Complex Numbers – Beyond Real Numbers

Complex Number in Argand Plane

Complex Numbers – Examples & Exercises

Square of any real number is either positive or zero, never negative. So, how to solve the equation x2+1=0? No number x exists in the set of real numbers such that 1 added to the square of x may equal zero. Hence, we need complex numbers, a further extension of the number system beyond the real numbers.

  1. Definition of Complex Numbers

An ordered pair of real numbers, written as (a, b) is called a complex number z. The real number a is called the real part and the real number b is called the imaginary part. That is, a number of the form a+ib where a, b \(\in\) R and i=\(\sqrt{-1}\) is called a complex number. Here, \(i\) is called iota. Examples: 5+3i, -1+i, 0+4i, 4+0i. The set of complex number is denoted by C. Symbolically, \[C = \left\{ x + iy : x \in R, y \in R, i = \sqrt{-1} \right\}\]

Equality of Complex Numbers

Two complex numbers z1=(x1 + iy1) and z2=(x2 + iy2) are equal if and only if their real and imaginary parts are separately equal, i.e.
z1 = z2 \(\Leftrightarrow\) x1= x2 and y1 = y2.

NOTE: (a+ib)> or < (c+id) is not defined. So, the statement (9+6i) > (3+2i) makes no sense.

PURELY REAL NUMBERS & PURELY IMAGINARY NUMBERS

The complex number (a, 0) is purely real number. We write it as a. The complex number (0, 1) is purely imaginary number. We write it as \(i\). All real numbers are in fact complex numbers. That is, the set of real numbers is a proper subset of the set of complex numbers.

As we see, (0,1).(0,1) = (-1,0), which is purely real and equals to -1. So, \(i \times i = -1\) \(\Rightarrow i = \sqrt{-1}\).

It solves x²+1=0. In general, the square root of any negative number can be written as the product of a real number and \(i\), i.e.\(\sqrt{-a} = \sqrt{a}i\).

  1. Algebra of Complex Numbers

Complex numbers obey the commutative and the associative rules for operations of addition and multiplication. Subtraction is not commutative. This set of numbers follows distributive law for multiplication over addition. Over subtraction? Find out yourself. The multiplicative inverse of a complex number x+iy is \[\frac{x}{x^2 + y^2} + i\frac{(-y)}{x^2 + y^2}\]

Power of \(i\)

Any positive integral power of i can have one of the only four values, viz. \(i, -1, -i, 1\). Evidently, two of these are real and the other two imaginary. The value of the negative integral power of i is one of the four values: \(-i, -1, i, 1\). These values of power follow the principle of periodicity.

THE OPERATORS

Let z1 and z2 be two complex numbers such that z1=a+ib and z2=c+id. Then,

z1+z2 = (a+ib)+(c+id) = (a+c)+i(b+d)

z1-z2 = (a+ib)-(c+id) = (a-c)+i(b-d)

z1.z2 = (a+ib)(c+id) = (ac-bd)+i(ad+bc)

Division of complex Numbers, z1 \(\div\) z2 \[ \frac{a + ib}{c + id} = \frac{(ac + bd)}{c^2 + d^2} + \frac{i(bc – ad)}{c^2 + d^2}\]

  1. Conjugate of a Complex Number

Every complex number a+ib has a corresponding complex number a-ib where each is called the conjugate of the other. The conjugate of \(z\) is denoted by \(\overline{z}\). Geometrically, conjugates are point or mirror images in real axis.

Properties of the Conjugates

  • \(\overline{(\overline{z})} = z\)
  • \(\overline{{z_1 + z_2}} = \overline{z_1}+ \overline{z_2}\)
  • \(\overline{{z_1 – z_2}} = \overline{z_1}- \overline{z_2}\)
  • \(\overline{{z_1.z_2.z_3…z_n}} = \overline{z_1}.\overline{z_2}.\overline{z_3}…\overline{z_n}\)
  • \(\overline{{\left( \frac{z_1}{z_2} \right)}} = \frac{\overline{{z_1}}}{\overline{{z_2}}}, z_2 \neq 0\)
  • \((\overline{z})^n = (\overline{{z^n}})\)
  • \(\newcommand{\repart}{\operatorname{Re}} z + \overline{z} = 2\repart(z) = 2\repart(\overline{z}) =\) purely real number
  • \(\newcommand{\impart}{\operatorname{Im}} z – \overline{z} = 2i\impart(z) =\) purely imaginary number
  • \(z\overline{z} = |z|^2 = \) purely real
  • \(z_1\overline{{z_2}} + \overline{{z_2}}z_1\)
  • \(\newcommand{\repart}{\operatorname{Re}} 2\repart((z_1\overline{{z_2}})) = 2\repart((\overline{{z_1}}z_2))\)

ARGAND PLANE

The Argand plane is a 2-D plane plotting complex numbers as points. Here, \(x\)-axis is real axis and \(y\)-axis is imaginary axis. Here, the radius represents the modulus |z| of the complex number \(z=x+iy\) and the angle \(\theta\) that the radius makes with the +ve x-axis represents the complex argument.

The significance of the Argand plane lies in the fact that it visualizes the “imaginary” part of the complex numbers. This visualization helped in their acceptance as a natural acceptance to –ve integers along the real line.  

  1. Modulus of Complex Numbers

Mod of a complex number z=a+ib is defined by a positive real number given by \(|z| = +\sqrt{a^2 + b^2}\), where a and b are real numbers. Geometrically, modulus is the distance of the complex number from the origin. Again, \(|z| = OP = \sqrt{a^2 + b^2}\)

NOTE: Since modulus of a complex number is always +ve, it is wrong to say that if \(z = 1 + i\tan \theta\), then \(|z| = \sqrt{1 + \tan^2\theta} = \sec \theta\), because \(\sec \theta\) may be negative for some values of \(\theta\). We have to specify range, hence.

Argument / Amplitude

Angle between positive vector of a complex number \(z\) with positive direction of x-axis is called amplitude or argument of \(z\). For a complex number, \(z = x + iy\), amplitude or argument is given by \(\arg z = \tan^{-1}\mkern-5mu\left( \frac{y}{x} \right).\)

PROPERTIES OF MODULUS

  • \(|z| \geq 0 \Rightarrow |z| = 0\) if \(z=0\) and \(|z| > 0\) if \(z \neq 0\)
  • \(|z| = |\overline{z}| = |-z| = |-\overline{z}| = |zi|\)
  • \(z\overline{z} = |z|^2 = |\overline{z}|^2\)
  • \(|z_1z_2z_3 ⋯ z_n| = |z_1||z_2||z_3| ⋯ |z_n|\)
  • \(\left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|}\) where \(z_2 \neq 0\)
  • \(|z^n| = |z|^n\) where \(n \in N\)
  • \(\newcommand{\repart}{\operatorname{Re}}|z_1 + z_2|^2 = (z_1 + z_2)(\overline{{z_1}} + \overline{{z_2}}) = |z_1|^2 + |z_2|^2 + (z_1\overline{{z_2}} + \overline{{z_1}}z_2) = |z_1|^2 + |z_2|^2 + 2\repart((z_1\overline{{z_2}}))\)
  • \(\newcommand{\repart}{\operatorname{Re}}|z_1 – z_2|^2 = (z_1 – z_2)(\overline{{z_1}} – \overline{{z_2}}) = |z_1|^2 – |z_2|^2 – (z_1\overline{{z_2}} – \overline{{z_1}}z_2) = |z_1|^2 – |z_2|^2 – 2\repart((z_1\overline{{z_2}}))\)
  • \(|z_1 + z_2|^2 + |z_1 – z_2|^2 = 2(|z_1|^2 + |z_2|^2)\)

PROPERTIES OF TRIANGLE INEQUALITY

  • \(|z_1 + z_2| \leq |z_1| + |z_2| \geq ||z_1| – |z_2||\)
  • \(|z_1 – z_2| \leq |z_1| + |z_2| \geq ||z_1| – |z_2||\)
  1. Cube Roots of Unity

\(x = \sqrt[3]{1}\)

\(\Rightarrow x^3 – 1 = 0\)

\(\Rightarrow (x – 1)(x^2 + x + 1) = 0\)

\(\therefore \sqrt[3]{1} = 1, \frac{-1 + i\sqrt{3}}{2}, \frac{-1 – i\sqrt{3}}{2}\)

Properties of Cube Roots of Unity

  • One of the three cube roots of unity is real and the other two are conjugate complex numbers.
  • Each of the complex roots is square of the other. If one of the non-real roots is \(\omega\), the other non-real root is \(\omega^2\) and vice-versa.
  • \(1 + \omega + \omega^2 = 0\)
  • \(\omega^{3n} = 1\)
  • \(\omega^{3n+1} = \omega\)
  • \(\omega^{3n+2} = \omega^2\)
  • \(1 + \omega^n + \omega^{2n} = 0\) or \(3\) {0 when n not a multiple of 3; 3 when n is a multiple of 3.}
  • Cube roots of -1 are \(-1, -\omega , -\omega^2.\)

PROPERTIES OF \(n^{th}\) ROOTS OF UNITY

  • The sum of all n roots of unity is zero
    i.e., \(1 + \alpha + \alpha^2 + ⋯ +\alpha^{n – 1} = 0\).
  • The product of all n roots of unity is \((-1)^{n – 1}\).
  1. De Moivre’s Theorem & Deductions

If \(n \in I\),

  • \((\cos \theta + i\sin \theta)^n = \cos n\theta + i \sin n\theta\)
  • \((\cos \theta – i\sin \theta)^n = \cos n\theta – i \sin n\theta\)
  • \((\cos \theta + i\sin \theta)^{-n} = \cos n\theta – i \sin n\theta\)
  • \((\cos \theta – i\sin \theta)^{-n} = \cos n\theta + i \sin n\theta\)
  • \((\sin \theta + i\cos \theta)^n = \cos n\mkern-5mu\left( \frac{\pi}{2} – \theta \right) + i\sin n\mkern-5mu\left( \frac{\pi}{2} – \theta \right)\)

Clark University – Dave’s short course on Complex Numbers

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