Further Mathematics

AS: Further Pure Mathematics

# Complex Numbers

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Complex Numbers

**Introduction to Complex Numbers**

- Complex numbers are an extension of the real numbers and contain all the numbers in the form
**a + bi**, where**a**and**b**are real numbers and**i**is the imaginary unit with the property that**i² = -1**. - Any complex number can be represented as a point in the complex plane, with the horizontal (real) axis representing the real part and the vertical (imaginary) axis representing the imaginary part.

**Arithmetic with Complex Numbers**

- The standard
**rules of algebra**apply to complex numbers, including commutativity, associativity, and distributivity. **Adding**and**subtracting**complex numbers works component-wise, i.e., real parts with real parts and imaginary parts with imaginary parts.- When
**multiplying**complex numbers, apply the FOIL method (first, outside, inside, last), remembering to substitute**i²**with**-1**where necessary.

**Modulus and Argument of Complex Numbers**

- The
**modulus**of a complex number, represented as**r**, is the distance from the number to the origin in the complex plane. It is calculated using Pythagoras’ theorem: r = sqrt(a² + b²). - The
**argument**of a complex number, represented as**θ**, is the angle that the line joining the number and the origin makes with the positive real axis, measured anti-clockwise. Applying trigonometry, tan(θ) = b/a.

**Conjugates and Division of Complex Numbers**

- The
**conjugate**of a complex number is the number with the same real part but with an imaginary part of the opposite sign. If a complex number is a + bi, its conjugate will be a - bi. **Dividing**complex numbers involves multiplying both the numerator and denominator by the conjugate of the denominator, simplifying where possible.

**Polar Form of Complex Numbers**

- A complex number can also be represented in
**polar form**: z = r(cos θ + i sin θ), where r is the modulus and θ is the argument. **Euler’s formula**provides a shortcut for transforming from Cartesian to polar form. According to this formula, e^(iθ) = cos θ + i sin θ.

**De Moivre’s Theorem**

**De Moivre's theorem**is powerful for making computations with complex numbers easier. It states that for any integer n, (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ).

**Complex Roots**

- Polynomials can have
**complex roots**, thanks to the Fundamental Theorem of Algebra. If a polynomial's real roots do not account for all its roots, the remaining roots will be complex. - To find these roots, set the polynomial equal to zero and solve for the variable. If you encounter a square root of a negative number, remember that this can be simplified using
**i**.