Complex Numbers: A Review

The standard form of a complex number is a + bi, where a and b are real numbers, a is called the real part, and b, the imaginary part. Rene Descartes first called ‘i’ the imaginary number because the equation cannot be satisfied by a real number. i always satiates the equation i2 = −1.  If b ≠ 0, a + bi is also called a complex number. We use the letter C to refer to the set of all complex numbers. If an equation has no solution using real numbers, you can easily insert imaginary numbers into the same to get an answer. 

Thus,

C = {a + bi}, a ∈ R and b ∈ R and i²= –1

The real and imaginary parts of a complex number C may be denoted by Re(C), and Im(C), respectively. 

For example, if (i) z = 7 + 8i, then Re(z) = 7, Im(z) = 8

                        (ii) z = –√3 – 11i, then Re(z) = –√3, Im(z) = –11

For adding and subtracting complex numbers, we need to first combine the real parts of the complex numbers together and then proceed to combine the imaginary parts of the complex numbers together and then finally apply the operations.

Addition of Complex Numbers

When we seek to add complex numbers, we combine the real parts of the complex numbers separately and imaginary parts separately and add them. In a geometric plane, if you wish to add two complex numbers, it can be done with the construction of a parallelogram. The addition of complex numbers is carried out using the following formula:

x1 + x2 = a + iy + c + iz

= (a + c) + (iy + iz)

= (a + c) + i(y + z)

Hence we have (a + iy) + (c + iz) = (a + c) + i(y + z)

Examples 

  1. Add (2 + 4i) + (3 – 2i) 

(2 + 4i) + ( 3 – 2i) = 2 + 4i + 3 – 2i = 2 + 3 + 4i – 2i = 5 + 2i

  2. Add 3 + (–1 – 3i) + (4 + i) + (5 – 2i)

3 + (–1 – 3i) + (4 + i) + (5 – 2i) = 3 – 1 – 3i + 4 + i + 5 – 2i = 11 – 4i

  3. Add (3 + √–49) + (9 – √–25)

(3 + √–49) + (9 – √–25) = (3 + 7i) + (9 – 5i)

= 3 + 7i + 9 – 5i = 3 + 9 + 7i – 5i = 12 + 2i

  1. Add (–3 + 7i√3) + (9 – 2i√3)

(–3 + 7i√3) + (9 – 2i√3) = –3 + 7i√3 + 9 – 2i√3

= –3 + 9 + 7i√3 – 2i√3 = 6 +5i√3

Subtraction of Complex Numbers

To subtract complex numbers, we separate the real and imaginary parts of the complex numbers and group them. Then we subtract the real and imaginary parts of one complex number from the real and imaginary parts, respectively, of the other complex number. Thus, subtraction of complex numbers is done by the following formula,

x1 – x2 = (a + iy) – (c + iz)

= a + iy – c – iz

= (a – c) + (iy – iz)

= (a – c) + i(y – z)

Thus we have (a + iy) – (c + iz) = (a – c) + i(y – z)

Examples

  1. Subtract (6 + 3i) – (4 – 4i)

(6 + 3i) – (4 – 4i) = 6 + 3i – 4 – (–4i) = 6 – 4 + 3i + 4i = 2 + 7i

Rule grouping: (6 + 3i) – ( 4 – 4i ) = (6 – 4) + ( 3 – (–4) )i = 2 + 7i

  1. Subtract ( 4–√–16 ) – ( 12–√–169 )

= ( 4 – 4i ) – ( 12 – 13i )

= 4 – 4i – 12 + 13i = 4 – 12 – 4i + 13i = –8 + 9i

  1. Subtract ( 5 + 7√5i ) – ( 6 + 12√5i )

= 5 + 7√5i – 6 – 12√5i

= (5 – 6) + (7√5 – 12√5)i

= –1 – 5√5i 

  1. (15 + √–81) – ( 10 – √–225 )

= ( 15 + 9i ) – ( 10 – 15i )

= 15 – 10 + 9i – (– 15i)

= ( 15 – 10 ) + ( 9 + 15 )i 

= 5 + 24i

Thus the steps for adding and subtracting complex numbers are as follows : 

  1. Segregation of the imaginary and the real part.
  2. First, the performance of the operation on the real part of the numbers.
  3. Second, the performance of the operation on the imaginary part of the numbers.
  4. Presentation of the final answer in the complex number i.e., a + ib format.

Further,

  • The addition and subtraction of complex numbers are similar to the addition and subtraction of binomials because we need to combine the like terms.
  • All real numbers are complex numbers, but all complex numbers are not necessarily real numbers.
  • Commutative law does not apply to the subtraction of complex numbers.
  • For the addition and subtraction of complex numbers in polar form, we assign the horizontal axis as real and vertical axis as imaginary. We then perform addition in the form of vectors. Then, we convert the final answer into polar form.

Properties in Addition and Subtraction of Complex Numbers

Following are the properties of addition and subtraction of complex numbers:

  • Commutative Property: The order in which the numbers are added does not affect the sum of the numbers. The addition of complex numbers is commutative, but complex numbers’ subtraction is not commutative.

   (a+ib)+(c+id)=(c+id)+(a+ib)

  • Associative Property: In an expression that contains two or more than two numbers and an associative operator in between, the rule is that the order in which the operations are conducted does not change the final result so long as we keep the sequence of the operands the same. Adding complex numbers is associative, but complex numbers’ subtraction is not associative.

(a+b)+c=a+(b+c)

Where a,b,c are complex numbers.

  • Additive Identity: The additive identity property, also known as the identity property of zero, states that adding zero to any number results in the number itself. This is because when we add zero to any number, it does not change the number, and thus the number retains its identity. Zero is the additive identity of the complex numbers, i.e., for a complex number z, we have z + 0 = 0 + z = z.
  • Additive inverse: Additive inverse is the number you add to a given number to make the sum zero. For a complex number z, the additive inverse in complex numbers is –z, i.e., z + (–z) = 0
  • Closure Property: The sum and difference of complex numbers are also complex numbers. Thus, closure property applies.