Complex Number

Complex number is a number that consists of real number and imaginary number. It is expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit (Here, i² = −1). So complex numbers are combination of real and imaginary number. For example,

5 + i,    68 + 3i,     0.8 − 0.5i,     −2 + πi,     √2 + i/2

Above numbers are example of complex number. All the numbers are in the form of a+bi.

Complex number helps us to solve certain types of equations that is not possible by real numbers.
Suppose ,the equation

 x² + 7 = − 9

It has no real solution, since the square of a real number cannot be negative. With the help of complex number we can reach to a solution. We know that imaginary unit is i where i² = −1. So, the equation can be written as,

x² = (− 9). (− 7) 

or x² = − 16 

or x² = (−1).(16)

or x² = i² . 16 

or x = 4i

Properties of complex quantities:
i)  If  a+bi = 0 then a=0, b=0

ii) If  a+bi = c+di  then a=c and b=d
So if two complex numbers are equal then real and complex part of first number will be equal to real and complex part of second number.

iii) The result of addition and multiplication of the two conjugate complex quantities is always real.
The complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude but opposite in sign. for example, the complex conjugate of  3 + 4i is 3 − 4i [From wikipedia]
 (a+bi) + (a-bi)= 2a      (a real number)
 (a+bi) . (a-bi)= a²+b²  (a real number)

iv) The result of addition, subtraction, multiplication and division of two complex numbers(not complex conjugate) will always be a complex number.

(a+bi) + (c+di)   = (a+c) + (b+d)i

(a+bi) − (c+di)  = (a−c) + (b−d)i

(a+bi) . (c+di)    = (ac-bd) + (bc+ad)i

(a+bi) / (c+di)    = (ac+bd/c²+d²) + (bc-ad/c²+d²)i

v) The root of a complex number is a complex one.

     ⁿ√a+bi = x 

 Cartesian Form of Complex Numbers

Complex number is used in various field of science,engineering as well as in mathematics to explain and understand scientific phenomena. In engineering, it is sometimes denoted by j.