complex number

Complex Numbers

Complex numbers are essentially a composite number comprising of a real and an imaginary components . The real component of a complex number is a real number which can be geometrically represented on a number line . The imaginary component on the other hand is an imaginary number which is obtained by multiplying a real number with square root of -1 , which is represented as i or sometimes as j as well , in this essay we will use the symbol i

or i2 -1 . Because , multiplying a number by -1 or i2

rotates it by 180o (counter clockwise ) in x-y plane , therefore , multiplying a number by i rotates it by 90o (counter clockwise . Thus a complex number can be geometrically represented in an x-y plane

along y-axis of length b (Figure 1

Figure 1 : Geometric representation of a complex number

If b 0 , then z a i0 a is purely real number and we find that real numbers are a subset or a special case of complex numbers , imaginary component of which is zero if a 0 , then z 0 ib ib is purely complex number

2 3i . It is worth mentioning here that two unequal complex numbers are not necessarily greater than or less than the other . The concept of greater or smaller is limited to real domain only and therefore , this concept is restricted only for those situations of complex numbers where it is either purely real or purely complex . For example 7 0i 2 0i and 0 7i 0 2i simply because 7 2 . But we cannot write 7 2i 5 i , despite the fact that both real as well as imaginary components of the former is greater than that of the latter

Addition of Complex numbers

When adding two or more of complex numbers , the real and the imaginary components are added separately and the resultant number is also a complex number . To take an example (a ib (c id (a c (b d )i (2 3i (4 8i (2 4 ( 3 8 )i 6 11i (5 2i (6 - 5i 11 - 3i . Because , summation is being carried out in real space only , therefore , addition of complex numbers is similar to that of real numbers , the only difference is that the real and the complex components has to be added separately

Geometrically , addition of complex numbers is vector addition of the vectors representing the complex numbers (Figure 2 . An example of addition of two complex numbers 4 i and 2 3i is geometrically represented in figure 2 below

Algebraically (4 i (2 3i (4 2 (1 3 )i 6 4i

Figure 2 : Geometric representation of addition of two complex numbers

Multiplication of Complex numbers

Multiplication of complex numbers (a ib ) and (c id ) is carried out in following manner (a ib (c id a (c id ib (c id ac iad ibc i2bd ac iad ibc (-1 )bd (a ib (c id (ac-bd i (ad bc

Geometrically , multiplication of two complex numbers (vectors ) yields another complex number...