Complex numbers

Complex numbers: A number of the form a + bi, where a and b are real numbers and i is the imaginary unit.
Real part: The a in a + bi. It’s the portion of the complex number that’s a real number.
Imaginary part: The b in a + bi. It’s the generally real number that multiplies the imaginary unit i in the complex number.
Imaginary unit (i): The square root of -1. It is denoted as i such that i² = -1.
Pure imaginary numbers: When a is 0 in a + bi, the resulting bi is a pure imaginary number.

Operations with Complex Numbers

Addition: Add the real parts and the imaginary parts separately.
Subtraction: Subtract the real parts and the imaginary parts separately.
Multiplication: Use the distributive property, keeping in mind that i² = -1.
Division: Multiply the numerator and denominator by the conjugate of the denominator, then simplify.

The Complex Plane

Complex plane: A coordinate system where the x-axis represents the real parts and the y-axis represents the imaginary parts of complex numbers.

Modulus of a complex number: The distance from the origin to the point on the complex plane representing the complex number. If z = a + bi, then the modulus of z, denoted

, is √(a² + b²).

Argument of a complex number: The angle the line connecting the origin to the point on the complex plane representing the complex number makes with the positive x-axis. It is often denoted as ‘arg’.

Complex Conjugate and Absolute Value

Complex conjugate: The complex number with the same real part and the negation of its imaginary part. The conjugate of a + bi is a - bi.
Properties of complex conjugates: The product of a complex number and its conjugate is always a real number. The sum and difference of a complex number and its conjugate are respectively real and imaginary numbers.
Absolute value or modulus: The absolute value of a complex number a + bi is always a non-negative real number and is calculated as √(a² + b²).

Polar Form of Complex Numbers

Polar form: Another way to express a complex number using its magnitude (r) and angle (θ). Written as r(cos θ + i sin θ).
Euler’s formula: States that e^(iθ) = cos θ + i sin θ, a critical bridge between trigonometry and complex analysis.

Conversion into polar form: If z = a + bi, the modulus **r =

= √(a² + b²), and the argument or angle **θ = atan(b/a). The polar form is then r (cos θ + i sin θ).

Multiplication in polar form: Multiply the moduli and add the angles.
Division in polar form: Divide the moduli and subtract the angles.

Powers and Roots of Complex Numbers

Powers of complex numbers: For a complex number in polar form r(cos θ + i sin θ), its n-th power is calculated by raising the modulus to the power and multiplying the angle by the power.
Roots of complex numbers: Take the n-th root of the modulus and divide the angle by n. Use the magnitude and angle to find the roots on the complex plane.
De Moivre’s theorem: A tool that simplifies the computation of powers and roots of complex numbers, particularly useful in combination with the polar form of complex numbers. It gives a formula to calculate the n-th power of a complex number.