Problem 13 - A Graphical Approach to Precalculus with Limits

Question

Perform operations and write the result in standard form. $$(-7 i)(1+i)$$

Step-by-Step Solution

Answer

The result in standard form is \(7 - 7i\).

Step 1: Distribute the Terms

To multiply two complex numbers, you need to use the distributive property, also known as the FOIL method for binomials. Here, multiply the expression \((-7 i)(1+i)\):- Multiply \(-7i\) with \(1\): \(-7i \times 1 = -7i\)- Multiply \(-7i\) with \(i\): \(-7i \times i = -7i^2\)

Step 2: Simplify the Complex Number

Now, simplify the multiplication results from Step 1. First, remember that \(i^2 = -1\). So, the term \(-7i^2\) simplifies to:\(-7i^2 = -7(-1) = 7\).Add this real component to the complex component obtained from the previous step:\(7 - 7i\).

Step 3: Write in Standard Form

The standard form for a complex number is \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. From the result of Step 2, we can conclude that the complex number is:\(7 - 7i\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, employed frequently when dealing with complex numbers. It allows us to multiply a monomial by a binomial, by distributing or 'spreading out' the single term over each term in the binomial. In simple terms, for any numbers or variables, the distributive property states that:\[a(b + c) = ab + ac\]Applying this to complex numbers is quite analogous. In the original exercise, we have the expression \((-7i)(1+i)\).To resolve this, distribute \(-7i\)to both \(1\) and \(i\):

• Multiply \(-7i\) with \(1\): which results in \(-7i \times 1 = -7i\).
• Multiply \(-7i\) with \(i\): resulting in \(-7i \times i = -7i^2\).

Understanding how to distribute terms correctly in complex expressions is crucial, as it lays the foundation for simplifying the result into its cleanest and most interpretable form.

Standard Form

The standard form of a complex number is a specific way of expressing it, always shown as:\[a + bi\]where \(a\) represents the real part and \(b\) represents the imaginary part involving the imaginary unit \(i\). This format aids in easily identifying both components of a complex number, allowing for seamless mathematical operations and further analysis.For instance, after simplification in the original problem, our complex number becomes:\(7 - 7i\).Here:

• The real part \(a\) is \(7\).
• The imaginary part \(b\) is \(-7\).

Arranging numbers in this format not only makes them more accessible for operations like addition, subtraction, and comparison among complex numbers but also provides clarity. It ensures that anyone interpreting the data knows exactly which part is real and which involves the imaginary unit.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a revolutionary concept in mathematics. It is defined by its property:\[i^2 = -1\]This means that \(i\) is the square root of \(-1\), a concept which is not possible within the set of real numbers. The imaginary unit allows us to solve equations and describe numbers that exceed the realm of the real number line.For example, in our original exercise, one key step involves using \(i^2 = -1\).We had a product:\(-7i^2\).This simplifies to:\(-7 \times (-1) = 7\).Recognizing \(i\) as the imaginary unit helps us perform such operations with ease. Understanding and using the imaginary unit enables mathematicians to expand their toolkit, providing solutions and approaches that were previously unattainable through traditional real number methods.