Problem 48 - Precalculus Student Solutions Manual 5th

Question

Find each sum or difference. Write the answer in standard form. $$(-4-i)-(2+3 i)+(-4+5 i)$$

Step-by-Step Solution

Answer

-10 + i

Step 1: Identify the complex numbers

The given complex numbers are 1. -4-i 2. 2+3i 3. -4+5i

Step 2: Distribute the subtraction

Apply the subtraction to the complex numbers: (-4-i)-(2+3i) + (-4+5i) = -4 - i -(2+3i) -4+5i .

Step 3: Group real and imaginary parts

Group the real parts together and the imaginary parts together: (-4 - 2 - 4) + (-i - 3i + 5i) .

Step 4: Simplify the real and imaginary parts

Simplify the real part: -4 - 2 - 4 = -10 . Simplify the imaginary part: -i - 3i + 5i = i .

Step 5: Write the answer in standard form

Combine the simplified real and imaginary parts to get the final answer in standard form: -10 + i .

Key Concepts

Complex Number Addition

Adding complex numbers is just like adding two polynomial expressions. You add the real parts together and the imaginary parts together. For example, if you have two complex numbers: \(1 + 2i\) and \(3 + 4i\), you would first add the real parts (1 and 3) to get 4. Then, you add the imaginary parts (2i and 4i) to get 6i. So, the sum is \(4 + 6i\). When doing addition of complex numbers, always look to combine like terms. This makes simplifying much easier and ensures you get the correct result quickly.

Complex Number Subtraction

Subtracting complex numbers follows a similar pattern to addition, but you must remember that subtraction distributes across both the real and imaginary parts. Take the example: \((5 + 6i) - (3 + 2i)\). Start by subtracting the real parts: 5 minus 3 equals 2. Then, subtract the imaginary parts: 6i minus 2i equals 4i. Thus, \((5 + 6i) - (3 + 2i) = 2 + 4i\). Be cautious about the minus sign and ensure it applies to both parts of the complex number being subtracted.

Standard Form of Complex Numbers

Writing a complex number in its standard form means arranging it in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. For instance, \(-1 + 2i\), where \(-1\) is the real part and \(2i\) is the imaginary part. This is helpful for clarity and ease of performing operations like addition and subtraction. When combining real and imaginary parts, always present your final answer in this form. For example, the result of the exercise, \(-10 + i\), is already in standard form, making it easy to understand and use for further calculations.