# Study of Complex Numbers With Their Properties

Complex numbers are the study of finding the square root of negative numbers. It was first studied bye by a Greek mathematician in the first century. He was finding the square root of negative numbers by adding the numeric root value. Then in the 16 century, an Italian mathematician was studying negative roots of cubic and quadratic polynomial expressions. The study of a complex number is used in scientific research processing of signals electromagnetism Quantum mechanism analysis of vibration and fluid dynamics.

A complex number is the sum of a real number and an imagery number. A complex number is written as a plus IB. It is denoted by z where a and b are the real numbers. Few examples are as follows 2 + 3i, -2 2 – 6i etc

I, the iota denoting the imaginary part of a complex number. Help to find the square root of negative numbers where the value of I square is equal to -1.

There are few properties of complex numbers to understand how complex numbers work in Arithmetic operation. Some of them are listed below-

- Conjugate of the complex number- In complex number the real part and the changing imagery part are the same as its additive inverse. If the sum and the product of two complex numbers are the numbers that are real then they are called conjugate complex numbers. For a complex number, z is equal to a + IB and its conjugate is minus z is equal to a minus b. Sum of conjugate and its complex number is z plus z^ is equal to (a + IB) plus (a – Ib) is equal to 2 a, in the same way, a product of complex number is z into z ^ is equal to (a + IB) x (a – ib) is equal to a square + b square.
- Reciprocal of the complex number- This is helpful in the division of complex numbers where you divide with another complex number that is equal to the product of one complex number with reciprocal of another complex number. Reciprocal of complex number z is equal to a + IB where z^- is equal to1 upon a + IB is equal to a minus ib upon a square + b square is equal to upon a square + b square plus I (- b) upon a square + b square where z is not equal to z^-
- Equality of complex numbers- Equality of complex numbers is the same as real numbers. Two complex numbers Z1 is equal to a 1 + I b 1 and Z2 is equal to A2 + iB2 are equal if the real part of both complex numbers is equal to A1 and A2 and the imagery part is equal to B1 and B2.
- Ordering of complex numbers- The ordering of complex numbers is never possible. Real numbers can be placed in order but complex numbers cannot be ordered.

The addition of complex numbers is done according to four laws-

Closure law – If two complex numbers Z1 Z2 you are added together then their sum is Z1 + Z2 is equal to Z2 plus Z1

Commutative law- If three complex numbers Z1, Z2, and, Z3 are added together, their addition is commutative where Z1 plus (Z2 + z3) is equal to (Z1 + Z2) plus z3.

Additive identity- Z is equal to a + IB where o is equal to O + Io such that z plus zero is equal to 0 + z is equal to zero

Additive inverse- In this z is equal to a + IB and minus that is equal to minus a minus IB, such that zed plus (minus z) is equal to (minus z) plus z is equal to zero and minus z its additive inverse.

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The study of a complex number is very important. It works according to loss and has q basic properties. A complex number is used in arithmetical operations. Complex numbers have been studied by math experts right from the first century.