Properties of operation of complex numbers

Complex numbers are made of a real number and an imaginary number. Complex numbers are denoted as a + bi, where a is the exact number and b is the imaginary number. The real number comes before the imaginary number. Complex numbers are different from simple numbers simply because they consist of two parts and form a complex.

Continue reading this to understand the meaning of complex numbers, learn the division of complex numbers with various examples, and essential FAQs to clear your doubts on complex numbers.

Complex numbers definition

A Complex number is a sum of an actual number and an imaginary number.

Complex Number = a + bi

The real number (a) in the complex number is any tangible value whose square is always positive. Real numbers comprise positive and negative integers, fractions, decimals, irrational and rational numbers.

All real numbers are complex numbers, with their imaginary part having a value of zero.

The imaginary number (bi) in the complex number is the number whose square is always negative. All imaginary numbers (bi) have two parts. In the imaginary number bi, b is a non-zero real number, and “i” is called the imaginary unit known as iota. The squared value of the bit is -1.

  √ (-1), √ (-40) √ (-4), √ (-81) are all examples of imaginary units (i) as their squares are negative. Note that the square of the mythical team will always be negative.

5 + 6i, 27 + 3i, and 8 + 9i are all examples of complex numbers.

Properties of operations of complex numbers

We can perform many arithmetic operations on complex numbers. But since complex numbers are different from real numbers, the operation properties on complex numbers differ from that of real numbers.

Properties of Addition of complex numbers

The addition of complex numbers happens component-wise. First, real numbers are added together, and the imaginary units are added together.

Given are two complex numbers (a + bi) and (c + di)

Then, their addition will be:

(a + c) + (b + d)i

For example, 

Determine the addition of two complex numbers 5 + 8i and 13 + 6i

Answer: 18 + 14i

The properties of complex numbers and real numbers are very similar.

Let a, b, and c be three complex numbers, then the property of addition of these complex numbers are as follows:

1) Property of closure– The addition of complex numbers is a complex number.

2) Property of associativity– ( a + b )+ c = a +( b + c )

3) Property of commutativity– a + b = b + a

4) Property of additive identity– A complex number 0 (0 + 0i), which gives the same value when added to a complex number.

0 + a = a

5) Additive inverse– Given any complex number a, there always exists a unique complex number -a, such that a + (-a) = 0

Properties of subtraction of complex numbers

Just like addition, subtraction of complex numbers is also done component-wise.

The real numbers are subtracted, and the imaginary numbers are separately deducted from each other.

Given are two complex numbers (a + bi) and (c + di)

Then, their subtraction will be:

(a – c) + (b – d)i

For example,

Determine the subtraction of two complex numbers 15 + 8i and 13 + 6i

Answer: 2 + 2i

Let a and b be two complex numbers. Then the property of subtraction of these numbers are as follows:

1) Property of closure- The subtraction of complex numbers will result in a complex number.

2)The subtraction of complex numbers does not hold commutative and associative properties.

Properties of multiplication of complex numbers

Multiplication of two complex numbers is similar to the multiplication between binomials. The corresponding real and imaginary numbers are multiplied together. 

Given are two complex numbers, a and b, then the following are the properties of multiplication of these numbers:

1) Commutative property– a*b = b*a

2) Associative property– (a*b)*c = a*(b*c)

3) Multiplicative identity– The multiplicative identity of a complex number (a) is another complex number, such that their multiplication results in the same number (a). a*1 = a

The multiplicative identity of the complex number is 1.

4) Multiplicative inverse– The multiplicative inverse of a complex number a is another complex number z, such that a*z = 1

The multiplicative inverse of a complex number is the reciprocal of that complex number.

Properties of the division of complex numbers 

Given are two complex numbers, a + bi, and c + di, then the properties of the division of these complex numbers are as follows:

1) Property of closure– The division of two complex numbers will result in a complex number.

2)Complex conjugate– To divide two complex numbers, the imaginary part in the denominator has to be eliminated. This is done by multiplying and dividing the denominator by its complex conjugate.

The complex conjugate of a complex number is formed by changing the sign of its imaginary part.

So, to divide (a + bi) by (c + di) such that the value of c and d is not zero, we need to multiply and divide it by the complex conjugate of the denominator (c +di), which is (c – di)

The division of complex numbers does not follow commutative and associative properties.

Properties of complex numbers examples

1) If 3 + 5i and 7 + 9i are two complex numbers, then calculate the value of

1. a) (3 + 5i) + ( 7 + 9i) 
2. b) ( 7 + 9i ) + (3 + 5i). Which property of addition of complex numbers does it prove?

Answer– a) (3 + 5i) + ( 7 + 9i) = 10 + 14i

1. b) ( 7 + 9i ) + (3 + 5i) = 10 + 14i

         This proves the property of commutativity in the addition of complex numbers.

2) If the given two complex numbers, 4 + 2i and 5 + 7i, are subtracted, test the commutativity property between them. 

 Answer– (4 + 2i) – (5 + 7i) = -1-5i

                (5 + 7i) – (4 + 2i) = 1+5i

 (4 + 2i) – (5 + 7i) is not equal to (5 + 7i) – (4 + 2i)

  Hence, The property of commutativity does not apply in the subtraction of complex numbers.

3) In the operation 3+4i/ 8- 2i, determine the complex conjugate.

Answer: A complex conjugate is a complex number made by changing the sign of the imaginary unit in the denominator. The complex conjugate in this equation is 8 + 2i.

Conclusion

The properties of operations of complex numbers are different for some functions only because of their imaginary part. However, all its operations remain the same except for division. The real numbers and the imaginary numbers are operated separately.