What is a complex number?
A complex number consists of two parts: a real part and an imaginary part.
Z=a+jb where a and b are real numbers, and j=−1.
Why is the symbol j used in electronics instead of i?
To avoid confusion with i, which usually denotes current.
What does the term ‘imaginary number’ refer to?
It refers to the part of a complex number multiplied by the imaginary unit j.
How is a complex number represented graphically?
On the complex (Argand) plane — the horizontal axis is the real part, and the vertical axis is the imaginary part.
What is another name for the complex plane?
Argand plane or Gaussian plane.
What is the rectangular (Cartesian) form of a complex number?
Z=a+jb where a is the horizontal position (real part) and b is the vertical position (imaginary part).
What is the magnitude r of a complex number?
r=√(a²+b²)
What is the angle θ (argument or phase angle)?
θ=tan−1(b/a).
How is a complex number written in trigonometric form?
Z=r(cosθ+jsinθ).
How do you express a and b in terms of r and θ?
a=rcosθ, b=rsinθ.
What is Euler’s relationship between trigonometric and exponential forms?
e^(jθ)=cosθ+jsinθ.
Using Euler’s formula, how can we express a complex number?
Z=re^(jθ).
What shorthand notation is used for the exponential form?
Z=r∠θ.
How to convert from rectangular to polar form?
r = √(a² + b²), θ = tan⁻¹(b/a)
How to convert from polar to rectangular form?
a = r cos(θ), b = r sin(θ)
How do you add or subtract complex numbers in rectangular form?
Z₁ ± Z₂ = (a₁ ± a₂) + j(b₁ ± b₂)
Why is rectangular form best for addition/subtraction?
Because real and imaginary parts can be added directly.
How do you multiply two complex numbers in polar form?
Z₁ Z₂ = r₁ r₂ ∠(θ₁ + θ₂)
How do you divide two complex numbers in polar form?
Z₁/Z₂ = r₁/r₂ ∠(θ₁ − θ₂)
What are the basic powers of j?
j = -1, j^2 = -1, j^3 = -j, j^4 = 1
What is the reciprocal of j?
1/j = -j
What is 1/(A + jB) in complex form?
1/(A + jB) = (A - jB) / (A^2 + B^2)
What is e^(j0°), e^(j90°), e^(j180°), and e^(j270°)?
e^(j0°) = 1, e^(j90°) = j, e^(j180°) = -1, e^(j270°) = -j
What is the magnitude (modulus) of Z?
|Z| = √((Re Z)^2 + (Im Z)^2)
