What are complex numbers


Do you want to know what are complex numbers? Today I want to discuss a special type of numbers which we won't use in general those numbers are called complex numbers. What are complex numbers? Where this complex numbers are used? Do complex numbers form an Abelian Group? I will discuss in this article.

In Mathematics we come across several numbers like the set of natural numbers N= {0, 1,2,3}, the set of odd numbers like 0= {1,3,5}, the notation of even numbers are indicated E = {0,2,4,6,...}, the arrangement of prime numbers signified by P= (3,7,....) genuine numbers are the set meant by R= {-6, square root 5, a division of 4/3,....} etc. All are utilized as a part of our everyday life and every number has some essential standard. On the off chance that you consider a prime number if a number is said to be prime, then it ought to be divisors of 1 and itself. For instance 3 =1*3 and 7= 1*7 like this. If you consider even numbers, the numbers which are divided by 2 say for instance 2/2 =0 4/2=0 like this, if you consider odd number 1/2!=0, 3/2! =0 like this that implies which are not divisible by 2. On the off chance that you consider rational numbers a square root of 2 which results 1.414 and square root of 3 is 1.732 etc are said to be rational numbers.

Introduction to Complex numbers

Complex numbers do not have a single digit like the ordinary numbers - they are in the form of a+ib where a is the real part and ib is the imaginary part. If C is the set of complex number denoted, then C= {2+3i,4+5i,4-5i,....}.

What is this imaginary part i

Complex numbers are a combination of real numbers and an imaginary number which is in the form of a+ib as i is the imaginary part. The i is defined like this: where i square =-1, i POW 3 =-i, i pow 4=1, i pow 5=i where POW denotes power raised.

A small tabular representation of the imaginary part i.
complex numbers

What is the purpose of complex numbers

In general complex numbers will not be used. However, there are some equations whose result cannot be determined by the numbers which we are using then we use this complex number theory to solve the equations because to solve that equation an imaginary part should exist.

Do complex numbers form an Abelian Group

A Group (G, *) under multiplication is said to be group and it is following axioms or properties a) Closure (a*b)
b) Associative a* (b*c) = (a*b) *c
c) Identity (a*1= 1*a)
d) Inverse (1/a is multiplicative inverse)
e) Commutative (a*b=b*a) If the last axiom is followed then it is called Abelian Group.

Now see how the complex numbers will follow the Abelian Group:
Closure under multiplication: (a+ib) * (c+id) = ac+aid+ibc-bed = a (c+id) +b (side) - the resultant which is a complex number holds closure axiom
Associative under multiplication: a* (b*c) = (a*b) *c - the resultant will be again a complex number, so it holds associative axiom. Identity under multiplication: 1+0i is the multiplicative identity, so it holds identity axiom
Inverse under multiplication: Z= a+ib then 1/z also exists - this is again a complex number, so inverse satisfied so holds inverse axiom Commutative under multiplication.

Note

a= 0+i0 b= 0+i0 let us apply closure axiom a*b= (0+i0) (0+i0) = 0+0+0+ (-1)0=0 is not a complex number, so for complex numbers without zero they form an Abelian Group. Hence we can conclude non zero complex numbers forms an Abelian group.

Final note

Complex numbers are used in the other subjects like Physics, Chemistry, Statistics to derive new equations..


Attachments

  • complex (169765-1-complex.bmp)
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