AcademyComplex Arithmetic
Academy
Complex Numbers
Level 1 - Math I (Physics) topic page in Complex Arithmetic.
Introduction to Complex Numbers
Complex numbers extend the real number system to solve equations that have no real solutions. A complex number combines a real part and an imaginary part.
Definition
A complex number \(z\) is written in standard form as:
StandardForm
\[z = a + bi\]
where:
- \(a\) is the real part (Re(z))
- \(b\) is the imaginary coefficient
- \(i\) is the imaginary unit with property \(i^2 = -1\)
Real and Imaginary Parts
RealPart
\[\text{Re}(z) = a\]
ImagPart
\[\text{Im}(z) = b\]
Modulus (Magnitude)
The modulus of \(z\) represents its distance from the origin in the complex plane:
Modulus
\[|z| = \sqrt{a^2 + b^2}\]
Complex Conjugate
The conjugate of \(z\) reflects it across the real axis:
Conjugate
\[z^* = a - bi\]
Euler's Formula Connection
Complex numbers can be expressed in polar form using Euler's formula:
EulerForm
\[z = re^{i\theta} = r(\cos\theta + i\sin\theta)\]
where:
- \(r = |z|\) is the modulus
- \(\theta = \arg(z)\) is the argument (angle)
This connection between algebraic and exponential forms is fundamental to complex analysis.