Mathematics 18 flashcards ~9 min

Complex Numbers

About this deck

Complex numbers extend the number system beyond the real numbers, and this deck of 18 flashcards builds up from the basics to more advanced applications. You'll start with the imaginary unit and basic arithmetic — addition, multiplication and division of complex numbers — then learn about the complex conjugate, modulus and argument. The deck introduces the Argand diagram for visualising complex numbers geometrically, before moving into polar and exponential form and De Moivre's theorem for raising complex numbers to a power. It finishes with how complex roots arise from quadratics with negative discriminants, the fundamental theorem of algebra, and real-world applications in electrical engineering and quantum mechanics. Suited to A-Level Further Maths and AP Precalculus, this deck is especially useful for students who find the geometric interpretation of complex numbers clicks better than the pure algebra alone.

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Defined by 𝑖 = − 1 i= −1 , so that 𝑖 2 = − 1 i 2 =−1 — it allows square roots of negative numbers to be expressed.

𝑧 = 𝑎 + 𝑏 𝑖 z=a+bi, where 𝑎 a is the real part and 𝑏 b is the imaginary part.

Add/subtract the real parts and imaginary parts separately: ( 𝑎 + 𝑏 𝑖 ) + ( 𝑐 + 𝑑 𝑖 ) = ( 𝑎 + 𝑐 ) + ( 𝑏 + 𝑑 ) 𝑖 (a+bi)+(c+di)=(a+c)+(b+d)i.

Use distributive expansion (FOIL), remembering 𝑖 2 = − 1 i 2 =−1: ( 𝑎 + 𝑏 𝑖 ) ( 𝑐 + 𝑑 𝑖 ) = ( 𝑎 𝑐 − 𝑏 𝑑 ) + ( 𝑎 𝑑 + 𝑏 𝑐 ) 𝑖 (a+bi)(c+di)=(ac−bd)+(ad+bc)i.

a−bi — the imaginary part's sign is flipped. Multiplying a complex number by its conjugate gives a real number: ( 𝑎 + 𝑏 𝑖 ) ( 𝑎 − 𝑏 𝑖 ) = 𝑎 2 + 𝑏 2 (a+bi)(a−bi)=a 2 +b 2 .

Multiply the numerator and denominator by the conjugate of the denominator, then simplify to eliminate 𝑖 i from the denominator.

arg ⁡ ( 𝑧 ) = 𝜃 = tan ⁡ − 1 ( 𝑏 / 𝑎 ) arg(z)=θ=tan −1 (b/a) (adjusted for the correct quadrant) — the angle the number makes with the positive real axis.

A graphical representation of complex numbers, plotting the real part on the horizontal axis and the imaginary part on the vertical axis.

𝑧 = 𝑟 ( cos ⁡ 𝜃 + 𝑖 sin ⁡ 𝜃 ) z=r(cosθ+isinθ), where 𝑟 r is the modulus and 𝜃 θ is the argument.

𝑧 = 𝑟 𝑒 𝑖 𝜃 z=re iθ , following Euler's formula 𝑒 𝑖 𝜃 = cos ⁡ 𝜃 + 𝑖 sin ⁡ 𝜃 e iθ =cosθ+isinθ.

[ 𝑟 ( cos ⁡ 𝜃 + 𝑖 sin ⁡ 𝜃 ) ] 𝑛 = 𝑟 𝑛 ( cos ⁡ 𝑛 𝜃 + 𝑖 sin ⁡ 𝑛 𝜃 ) [r(cosθ+isinθ)] n =r n (cosnθ+isinnθ) — used to raise complex numbers to a power efficiently.

Complex conjugate roots of the form 𝑎 ± 𝑏 𝑖 a±bi, since the square root of a negative discriminant introduces 𝑖 i.

Every non-constant polynomial equation with complex coefficients has at least one complex root, and a degree-n polynomial has exactly n roots (counting multiplicity).

They occur in conjugate pairs — if 𝑎 + 𝑏 𝑖 a+bi is a root, then 𝑎 − 𝑏 𝑖 a−bi is also a root.

Rotates it 90° counterclockwise around the origin on the Argand diagram.

𝑖 1 = 𝑖 i 1 =i, 𝑖 2 = − 1 i 2 =−1, 𝑖 3 = − 𝑖 i 3 =−i, 𝑖 4 = 1 i 4 =1 — the pattern then repeats every four powers.

arg ⁡ ( 𝑧 1 𝑧 2 ) = arg ⁡ ( 𝑧 1 ) + arg ⁡ ( 𝑧 2 ) arg(z 1 z 2 )=arg(z 1 )+arg(z 2 ) — arguments add when complex numbers are multiplied.

Electrical engineering (AC circuit analysis), quantum mechanics, signal processing, and control theory, among others.