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Arithmetic operations

Basics of complex numbers

The main idea behind complex numbers is the square root of a negative number.

In real numbers, negative numbers can't have a square root. Hence, complex numbers denote a special number, \(i=\sqrt{-1}\).

Complex number: a complex number is of the form \(z=a+ib\).

Where,

  • \(a\) - Real part (\(\text{Re}\,z\))
  • \(b\) - Imaginary part (\(\text{Im}\,z\))

Addition of complex numbers

Let \(z_1=a+ib\) and \(z_2=c+id\) be two complex numbers. Then, $$ z_1+z_2=(a+c)+i(b+d)$$

Tip

To add 2 complex numbers, add the real parts and the imaginary parts.

  • \(\text{Re}\,(z_1+z_2) = \text{Re}\,z_1+\text{Re}\,z_2\)
  • \(\text{Im}\,(z_1+z_2) = \text{Im}\,z_1+\text{Im}\,z_2\)

Properties:

  1. Closure law: If \(z_1\) and \(z_2\) are complex numbers, then \(z_1+z_2\) is also a complex number.
  2. Commutative law: \(z_1+z_2=z_2+z_1\)
  3. Associative law: \((z_1+z_2)+z_3=z_1+(z_2+z_3)\)
  4. Existence of additive identity: \(0+i0\), or simply \(0\), is the additive identity or zero complex number. \([z+0=z]\)
  5. Existence of additive inverse: If \(z=a+ib\), then \(-z=-a+i(-b)\) is the additive inverse or negative of \(z\). \([z+(-z)=(-z)+z=0]\)

Difference of complex numbers:

Let \(z_1=a+ib\) and \(z_2=c+id\) be two complex numbers. Then, $$ z_1-z_2=(a-c)+i(b-d)$$

Info

This is also considered as \(z_1-z_2=z_1+(-z_2)\). (Sum of \(z_1\) and negative of \(z_2\))

Tip

To subtract 2 complex numbers, subtract the real parts and the imaginary parts.

  • \(\text{Re}\,(z_1-z_2) = \text{Re}\,z_1-\text{Re}\,z_2\)
  • \(\text{Im}\,(z_1-z_2) = \text{Im}\,z_1-\text{Im}\,z_2\)

Multiplication of complex numbers:

Let \(z_1=a+ib\) and \(z_2=c+id\) be two complex numbers. Then, $$ z_1\,z_2=(ac-bd)+i(ad-bc)$$

Tip

You can get this by simply expanding the product.

\[ \begin{aligned} z_1 z_2 &= (a+ib)(c+id) \\ &= ac + a(id) + (ib)c + (ib)(id) &\\ &= ac + i(ad) + i(bc) + i^2(bd) &\\ &= ac + i(ad + bc) - bd \quad &[\because i^2 = -1]\\ &= (ac - bd) + i(ad + bc) \end{aligned} \]

Properties:

  1. Closure law: If \(z_1\) and \(z_2\) are complex numbers, then \(z_1\,z_2\) is also a complex number.
  2. Commutative law: \(z_1\,z_2=z_2\,z_1\)
  3. Associative law: \((z_1\,z_2)\,z_3=z_1\,(z_2\,z_3)\)
  4. Existence of multiplicative identity: \(1+i0\), or simply \(1\), is the multiplicative identity. \([z.1=z]\)
  5. Existence of multiplicative inverse: If \(z=a+ib\), then \(z^{-1}\) or \(\frac{1}{z}\) is the multiplicative inverse of \(z\). \([z.z^{-1}=1]\)
  6. Distributive law:
    1. \(z_1\,(z_2+z_3) = z_1\,z_2+z_1\,z_3\)
    2. \((z_1+z_2)\,z_3 = z_1\,z_3+z_2\,z_3\)

Finding multiplicative inverse:

Let \(z=a+ib\) is a complex number, then it's inverse is

\[z^{-1}=\frac{a}{a^2+b^2}+i\frac{-b}{a^2+b^2}\]

Division of complex numbers:

Let \(z_1\) and \(z_2\) be two complex numbers. Then, to find \(\frac{z_1}{z_2}\),

  • Step 1: Find \(z_2^{-1}\).
  • Step 2: Multiply \(z_1\) by \(z_2^{-1}\).

Powers of \(i\):

Let \(k\) be an integer, then the powers of \(i\) are:

  • \(i^{4k+1}=i^5=i^{-3}=i\)
  • \(i^{4k+2}=i^2=i^{-2}=-1\)
  • \(i^{4k+3}=i^3=i^{-1}=-i\)
  • \(i^{4k}=i^0=i^4=i^{-4}=1\)

Identities:

The following identities from read numbers can also be used for complex numbers.

  1. \((z_1+z_2)^2=z_1^2+z_2^2+2\,z_1\,z_2\)
  2. \((z_1-z_2)^2=z_1^2+z_2^2-2\,z_1\,z_2\)
  3. \((z_1+z_2)^3=z_1^3+3\,z_1^2\,z_2+3\,z_1\,z_2^2+z_2^3\)
  4. \((z_1-z_2)^3=z_1^3 - 3\,z_1^2\,z_2 + 3\,z_1\,z_2^2 - z_2^3\)
  5. \(z_1^2-z_2^2=(z_1+z_2)(z_1-z_2)\)