bi is |z| = √(a² + b²), representing the distance from the origin in the complex plane. The argument (angle) is θ = arctan(b/a).
The conjugate of z = a + bi is z̄ = a − bi. Conjugates are critical in division: multiplying numerator and denominator by the conjugate eliminates the imaginary part from the denominator, yielding a standard-form result.
The polar form expresses a complex number as z = r(cos θ + i sin θ), or compactly as z = re^(iθ) using Euler's formula, where r = |z| is the modulus and θ is the argument (angle). Polar form makes multiplication and division especially elegant.
To find √(a + bi), express the number in polar form: r = √(a² + b²), θ = arctan(b/a). The square root is then √r × e^(iθ/2) = √r(cos(θ/2) + i sin(θ/2)). There are always two square roots for any nonzero complex number.
Euler's formula states e^(iθ) = cos θ + i sin θ. It creates a deep connection between exponential functions and trigonometry through the complex plane. The special case θ = π gives Euler's identity: e^(iπ) + 1 = 0.
Absolutely. Complex numbers underlie JPEG and MP3 compression (via Fourier transforms), the design of AC electrical systems, radar and wireless communication signal processing, and the stability analysis of control systems in aircraft and robots.
The imaginary unit **i** is defined as the square root of −1, so i² = −1. It is a mathematical construct that extends real numbers and enables solutions to equations like x² + 1 = 0, which have no real solutions.
To divide z₁ by z₂, multiply both the numerator and denominator by the **complex conjugate** of z₂ (which is c − di). This eliminates the imaginary term in the denominator. The result is [(ac + bd) / (c² + d²)] + [(bc − ad) / (c² + d²)]i.
Yes. If b = 0, then a + 0i = a, which is simply a real number. All real numbers are a special case of complex numbers with no imaginary component.
The modulus of z = a + bi is |z| = √(a² + b²). It represents the distance from the origin to the point (a, b) on the complex plane and is always a non-negative real number.
Electrical engineers use **j** to avoid confusion with **i**, which is the conventional symbol for electrical current. The mathematics is identical — j² = −1 just like i² = −1.
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