The Set of Complex Numbers: Definitions, Operations, Modulus, and Geometric Representation

Definition and Set of Complex Numbers

• A complex number $z$ is defined as any ordered pair $(a, b)$ of real numbers.

• Formal notation:

  • $z = (a, b)$ with $a, b \in \mathbb{R}$.

• The set of all complex numbers is designated by $\mathbb{C}$.

• Null element identity and property:

  • $(a, 0) = a(1, 0) = 1(0, 0) = 0$.

Cartesian Form of a Complex Number

• Vector decomposition of the ordered pair:

  • For any complex number $z = (a, b)$, the pair decomposes as:

  • $z = (a, 0) + (0, b) = a(1, 0) + b(0, 1) = a + b(0, 1)$.

• Introduction of the imaginary unit:

  • By defining the imaginary unit as $i = (0, 1)$, substitution yields the Cartesian or algebraic form:

  • $z = a + bi$.

• Components of the Cartesian form:

  • Real part: The real number $a$ is called the real part of $z$ and is denoted by $\operatorname{Re}(z)$.

  • Imaginary part: The quantity $bi$ is called the imaginary part of $z$ and is denoted by $\operatorname{Im}(z)$.

  • Imaginary unit: The symbol $i$ denotes the imaginary unit.

  • Full expression in terms of parts:

  • $z = a + bi = \operatorname{Re}(z) + \operatorname{Im}(z)$.

• Classification of complex numbers:

  • Real numbers:

  • A complex number $z$ is real if and only if its imaginary part is zero:

  • $\operatorname{Im}(z) = 0$.

  • Examples of real numbers:

    • $z = 2$

    • $z = -143$

    • $z = \sqrt{3}$

  • Purely imaginary numbers:

  • A complex number $z$ is purely imaginary if and only if its real part is zero:

  • $\operatorname{Re}(z) = 0$.

  • Examples of purely imaginary numbers:

    • $z = 3i$

    • $z = \sqrt{3}i$

    • $z = i$

• Fundamental operational convention:

  • All standard arithmetic calculation rules apply to complex numbers, under the single governing convention:

  • $i^2 = -1$.

Integer Powers of the Imaginary Unit

• Integer powers of the imaginary unit $i$ evaluated sequentially:

  • Second power:

  • $i^2 = -1$

  • Third power:

  • $i^3 = i^2 \cdot i = (-1)i = -i$

  • Fourth power:

  • $i^4 = (i^2)^2 = (-1)^2 = 1$

  • Fifth power:

  • $i^5 = i^4 \cdot i = (i^2)^2 \cdot i = (-1)^2 \cdot i = 1 \cdot i = i$

  • Sixth power:

  • $i^6 = (i^2)^3 = (-1)^3 = -1$

  • Seventh power:

  • $i^7 = i^6 \cdot i = (i^2)^3 \cdot i = (-1)^3 \cdot i = (-1) \cdot i = -i$

  • Eighth power:

  • $i^8 = (i^2)^4 = (-1)^4 = 1$

  • Ninth power:

  • $i^9 = i^8 \cdot i = (i^2)^4 \cdot i = (-1)^4 \cdot i = 1 \cdot i = i$

Algebraic Operations in the Set of Complex Numbers

• General validity:

  • All algebraic operations defined in $\mathbb{R}$ remain valid in $\mathbb{C}$.

• Opposite of a complex number:

  • For any complex number $z = a + bi$, its additive opposite is denoted $-z$ and defined as:

  • $-z = -a - bi$.

• Complex conjugate:

  • For a complex number $z = a + bi$, its complex conjugate is denoted $\overline{z}$ and defined as:

  • $\overline{z} = a - bi$.

• Multiplicative inverse of a non-zero complex number:

  • For any non-zero complex number $z = a + bi$, the inverse $z^{-1}$ is obtained by multiplying numerator and denominator by the conjugate $\overline{z} = a - bi$:

  • $z^{-1} = \frac{1}{z}$

  • $z^{-1} = \frac{1}{a + bi}$

  • $z^{-1} = \frac{1 \cdot (a - bi)}{(a + bi)(a - bi)}$

  • $z^{-1} = \frac{a - bi}{a^2 - (bi)^2}$

  • $z^{-1} = \frac{a - bi}{a^2 - b^2 i^2}$

  • $z^{-1} = \frac{a - bi}{a^2 - b^2(-1)}$

  • $z^{-1} = \frac{a - bi}{a^2 + b^2}$

  • Decomposed into real and imaginary parts:

  • $z^{-1} = \frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i$.

• Addition and subtraction:

  • For two non-zero complex numbers $z_1 = a + bi$ and $z_2 = c + di$:

  • $z_1 \pm z_2 = (a \pm c) + (b \pm d)i$.

• Multiplication:

  • For two complex numbers $z_1 = a + bi$ and $z_2 = c + di$, multiplication is performed by distribution:

  • $z_1 \cdot z_2 = (a + bi)(c + di)$.

Modulus of a Complex Number

• Definition of modulus:

  • For any complex number $z = a + bi$, the modulus is the non-negative real number denoted by $|z|$ or $r$.

  • Derivation via the product of a complex number with its conjugate:

  • $|z| = \sqrt{z\overline{z}}$

  • $|z| = \sqrt{(a + bi)(a - bi)}$

  • $|z| = \sqrt{a^2 - (bi)^2}$

  • $|z| = \sqrt{a^2 - b^2 i^2}$

  • General modulus formula:

  • $|z| = \sqrt{a^2 + b^2}$.

• Example 1:

  • Given complex number:

  • $z = 3 + 4i$

  • Real and imaginary parts:

  • $a = 3$

  • $b = 4$

  • Modulus calculation:

  • $r = \sqrt{a^2 + b^2}$

  • $r = \sqrt{3^2 + 4^2}$

  • $r = \sqrt{9 + 16}$

  • $r = \sqrt{25} = 5$.

• Example 2:

  • Given complex number:

  • $z = -3i = 0 - 3i$

  • Real and imaginary parts:

  • $a = 0$

  • $b = -3$

  • Modulus calculation:

  • $r = \sqrt{a^2 + b^2}$

  • $r = \sqrt{0^2 + (-3)^2}$

  • $r = \sqrt{0 + 9}$

  • $r = \sqrt{9} = 3$.

• Example 3:

  • Given complex number:

  • $z = \frac{1}{2} + \frac{\sqrt{3}}{2}i$

  • Real and imaginary parts:

  • $a = \frac{1}{2}$

  • $b = \frac{\sqrt{3}}{2}$

  • Modulus calculation:

  • $r = \sqrt{a^2 + b^2}$

  • $r = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2}$

  • $r = \sqrt{\frac{1}{4} + \frac{3}{4}}$

  • $r = \sqrt{\frac{1 + 3}{4}}$

  • $r = \sqrt{\frac{4}{4}} = \sqrt{1} = 1$.

Geometric Representation of Complex Numbers

• The complex plane (Gauss plane):

  • Let $\pi(O, \vec{j}, \vec{k})$ be a geometric plane mapped to an orthonormal coordinate frame $(O, \vec{j}, \vec{k})$.

  • For every point $P$ in the plane, the position vector $\vec{OP}$ is expressed as:

  • $\vec{OP} = a\vec{j} + b\vec{k}$ with $(a, b) \in \mathbb{R}^2$.

  • The point $P$ is determined by coordinates $(a, b)$ and denoted $P(a, b)$.

  • Plane identification: The plane $\pi$ identified with the set $\mathbb{C}$ is called the complex plane or Gauss plane.

• Affix and image:

  • Image: The point $P(a, b)$ that represents the complex number $z = a + bi$ is called the image of $z$.

  • Affix: The complex number $z$ is called the affix of the point $P$.

• Coordinate axes:

  • Real axis: Real complex numbers are represented by points along the $Ox$ axis, called the real axis.

  • Pure imaginary axis: Purely imaginary complex numbers are represented by points along the $Oy$ axis, called the axis of pure imaginaries.

• Determination of the geometric position of the image:

  • The position of the image of $z$ is fixed by two quantities:

  • The length of the position vector $\vec{OP}$, which equals the modulus of $z$ ($|z|$ or $r$).

  • The measure of the oriented angle $\theta = \widehat{AOP}$, which constitutes the argument of $z$.