# The language of complex numbers

## The language of complex numbers

The Fundamentals of Complex Numbers

• Describe complex numbers as all numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying i^2 = -1.
• Use the term real part to refer to the number a and imaginary part to the number b in a complex number.
• Understand that real numbers are a subset of complex numbers where the imaginary part is zero.
• Define imaginary numbers as numbers that can be written as a real number multiplied by i.

Operations with Complex Numbers

• Perform addition, subtraction, multiplication and division of complex numbers. Recall that both the real and imaginary parts are treated separately when performing these operations.
• Demonstrate the conjugate of a complex number, a + bi, is a - bi. Note the effect of multiplying a complex number by its conjugate: it produces a real number.
• Understand the modulus of a complex number as a measure of its ‘size’ or ‘magnitude’, computed as the square root of the sum of the squares of its real and imaginary parts.
• Use De Moivre’s theorem to find integral powers of complex numbers.
• Calculate the argument of a complex number, which is the angle it makes with the positive real axis in the Argand diagram.

Representations of Complex Numbers

• Display complex numbers graphically using the Argand diagram, which represents complex numbers as points in the plane, with the x-axis corresponding to real parts and the y-axis to the imaginary parts.
• Express complex numbers in polar form (rcosθ + irsinθ), which can be more convenient for calculations involving multiplication and division.
• Acknowledge Euler’s formula e^(iθ) = cosθ + isinθ, that plays a crucial role in transforming complex numbers between rectangular and polar form.

Complex Roots

• Understand complex roots as roots of polynomials that are complex numbers.
• Use De Moivre’s theorem to derive the nth roots of unity and compute roots of a complex number.
• Demonstrate that complex roots always appear in conjugate pairs whenever the coefficients of the polynomial are real.

Review these points thoroughly for a strong foundation in understanding the language of complex numbers. Surface-level understanding will not suffice; immerse yourself in these concepts to become proficient in these topics.