91Ó°ÊÓ

Complex numbers are numbers that have a real part and an imaginary part. The general form of a complex number is written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. To add complex numbers, we simply combine their real parts and their imaginary parts separately.

Consider the example given in the exercise: \( (4 + i) + 3 + 2i \). To perform the addition:

• First, combine the real parts: \(4 + 3 = 7\).


• Next, combine the imaginary parts: \(i + 2i = 3i\).

So, the sum of the complex numbers \( 4 + i \) and \( -3 - 2i \) results in \ 7 + 3i \. This is your final answer. Always ensure to express your result in standard form \(a + bi\).

Subtraction of complex numbers follows a similar approach to their addition but with an additional step of distributing the negative sign. Here’s the step-by-step breakdown from the example in the exercise:

The initial expression is \( (4 + i) - (-3 - 2i) \). Now, follow these steps:

• Distribute the minus sign: \ (4 + i) + 3 + 2i \.


• Now, combine the real parts: \( 4 + 3 = 7 \).


• Then, combine the imaginary parts: \ (i + 2i = 3i) \.

Hence, after distribution and subtraction, you end up with the same final result as in the addition: \ 7 + 3i \.
Remember, always handle the minus sign first before combining the parts.

The standard form of complex numbers is essential for clarity and consistency. This form is written as \( a + bi \), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\).

Using the example from the exercise, after performing the operations, the answer was \ 7 + 3i \. This is already in standard form. Here’s a quick guide to ensure your result is in standard form:

• The real part (\(a\)) should come first.
• Followed by the imaginary part (\(bi\)).

For example, \ (4 + 3) + (i + 2i) \ simplifies to \ 7 + 3i \. Both parts are in their required positions.

Always double-check, especially when combining terms, to ensure your final result adheres to this format. This consistency helps in clearer communication and understanding in mathematics.