Problem 5 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Answer: The product of the complex numbers (1+i)(1-i) is 2. This number is a real number, so it can be plotted on the real axis, at the point (2,0) where 2 is the real part and 0 is the imaginary part.

Step 1 : Multiply the complex numbers

We must multiply the given complex numbers \((1+i)\) and \((1-i)\): $$(1+i)(1-i) = 1 - i^2 = 1 - (-1) = 1+1=2$$ The simplified product is 2, which is a real number.

Step 2 : Plot the point in the complex plane

Now, we plot the point 2 on the complex plane. Being a real number with no imaginary part, it will be on the real axis. So, mark the point (2, 0) on the complex plane, where 2 corresponds to the real part and 0 corresponds to the imaginary part.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used to graphically represent complex numbers. This plane has a horizontal axis, called the "real axis," and a vertical axis, known as the "imaginary axis." Each point on this plane corresponds to a unique complex number. For example, if you have the complex number \(a + bi\), it can be represented on the complex plane by the point \((a, b)\). This means \(a\) units along the real axis and \(b\) units along the imaginary axis.
To plot a point on the complex plane, simply identify the real and imaginary components of the complex number:

• The real part determines the position along the horizontal axis.
• The imaginary part determines the position along the vertical axis.

In the exercise provided, the product of \((1+i)(1-i)\) simplifies to \(2\), a pure real number. Therefore, the point \((2, 0)\) is plotted, as it lies directly on the real axis due to having no imaginary component.

Multiplying Complex Numbers

Multiplying complex numbers involves expanding the expression similar to algebraic multiplication using the distributive property. The multiplication of two complex numbers, like \((1+i)(1-i)\), can be approached by treating the complex numbers as binomials:

1. First, apply the distributive property:
\((1+i)(1-i) = 1(1) + 1(-i) + i(1) + i(-i).\)
2. Simplify the terms:

• \(1 \cdot 1 = 1\)
• \(1 \cdot (-i) = -i\)
• \(i \cdot 1 = i\)
• \(i \cdot (-i) = -i^2 = 1\)

The expression simplifies to \(1 - i + i + 1 = 2.\)
Notice that the imaginary components cancel out, leaving a real number.
This result is typical for multiplying conjugate pairs (complex numbers of the form \((a+bi)(a-bi)\)).
These conjugate pairs always result in a real number, specifically \(a^2 + b^2.\)

Imaginary Unit

The imaginary unit \(i\) is a fundamental concept in complex numbers. It is defined as the square root of \(-1\), denoted \(i = \sqrt{-1}\).
As such, it allows for the extension of the real numbers into the complex plane by giving meaning to the square roots of negative numbers.
In multiplication, the defining characteristic of the imaginary unit is \(i^2 = -1\).

This property is crucial when multiplying complex numbers, especially since it involves:

• Squaring \(i\) to produce \(-1\) when necessary.
• Determining the real or imaginary nature of the result.

In the example, the multiplication \(i(-i) = -i^2\), which simplifies to \(1\) because \(i^2 = -1.\)
Understanding how to manipulate \(i\) correctly is key to performing operations with complex numbers and translating them onto the complex plane.