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:
- Closure law: If \(z_1\) and \(z_2\) are complex numbers, then \(z_1+z_2\) is also a complex number.
- Commutative law: \(z_1+z_2=z_2+z_1\)
- Associative law: \((z_1+z_2)+z_3=z_1+(z_2+z_3)\)
- Existence of additive identity: \(0+i0\), or simply \(0\), is the additive identity or zero complex number. \([z+0=z]\)
- 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.
Properties:
- Closure law: If \(z_1\) and \(z_2\) are complex numbers, then \(z_1\,z_2\) is also a complex number.
- Commutative law: \(z_1\,z_2=z_2\,z_1\)
- Associative law: \((z_1\,z_2)\,z_3=z_1\,(z_2\,z_3)\)
- Existence of multiplicative identity: \(1+i0\), or simply \(1\), is the multiplicative identity. \([z.1=z]\)
- Existence of multiplicative inverse: If \(z=a+ib\), then \(z^{-1}\) or \(\frac{1}{z}\) is the multiplicative inverse of \(z\). \([z.z^{-1}=1]\)
- Distributive law:
- \(z_1\,(z_2+z_3) = z_1\,z_2+z_1\,z_3\)
- \((z_1+z_2)\,z_3 = z_1\,z_3+z_2\,z_3\)
Finding multiplicative inverse:
Let \(z=a+ib\) is a complex number, then it's inverse is
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.
- \((z_1+z_2)^2=z_1^2+z_2^2+2\,z_1\,z_2\)
- \((z_1-z_2)^2=z_1^2+z_2^2-2\,z_1\,z_2\)
- \((z_1+z_2)^3=z_1^3+3\,z_1^2\,z_2+3\,z_1\,z_2^2+z_2^3\)
- \((z_1-z_2)^3=z_1^3 - 3\,z_1^2\,z_2 + 3\,z_1\,z_2^2 - z_2^3\)
- \(z_1^2-z_2^2=(z_1+z_2)(z_1-z_2)\)