Problem 22 - Intermediate Algebra

Question

Add or subtract as indicated. $$ (-2-3 i)-(-4-14 i) $$

Step-by-Step Solution

Answer

The result of the operation is \((2 + 11i)\).

Step 1: Identify the Expression

We need to find the difference: \((-2-3 i)-(-4-14 i)\).

Step 2: Perform the Subtraction

We subtract by aligning place values and borrowing as needed.

Step 3: State the Result

The result of the operation is \((2 + 11i)\).

Key Concepts

Subtraction of complex numbers

Subtracting complex numbers involves dealing with both the real and imaginary parts separately. When you see a subtraction expression like \(-2-3i - (-4-14i)\), you need to break it down by components: the real numbers and the imaginary numbers.
The key is:

• Subtract the real parts.
• Subtract the imaginary parts.

For our example: \[ (-2 - 3i) - (-4 - 14i) \].* Real parts: - starts with -2, from the first complex number. - then subtract -4, so -2 - (-4) = -2 + 4 = 2.* Imaginary parts: - starts with -3i, from the first complex number. - then subtract -(-14i), so -3i - (-14i) = -3i + 14i = 11i.
Combine the results to have the final answer: \(2 + 11i\).
Remember, subtraction follows similar principles as addition in terms of grouping real and imaginary parts but requires us to watch out for the changes in signs.

Imaginary unit

The imaginary unit, represented as \(i\), is vital in understanding complex numbers. The unique quality of the imaginary unit is that its square equals -1. This means: \[ i^2 = -1 \].
This property is what distinguishes imaginary numbers from real numbers.

• The imaginary unit allows for representation of numbers that are not real: numbers involving the square roots of negative numbers.
• It extends our number system and provides a way to solve equations like \(x^2 + 1 = 0\).

In the context of complex numbers, * The imaginary part, - is a real number multiplied by \(i\). - Each complex number has an imaginary part which corresponds to this definition.
It's essential to grasp this concept for managing and solving mathematical queries that involve complex numbers.With the imaginary unit \(i\), we delve into a broader understanding of numbers.

Real and imaginary parts of complex numbers

A complex number includes two parts: the real part and the imaginary part. In the form \(a + bi\),
- \(a\) is the real part,- \(bi\) is the imaginary part where \(b\) is a real number and \(i\) is the imaginary unit.
When we encounter a complex number, it's important to identify these components separately.
For our example in subtraction,

• First complex number: \(-2 - 3i\) - Real part: -2 - Imaginary part: -3i
• Second complex number: \(-4 - 14i\) - Real part: -4 - Imaginary part: -14i

Understanding these parts is crucial when performing operations like addition, subtraction, and multiplication on complex numbers. It allows us to correctly handle the numbers, ensuring proper separation and convergence of real and imaginary parts. Recognizing and isolating these elements helps simplify many mathematical operations involving complex numbers.