The complex numbers have some properties to simplify an equation. The definition of a complex number in short can be described, as it is a composition of a real number and an imaginary number. It can be described better with an example: z=15+7i. This equation consists of a real part, 15, and an imaginary part 7i. Therefore, it can be considered a complex number and the value of ‘i’ is equal to the square root of (-1). Two complex numbers cannot be compared as ‘greater than or less than’ until their properties are compared. The complex numbers only can be equal if b=d and a=c.

The Properties of Complex Numbers

So many properties of a complex number help in rearranging any equation consisting of a complex number. One of them is the properties of the addition of complex numbers.

• The properties of addition of complex numbers: If, z₁=a+ib and z₂=c+id are two of the complex numbers, therefore, z₁+z₂=(a+c)+i(b+d).
• The properties of subtraction of a complex number: For the same equations, the subtraction of z₁ and z₂ will be,  z₁–z₂=(a+c)-i(b+d).
• The properties of multiplication: z₁.z₂=(a+ib)(c+id)=(ac–bd)+i(ad+bc)
• Conjugation: For the two examples and the conjugate of the complex number as▁z=a–ib, the conjugation will be prioritised. It means that the properties under the conjugation mark will have to be calculated at first.
• Modulus: The modulus of z,|z|, will always give the positive value of z. As, |-z|=z.

The Theorems on Complex Numbers

There are many theorems on the properties of a complex number and some of the theorems on complex numbers are as follows.

• De Moivre’s Theorem- De Moivre’s Theorem is one of the theorems on complex numbers, which is based on the nth value of the properties of addition of complex numbers. According to this theorem, if,z=r[cos(nθ)+isin(nθ)], z n = r n [cos(nθ)+isin(nθ)]. This theorem can be used in the properties of addition of complex numbers, simplifying, or proving some trig identities.
• Pythagoras’ Theorem- It is one of the theorems on complex numbers, which is based on the factor theorem. According to the Pythagoras Theorem, in a right triangle, the hypotenuse’s square will always be equal to the square of the sum of the other two sides. The properties of the addition of complex numbers can also be used in terms of simplifying this theorem in applications. This theorem is used to calculate the complex number’s absolute value. As per this theorem, the absolute value of a complex number is the distance between the point of the complex number on the complex plane and the origin.

Commutative Law for Addition

The Commutative Law for Addition is all about ordering any complex number’s equation.  The properties of the addition of complex numbers are a part of the Commutative Law for Addition and are very important for multiplication, subtraction, or division. For example, 3+6i can be taken, and as per the Commutative Law for Addition, 6i+3 is also equal to the example. Some of the examples are shown below to describe the functions of the Commutative Law for Addition:

1. (2+3i)+(2+3i)=2(2+3i)=4+6i
2. (2+3i)+(4-2i)=6+i

The commutative law also plays important roles in calculating and ordering the equations of mathematics. The properties of complex numbers also help in simplifying the equations. These also help to improve student’s knowledge and explore more sites of mathematics.

Conclusion

The complex numbers and their properties are so important to a student’s life. The complex number can be ignored in any aspect of science. It plays a very important role in the world of mathematics, physics, and other science topics. The complex numbers give a simplified knowledge of some complex phonemes’. Therefore, the properties of the addition of complex numbers, the theorems on complex numbers, and the Commutative Law for Addition are the toolkits to solve and simplify the equations of a complex number. The complex numbers and their properties have contributions in terms of simplifying the complex topics and data of the scientific world.