Complex Number-Operations on Complex Number

Complex numbers are the numbers that are expressed in the form of ; a+ib

The history of mathematics shows the man has been developing and enlarging his concept of number according to the saying that, Necessity is the number of invention.

in the remote past they started with the set of counting numbers and invented, by stages, the negative numbers, rational numbers and irrational numbers.

Since square of a positive as well as negative number is a positive number, the square root of a negative number does not exist in the realm of real numbers. Therefore square root of negative numbers were given no attention for centuries together.

However, recently, properties of numbers involving square roots of negative number have also been discussed in detail and such numbers have been found useful and have been applied in many branches of pure and applied mathematics.

Definition of the Complex Number.

The number of the form z=x+iy where                           

 x, y ∈R and i=√(-1) are called complex numbers.

here x is called real part and y is called imaginary part of the complex number. Complex number is denoted by (Z)

(R) is denoted by real number.

For example:

• 3+5i

3 is called real part

5 is called imaginary part

• -17/2-7/2 i

Real part=-17/2

Imaginary part=-7/2

• Z=4+6i

Real part=4

Imaginary part=6

Does every real number a complex number?

Yes, every real number of complex number with ‘0’ as its imaginary part.

For example:

Z=7+0i is a complex number with 0 is imaginary part. And 7 is a real part.

∀ a ∈R 

Z=a+0i is a complex number with ‘0’ as it imaginary part. a is a real part.

Definition of imaginary number

Let us start with is considering the equation.

X²+1=0

X²=-1

Square roots of both sides

√x²=√ (-1)

X=±√ (-1)

√ (-1) does not belong to the set of real numbers. We therefore, √ (-1) is called it imaginary number. Imaginary number is denoted by I (read as iota).

The product of a real number and i(iota) is also called an imaginary number.

For example:

3i,-5i,√7 i ,9/2 i  are  all  imaginary numbers .i is also imaginary number.

Powers of i(iota):

i=√ (-1)

(i)²=-1       (by definition )

(i)³=(i)(i)²=(i)(-1)=-i

(i)⁴=(i)²(i)²=(-1)(-1)=1

(i)⁵=((i)²)²(i)=(-1)²(i)=(1)(i)=i

(i)¹³=(i²)⁶(i)=(-1)⁶(i)=(1)(i)=i

Thus any power of i(iota) must be equal to’1,-1,i,-i.

Conjugate of Complex Numbers

Complex numbers of the form (a+bi) and (a-bi) which have the same real parts and whose imaginary parts differ in sign only, are called conjugates of each other.

Conjugate of complex number is denoted by z .

Example:

Thus 5+4i and 5-4i, -2+3i and -2-3i are two pairs of conjugate numbers.

A real number is self-conjugate.

Operations on Complex Number

With a view to develop algebra of complex numbers, we state a few definitions. The symbols a, b, c, d, k, where used, represent real numbers.

• a+bi=c+di

a=c and b=d

• addition:(a+bi)=(c+di)=(a + c)+(b + d)i
• k(a+bi)=ka+kbi
• (a+bi)-(c+di)=(a-c)+(b-d)i
• (a+bi).(c+di)=(ac-bd)+(ad+bc)i

Complex Numbers as Ordered Pairs of Real Numbers

We can define complex numbers also by using ordered pairs.

Let C be the set of ordered pairs belonging to R x R which are subject to the following properties:-

1. (a , b)=(c , d)

iff a=c and b=d.

1. (a,b)+(c,d)=(a+c, b+d)
2. If K is any real number ,than k(a,b)=(ka,kb)
3. (a,b)(c,d)=(ac-bd,ad+bc)

Than C is called the set of complex numbers.

it is easy to see that.

(a,b)-(c,d)=(a-c,b-d).

There is no sense in saying that one complex number is greater than or less than .thus the set of C complex numbers does not satisfy the order axioms.

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