Further Mathematics
Complex numbers, matrices, polar, hyperbolic and differential equations.
26 formulas
Complex Numbers
Modulus of a complex number
Distance of z = a + bi from the origin in the Argand diagram.
Argument of a complex number
Principal argument of z = a + bi, returned in (−π, π]. Use atan2 to handle every quadrant correctly.
De Moivre's Theorem
Raising a complex number in polar form to the power n: modulus is raised to the n-th power, argument multiplied by n.
Modulus of a product
Multiplying complex numbers multiplies their moduli and adds their arguments.
Sum of first n integers
Closed form for 1 + 2 + … + n.
Sum of first n squares
Closed form for 1² + 2² + … + n².
Complex modulus
Complex modulus
Complex argument
Complex argument
de Moivre's theorem
de Moivre's theorem
Matrices & Determinants
2×2 Determinant
Determinant of a 2×2 matrix. Equals the signed area-scale factor of the linear map.
2×2 Matrix inverse
Inverse of a non-singular 2×2 matrix; swap a and d, negate b and c, divide by the determinant.
3×3 Determinant (cofactor expansion)
Cofactor expansion along the first row of a 3×3 matrix.
2×2 determinant
2×2 determinant
Polar & Hyperbolic
Polar → Cartesian (x-coordinate)
Cartesian x of a point given polar (r, θ).
Polar → Cartesian (y-coordinate)
Cartesian y of a point given polar (r, θ).
Hyperbolic sine
Definition of sinh in terms of exponentials. Odd function: sinh(−x) = −sinh x.
Hyperbolic cosine
Definition of cosh. Even function with cosh x ≥ 1 for all real x.
Hyperbolic identity
Fundamental hyperbolic identity (note the MINUS sign — different from sin² + cos² = 1).
Hyperbolic identity
Hyperbolic identity
Polar area
Polar area