Problem 6 - Contemporary Precalculus

Question

In Exercises \(1-8,\) plot the point in the complex plane corresponding to the number. $$(2+i)(1-2 i)$$

Step-by-Step Solution

Answer

A) 4 - 3i

Step 1: Find the product of the complex numbers

Multiply the given complex numbers \((2+i)(1-2i)\) using the distributive property (FOIL). Remember that \(i^2 = -1\). \((2+i)(1-2i) = 2(1) + 2(-2i) + i(1) + i(-2i)\) \(= 2 - 4i + i - 2i^2\) Now, replace \(i^2\) with \(-1\) and simplify: \(= 2 - 4i + i + 2\) \(= (2 + 2) + (-4i + i)\) \(= 4 - 3i\)

Step 2: Express the result in the form \(a+bi\)

The product is already in the form \(a + bi\), with \(a = 4\) and \(b = -3\). So, the point we need to plot on the complex plane is \((4, -3)\).

Step 3: Plot the point on the complex plane

To plot the point \((4, -3)\) on the complex plane, locate \(4\) units in the positive horizontal (real) direction, and \(3\) units in the negative vertical (imaginary) direction, and mark the point. This point corresponds to the complex number \(4-3i\) in the complex plane.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane where each point represents a complex number. It consists of:

• A horizontal axis, known as the real axis, which represents the real part of the complex number.
• A vertical axis, known as the imaginary axis, which corresponds to the imaginary part.

To plot a complex number, you determine its coordinates from its form \(a + bi\). The real component \(a\) is the coordinate along the real axis, while \(b\) is the coordinate along the imaginary axis.
For example, to plot the complex number \(4 - 3i\) on the complex plane, you start from the origin:

• Move 4 units right along the real axis (positive \'a\' value).
• Move 3 units down along the imaginary axis, as the imaginary part \(b = -3\) is negative.

By marking this position, you represent the location of the complex number in the complex plane.

FOIL Method

The FOIL method is a mnemonic tool used to multiply two binomials. It's especially handy when dealing with complex numbers of the form \((a+bi)(c+di)\). FOIL stands for:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outer terms of the binomials.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

Applying this to \((2+i)(1-2i)\), we calculate:

1. First: \(2 \times 1 = 2\)
2. Outer: \(2 \times -2i = -4i\)
3. Inner: \(i \times 1 = i\)
4. Last: \(i \times -2i = -2i^2\)

Combine these results: \(2 - 4i + i - 2i^2\). Remember that \(i^2 = -1\), therefore, \(-2i^2 = 2\). Simplifying gives \(4 - 3i\), showing how FOIL is applied to achieve the final product.

Imaginary Unit

The imaginary unit, denoted as \(i\), is the foundation upon which complex numbers are built. It is defined by the property:

• \(i^2 = -1\)

This definition solves the otherwise unsolvable square root of negative one. Complex numbers use \(i\) to express numbers that extend beyond the real number line. When multiplying complex numbers and applying the FOIL method, remembering that \(i^2 = -1\) is crucial in simplifying the expression.
For instance, during the multiplication of \((2+i)(1-2i)\), the calculation \(i \times -2i\) results in \(-2i^2\). By substituting \(i^2\) with \(-1\), it becomes \(2\), which allows the correct simplification of the expression, ultimately leading to the simplified form \(4 - 3i\).
Understanding \(i\) is key to working with complex numbers and performing arithmetic on them effectively.