## 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
$$(a+bi)±(c+di)=(a±c)+(b±d)i$$
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
$$\displaystyle ={15+41i+14i^2\over 5^2+2^2}$$
$$\displaystyle ={1+41i\over29}$$
$$\displaystyle ={1\over29}+{41\over29} i$$