Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the addition or subtraction and write the result in standard form. $$13 i-(14-7 i)$$

Step-by-Step Solution

Answer

The result of the operation in standard form is \(-14 + 20i\).

Step 1: Break down the terms

Seperate the given expression into real and imaginary parts: We have 0 as the real part (since there is no standalone real number with 13i) and 13i as the imaginary part from the first term. For the second term, we have 14 as the real part and -7i as the imaginary part. So, we write it as: \(13i - (14-7i)\)

Step 2: Distribute the negative sign

The negative sign in front of the brackets will be distributed to the contained terms. This changes the signs of the contained real and imaginary numbers. This results to \(13i - 14 + 7i\)

Step 3: Combine like terms

The like terms in this context are the ones containing 'i', the imaginary unit. We add the real parts together and the imaginary parts together. This results to \(-14 + 20i\)

Key Concepts

Imaginary Unit

The imaginary unit is a fundamental concept in the realm of complex numbers. It is denoted by the letter 'i' and is defined as the square root of -1. This means that

• i squared equals -1: \[ i^2 = -1 \]

When dealing with complex numbers, the imaginary unit allows us to represent numbers that are not on the real number line. These numbers include a component that involves the imaginary unit 'i'.

For example, a basic complex number such as 3 + 4i consists of two parts: 3 is the real part, and 4i is the imaginary part, where 'i' represents the imaginary portion.

Standard Form

The standard form of a complex number is expressed as a combination of its real and imaginary parts. It is typically written as:

• \( a + bi \)

where 'a' represents the real part, and 'b' represents the imaginary coefficient that multiplies the imaginary unit 'i'. This form is crucial because it provides a clear and structured way to present complex numbers.

The process of deriving a complex number into its standard form is essential when performing operations such as addition and subtraction. After computation, rearranging numbers into this form ensures coherence, which simplifies further calculations and interpretations.

Addition and Subtraction of Complex Numbers

Adding and subtracting complex numbers is quite straightforward once you understand the role of real and imaginary parts. Here’s how to do it step by step:

**Step 1: Identify Real and Imaginary Parts**
Look at each complex number and separate the real parts from the imaginary parts. For instance, in an expression like 3 + 4i and 5 - 2i, 3 and 5 are the real parts, while 4i and -2i are the imaginary parts.

**Step 2: Combine Like Terms**
Perform the addition or subtraction separately for the real and imaginary components. Using our example:

• Add real parts: \( 3 + 5 = 8 \)
• Add imaginary parts: \( 4i - 2i = 2i \)

This results in the new complex number: 8 + 2i.

Thus, when combining complex numbers, always ensure to rearrange your final answer into standard form for clarity and consistency.