1 The Complex Number System and its Properties
Definition 1.1. A complex number is any number of the form \(z = a + ib\). Here \(a\) is called the real part of \(z\) and is denoted by \(\operatorname {Re}(z)\). \(b\) is called the imaginary part of \(z\) and is denoted by \(\operatorname {Im} (z)\).
Definition 1.2. Complex numbers \(z_1 = a_1 + ib_1\) and \(z_2 = a_2 + ib_2\) are equal if \(a_1 = b_1\) and \(a_2 = b_2\).
Arithmetic Operations
- 1.
- \(z_1 + z_2 = \big (a_1 + ib_1\big ) + \big (a_2 + ib_2\big ) = \big (a_1+a_2\big ) + i\big (b_1 + b_2\big )\)
- 2.
- \(z_1 - z_2 = \big (a_1 + ib_1\big ) - \big (a_2 + ib_2\big ) = \big (a_1+a_2\big ) - i\big (b_1 + b_2\big )\)
- 3.
- \(z_1z_2 = \big (a_1 + ib_1\big )\big (a_2 + ib_2\big ) = \big (a_1a_2 - b_1b_2\big ) + i \big (a_1b_2 + a_2b_1\big )\)
- 4.
- \(\frac {z_1}{z_2} = \frac {a_1 + ib_1}{a_2 + ib_2} = \frac {a_1a_2 + b_1b_2}{a_2^2 + b_2^2} + i \,\frac {b_1a_2 - a_1b_2}{a_2^2 + b_2^2}\quad , \quad a_2\neq 0\,,\, b_2\neq 0\)
The usual commutative, associative and distributed laws holds for complex numbers.
Check!!
The zero of the complex number system is \(\, 0 + 0i\,\) and the unity is \(\, 1 + 0i\). The zero and the unity are denoted by \(0\)
and \(1\) respectively.
The conjugate of a complex number \(z = a + ib\) is denoted by \(\overline {z}\) and is the complex number \(\overline {z} = a - ib\).
The conjugate has the following properties
- i).
- \(\overline {z_1 + z_2} = \overline {z_1} + \overline {z_2}\)
- ii).
- \(\overline {z_1 - z_2} = \overline {z_1} - \overline {z_2}\)
- iii).
- \(\overline {z_1 \cdot z_2} = \overline {z_1} \cdot \overline {z_2}\)
- iv).
- \(\overline {\Bigg (\frac {z_1}{z_2}\Bigg )} = \frac {\overline {z_1}}{\overline {z_2}}\)
- v).
- \(\overline {\overline {z}} = z\)
Moreover, the following are true for any complex number \(z = a + ib\)
- i).
- \(z + \overline {z} = \big ( a + ib\big ) + \big ( a - ib\big )= 2a\)
- ii).
- \(z\overline {z} = \big ( a + ib\big )\big ( a - ib\big ) = a^2 + b^2\)
- iii).
- \(z - \overline {z} = \big ( a + ib\big ) - \big ( a - ib\big ) = 2ib\)
Hence we get \(\,\operatorname {Re}(z) = \frac {z + \overline {z}}{2}\,, \quad \operatorname {Im} (z) = \frac {z - \overline {z}}{2i}\)
Complex numbers also have inverse. The inverse of a complex number \(z = a + ib\,\) is
\(\,-z = - a - ib\). Also, the reciprocal of a complex number \(z\) is \(\frac {1}{z}\)
Remark 1.3. Unlike real numbers, complex numbers are not ordered. Thus concepts like \(\, z_1 < z_2\,\) do not hold in the complex numbers.
1.2 The Triangle Inequality
1.3 Polar Coordinates
1.3.1 Multiplication and Division
1.4 Powers and Roots
1.5 The Exponential Representation
1.6 Lines
1.7 Practice Problems
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