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Mathematics LibreTexts

1.3: Terminology and Basic Arithmetic

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    Definition

    Complex numbers are defined as the set of all numbers

    \(z = x + yi\),

    where \(x\) and \(y\) are real numbers.

    • We denote the set of all complex numbers by \(\mathbb{C}\).
    • We call \(x\) the real part of \(z\). This is denoted by \(x = \text{Re} (z)\).
    • We call \(y\) the imaginary part of \(z\). This is denoted by \(y = \text{Im} (z)\).

    Important: The imaginary part of \(z\) is a real number. It does not include the \(i\).

    The basic arithmetic operations follow the standard rules. All you have to remember is that \(i^2 = -1\). We will go through these quickly using some simple examples. It almost goes without saying that it is essential that you become fluent with these manipulations.

    • Addition: \((3 + 4i) + (7 + 11i) = 10 + 15i\)
    • Subtraction: \((3 + 4i) - (7 + 11i) = -4 - 7i\)
    • Multiplication:

    \((3 + 4i)(7 + 11i) = 21 + 28i + 33i + 44i^2 = -23 + 61i.\)

    Here we have used the fact that \(44i^2 = -44\).

    Before talking about division and absolute value we introduce a new operation called conjugation. It will prove useful to have a name and symbol for this, since we will use it frequently.

    Definition: Complex Conjugation

    Complex conjugation is denoted with a bar and defined by

    \(\overline{x + iy} = x - iy\).

    If \(z = x + iy\) then its conjugate is \(\bar{z} = x - iy\) and we read this as "z-bar = \(x - iy\)".

    Example \(\PageIndex{1}\)

    \(\overline{3 + 5i} = 3 - 5i\).

    The following is a very useful property of conjugation: If \(z = x + iy\) then

    \(z\bar{z} = (x + iy)(x - iy) = x^2 + y^2\)

    Note that \(z\bar{z}\) is real. We will use this property in the next example to help with division.

    Example \(\PageIndex{2}\) (Division).

    Write \(\dfrac{3 + 4i}{1 + 2i}\) in the standard form \(x + iy\).

    Solution

    We use the useful property of conjugation to clear the denominator:

    \(\dfrac{3 + 4i}{1 + 2i} = \dfrac{3 + 4i}{1 + 2i} \cdot \dfrac{1 - 2i}{1 - 2i} = \dfrac{11 - 2i}{5} = \dfrac{11}{5} - \dfrac{2}{5} i\).

    In the next section we will discuss the geometry of complex numbers, which gives some insight into the meaning of the magnitude of a complex number. For now we just give the definition.

    Definition: Magnitude

    The magnitude of the complex number \(x + iy\) is defined as

    \(|z| = \sqrt{x^2 + y^2}\).

    The magnitude is also called the absolute value, norm or modulus.

    Example \(\PageIndex{3}\)

    The norm of \(3 + 5i = \sqrt{9 + 25} = \sqrt{34}\).

    Important. The norm is the sum of \(x^2\) and \(y^2\). It does not include the \(i\) and is therefore always positive.