Subtraction of Complex Numbers

Subtracting complex numbers involves finding the difference between their real parts and their imaginary parts separately.

A complex number is expressed in the form a + bi, where a is the real part and b is the imaginary part.

The subtraction of two complex numbers (a + bi) and (c + di) is given by (a - c) + (b - d)i.

When subtracting complex numbers, follow these steps:

Subtract real parts: Take the real part from the first number and subtract the real part of the second number.

Subtract imaginary parts: Take the imaginary part from the first number and subtract the imaginary part of the second number.

Combine results: Combine the differences of the real and imaginary parts to form the resulting complex number.

The following methods can be used for subtracting complex numbers:

Method 1: Direct Subtraction

To apply direct subtraction for complex numbers, use the following steps.

Step 1: Identify the real and imaginary parts of both complex numbers.

Step 2: Subtract the real parts and the imaginary parts separately.

Step 3: Combine the results to form the final complex number.

Example: Subtract (3 + 4i) from (5 + 7i)

Step 1: Identify real and imaginary parts: Real parts: 5 and 3; Imaginary parts: 7i and 4i

Step 2: Subtract real parts: 5 - 3 = 2; Subtract imaginary parts: 7i - 4i = 3i

Step 3: Combine: 2 + 3i Answer: 2 + 3i

Method 2: Column Method

When using the column method for subtraction, write the complex numbers one below the other, aligning real and imaginary parts. Subtract each column separately.

Example: Subtract (2 - 3i) from (6 + 5i)

Solution: 6 + 5i ← Minuend (from which we subtract) - 2 - 3i ← Subtrahend (what we subtract) -------------- 4 + 8i Therefore, upon subtracting, we get 4 + 8i.

Subtraction of complex numbers has characteristic properties:

Subtraction is not commutative

In subtraction, changing the order of the numbers changes the result, i.e., (a + bi) - (c + di) ≠ (c + di) - (a + bi).

Subtraction is not associative Unlike addition, we cannot regroup in subtraction.

When three or more complex numbers are involved, changing the grouping changes the result.

((a + bi) − (c + di)) − (e + fi) ≠ (a + bi) − ((c + di) − (e + fi))

Subtraction is the addition of the opposite sign

Subtracting a complex number is the same as adding its opposite, so to make calculations easier, you can convert subtraction into addition by changing the signs of the second term.

(a + bi) − (c + di) = (a + bi) + (−c − di)

Subtracting zero from a complex number leaves the number as is Subtracting zero from any complex number results in the same complex number: (a + bi) - 0 = a + bi.

Here are some tips and tricks to efficiently subtract complex numbers:

Tip 1: Pay close attention to signs when subtracting, especially the imaginary unit 'i'.

Tip 2: Double-check the subtraction of real and imaginary parts separately to avoid errors.

Tip 3: For visualization, consider using the geometric representation of complex numbers on the complex plane, which can help in understanding the difference as a vector displacement.

Subtraction in complex numbers can be tricky, leading to common mistakes. However, being aware of these errors can help students avoid them.