Problem 13 - A Graphical Approach to College Algebra

Question

Write each expression in standard form. Do not use a calculator. $$(-7 i)(1+i)$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We are given the expression \((-7i)(1+i)\) and need to write it in standard form. The standard form of a complex number is \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

Step 2: Distribute the Expression

Apply the distributive property to expand the product \((-7i)(1+i)\). Distribute \(-7i\) across both terms inside the parenthesis:\[(-7i) \cdot 1 + (-7i) \cdot i.\]

Step 3: Calculate Each Term

Calculate the results of the distribution:1. \((-7i) \cdot 1 = -7i\).2. \((-7i) \cdot i = -7i^2\). Recall \(i^2 = -1\), so the second term becomes:\(-7(-1) = 7\).

Step 4: Combine the Results

Combine the real and imaginary terms from Step 3:\[(-7i) \cdot 1 + (-7i) \cdot i = -7i + 7.\]Reorder the terms to match standard form, \(a + bi\):\[7 - 7i.\]

Step 5: Finalize in Standard Form

The expression \((-7i)(1+i)\) expressed in standard form is thus:\[7 - 7i.\]

Key Concepts

Imaginary Unit

In the world of complex numbers, understanding the imaginary unit is crucial. Represented as \(i\), the imaginary unit is defined by its fundamental property that \(i^2 = -1\). This property is what distinguishes imaginary numbers from real numbers, allowing mathematicians to perform calculations that are impossible within the realm of real numbers alone.

• The imaginary unit is often used when dealing with square roots of negative numbers, transforming these into expressions involving \(i\).
• For instance, \(\sqrt{-1} = i\) and \(\sqrt{-4} = 2i\).

The imaginary unit serves as the building block for complex numbers, which can be expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. This unique feature allows complex numbers to solve equations that have no real solutions, thereby expanding the possibilities for solutions in mathematics.

Standard Form

The standard form of a complex number is a neat way to express any complex number. It's written as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) signifies the imaginary unit.

• Here, \(a\) represents the real part of the complex number.
• The \(bi\) component stands for the imaginary part, where \(b\) is a real number coefficient.

An important aspect of the standard form is the order of terms. The real part always comes first, followed by the imaginary part. This format not only ensures clarity but facilitates arithmetic operations, such as addition, subtraction, multiplication, and division, with complex numbers. In our given example, the product \((-7i)(1+i)\) was simplified to \(7 - 7i\), neatly presenting the number in standard form.

Distributive Property

The distributive property is a fundamental principle used in algebra to multiply one quantity across multiple terms within a parenthesis. This property is especially useful when dealing with expressions involving complex numbers.

• In mathematical terms, if you have a term \(a\) multiplied by an expression \((b + c)\), the distributive property allows you to express it as \(a \cdot b + a \cdot c\).
• This ensures that each component within the parenthesis is multiplied by the term outside, breaking down complex calculations into simpler, manageable steps.

For the expression \((-7i)(1+i)\) in our exercise, applying the distributive property involves multiplying \(-7i\) by each term inside the parenthesis: 1 and \(i\). Performing these steps leads to a result in standard form, making it easier to interpret and further manipulate the complex number.