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Chapter 10: Problem 34

Add or subtract as indicated. Write your answers in the form \(a+b i\) $$ (9+i)-(3+2 i) $$

Short Answer

Expert verified
6 - i

Step by step solution

01

Distribute the negative sign

We need to distribute the negative sign to both terms inside the parentheses: o \( (9+i) - (3+2i) = 9 + i - 3 - 2i \)
02

Combine real parts

Add the real parts of the expression: o \(9 - 3 = 6\)
03

Combine imaginary parts

Add the imaginary parts of the expression: o \(i - 2i = -i\)
04

Write the final answer

Combine the results from Step 2 and Step 3 to get the final answer in the form \( a + bi \): o \(6 - i\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributing Negative Sign
When subtracting complex numbers, it is essential to distribute the negative sign properly before performing the addition or subtraction. This process involves applying the negative sign to each term inside the parentheses. Let's take the given problem as an example:
\[(9+i)-(3+2i)\]
We distribute the negative sign to both terms inside the second set of parentheses:
\[9 + i - 3 - 2i\]
This step ensures all terms are correctly signed and ready for further combination. Remember, failing to distribute the negative sign can lead to errors in your final result.
Combining Real Parts
Once the negative sign is properly distributed, the next step is to combine the real parts of the complex numbers. Real parts are the numbers without the imaginary unit \(i\). Using our example problem after distributing the negative sign:
\[9 + i - 3 - 2i\]
The real parts here are 9 and -3. Adding these together:
\[9 - 3 = 6\]
Always ensure you handle the real numbers first, separating them from the imaginary parts.
Combining Imaginary Parts
With real parts combined, the next focus is on the imaginary parts. Imaginary parts include the terms with the imaginary unit \(i\). From our example:
\[9 + i - 3 - 2i\]
The imaginary parts are \(i\) and \(-2i\). Adding these together gives:
\[i - 2i = -i\]
This step completes the combination of imaginary parts, making sure every imaginary unit term is accounted for. Remember to respect the signs (+ or -) during addition or subtraction.

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