Problem 47 - Precalculus Student Solutions Manual 5th

Question

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

Step-by-Step Solution

Answer

1 - 8i

Step 1: Distribute the Negative Sign

First, distribute the negative sign across the parentheses: }} This will change the expression to: (2-5i) - 3 - 4i - (-2 + i) which simplifies to: (2-5i) - 3 - 4i + 2 - i by flipping the signs inside the third parentheses.

Step 2: Combine Like Terms

Now combine the real numbers (2, -3, and 2) and the imaginary numbers (-5i, -4i, and i): Real part: 2 - 3 + 2 = 1 Imaginary part: -5i - 4i + i = -8i

Step 3: Write in Standard Form

Express the result as a complex number in standard form, which is a + bi: The result is 1 - 8i

Key Concepts

Complex Numbers

Complex numbers are a special type of number that combines a real part and an imaginary part. You can think of them as numbers that lie on a 2-dimensional plane, unlike the regular (real) numbers that lie on a 1-dimensional line.
The general form of a complex number is written as \(a + bi\), where:

• \(a\) is the real part
• \(b\) is the imaginary part
• \(i\) is the imaginary unit, defined as \(i = \sqrt{-1}\)

For example, in the complex number \(3 + 2i\), 3 is the real part, and 2i is the imaginary part.

Standard Form

Writing complex numbers in standard form helps to clearly see both the real and imaginary parts. The standard form is \(a + bi\), and it's crucial for performing operations like addition and subtraction.
For instance, if you have two complex numbers like \(2 - 5i\) and \(3 + 4i\), writing them in standard form makes it easier to process steps to find the sum or difference.

Imaginary Numbers

Imaginary numbers represent a whole new dimension of numbers beyond the real number line. The basic unit is \(i = \sqrt{-1}\), an imaginary number that satisfies the equation \(i^2 = -1\).
When you perform arithmetic with imaginary numbers, treat the \(i\) part separately from the real part. For example, in the expression \(-5i - 4i + i\), you only combine the imaginary coefficients (-5, -4, and 1).
When adding or subtracting, think of \(i\) as a variable, like \(x\) in algebra, but you have to remember that \(i^2 = -1\).

Combining Like Terms

Combining like terms in complex numbers follows the same principle as in algebra. You group the real parts together and the imaginary parts together.

• Real parts: These are the numbers without the imaginary unit \(i\). For example, in the expression \(2 - 3 + 2\), the real parts are 2, -3, and 2. When you add them, you get 1.
• Imaginary parts: These are the numbers multiplied by \(i\). For instance, in \(-5i - 4i + i\), the coefficients are -5, -4, and 1. Adding them gives \(-8i\).

After combining, you write the result in standard form. For example, the final answer for the given exercise is \(1 - 8i\).