COMPLEX NUMBERS

Complex numbers are written in the form \(a+ib\) where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit such that

\(i=\sqrt{-1}\) or \(i^2=-1\)
Sometimes the letter \(z\) is used to denote a complex number, \(z: z=a+ib\). A complex number can also be written as an ordered pair of its real numbers, \((a, b)\).
\(a\) is also known as the real part i.e. \(Re(z)=a\)
\(b\) is also known as the imaginary part i.e. \(Im(z)=b\)
SQUARE ROOT OF NEGATIVE NUMBERS
Thus complex numbers can be used to find the square roots of negative numbers. Examples
\(\sqrt{-16}=\sqrt{(-1)16}=\sqrt{-1} \sqrt{16}=4i\)
\(\sqrt{-18}=\sqrt{(9)(2)(-1)}=3\sqrt{2} i\)
With this extension of the number system we can now solve equations which were once unsolvable.

OPERATIONS ON COMPLEX NUMBERS
ADDING AND SUBTRACTING COMPLEX NUMBERS
\((a+bi)±(c+di)=(a±c)+(b±d)i\)
For example,
(i) \((6+2i)+(5-4i)=(6+5)+(2-4)i\)
\(=11-2i\)
(ii) \((-1-i)-(8-2i)=(-1-8)+(-1+2)i\)
\(=-9+i\)

MULTIPLYING COMPLEX NUMBERS
(i) \(2i(3i)=6i^2=6(-1)=-\)
(ii) \(-2i(4+3i)=-8i-6i^2=6-8i\)
(iii) \((3+2i)(5-4i)=15-12i+10i-8i^2\)
\(=15+8-12i+10i\)
\(=23-2i\)
(iv) \((a+bi)(a-bi)=a^2-b^2 i^2=a^2+b^2\)
DIVIDING COMPLEX NUMBERS
If \(z=a+bi\) then \(\overline{z}=a-bi\) is its conjugate and vice versa. It is also important to note that the product of a complex number and its conjugate is \(a^2+b^2\) which is always a Real number. The conjugate is also denoted \(z*\).
(i) \(\displaystyle {3i\over 2+i}\)
  • First we find the conjugate of the denominator
\(z=2+i→\overline{z}=2-i\)
  • Multiply the numerator and the denominator by the conjugate
\(\displaystyle {3i(2-i)\over(2+i)(2-i)}={6i-3i^2\over 2^2+1^2}\)
\(\displaystyle ={6i+3\over5}\)
\(\displaystyle ={3\over5}+{6i\over5}\)

(ii) Express \(\displaystyle {3+7i\over 5-2i}\) in the form \(a+bi\)
  • Multiply numerator and denominator by the conjugate of the denominator.
\(\displaystyle {3+7i\over 5-2i}={(3+7i)(5+2i)\over(5-2i)(5+2i)}\)
  • Simplify
\(\displaystyle ={15+41i+14i^2\over 5^2+2^2}\)
\(\displaystyle ={1+41i\over29}\)
\(\displaystyle ={1\over29}+{41\over29} i\)