Subtraction of Two Complex Numbers

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

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

To subtract two complex numbers, subtract the real parts and the imaginary parts separately.

When subtracting two complex numbers, follow these steps:

Separate real and imaginary parts: Write each complex number in the form a + bi.

Subtract real parts: Subtract the real part of the second complex number from the real part of the first.

Subtract imaginary parts: Subtract the imaginary part of the second complex number from the imaginary part of the first.

Combine results: Combine the results to form a new complex number.

The following are methods to subtract complex numbers:

Method 1: Horizontal Method

To apply the horizontal method for subtracting complex numbers, use these steps:

Step 1: Write both complex numbers in a horizontal line with a minus sign between them.

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

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

Example: Subtract (3 + 2i) from (5 + 4i).

Step 1: (5 + 4i) - (3 + 2i)

Step 2: (5 - 3) + (4i - 2i)

Step 3: 2 + 2i

Method 2: Column Method

Using the column method, write the complex numbers one below the other, aligning like parts vertically. Then subtract the corresponding parts.

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

Solution: Align the parts vertically: 6 + 5i ← Minuend - 2 + 3i ← Subtrahend --------- 4 + 2i Therefore, the result is 4 + 2i.

Subtraction of complex numbers has the following properties:

Subtraction is not commutative Changing the order of the numbers changes the result, i.e., (a + bi) - (c + di) ≠ (c + di) - (a + bi)

Subtraction is not associative When more than two 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. (a + bi) − (c + di) = (a + bi) + (−c − di)

Subtracting zero from a complex number leaves the number unchanged

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

Here are some tips for dealing with the subtraction of complex numbers efficiently:

Tip 1: Keep track of both real and imaginary parts separately.

Tip 2: Write complex numbers in the standard form to simplify subtraction.

Tip 3: Use parentheses to avoid confusion with signs, especially with negative complex numbers.

Subtraction in complex numbers can lead to common mistakes, but being aware of these can help avoid them.