Problem 127 - Intermediate Algebra

Question

What two real numbers are their own reciprocals?

Step-by-Step Solution

Answer

The two real numbers are 1 and -1.

Step 1: Understand the definition of a reciprocal

A real number is its own reciprocal if multiplying it by itself yields 1. This means if a real number is represented by \( x \), then \( x \times x = 1 \).

Step 2: Set up the equation for a reciprocal property

We need to find \( x \) such that \( x = \frac{1}{x} \). We can also write this as \( x^2 = 1 \).

Step 3: Solve the equation

By solving \( x^2 = 1 \), we take the square root of both sides. This gives us two potential solutions: \( x = 1 \) and \( x = -1 \).

Step 4: Verify the solutions

Check each solution: \( x = 1 \) gives \( rac{1}{1} = 1 \), and \( x = -1 \) gives \( rac{1}{-1} = -1 \). Thus, both solutions satisfy the condition that each number is its own reciprocal.

Key Concepts

Real Numbers

Real numbers are a broad category of numbers that are incredibly important in mathematics. They include many different types of numbers that you encounter regularly such as:

• Whole numbers like 0, 1, 2, 3, etc.
• Negative numbers such as -1, -2, -3, etc.
• Fractions and decimals like 0.5, 2.75, or -3.4.
• Irrational numbers such as pi (\( \pi \approx 3.14159\)) and the square root of 2 (\( \sqrt{2} \approx 1.414\)).

Real numbers form a continuous line without any gaps, known as the number line. Every point on this line corresponds to a real number, and every real number has a point on this line. This makes them essential for measuring, comparing, and understanding the world in a mathematical sense. When solving equations or comparing numbers, you're often working with real numbers.

Solving Equations

Solving equations is a fundamental skill in mathematics that allows us to find unknown values. To solve an equation means finding all the values (solutions) that make the equation true.
For instance, in the equation \( x^2 = 1 \), we are looking for numbers that can replace \( x \) and still satisfy this equation.
The process typically involves:

• Identifying what you need to find, such as the variable \( x \).
• Using algebraic operations like addition, subtraction, multiplication, or division to manipulate both sides of the equation.
• Simplifying until you isolate the variable.
• Checking your solutions by substituting them back into the original equation to ensure they work.

This allows us to verify that our solutions are correct. Equations are everywhere in science, engineering, finance, and many fields, as they help us predict and quantify different phenomena.

Reciprocal Property

The reciprocal of a number is a concept that simplifies division and multiplication. It's calculated as the inverse of the number. Simply put, the reciprocal of a number \( x \) is \( \frac{1}{x} \).
This means if you multiply a number by its reciprocal, you will always get \( 1 \). This property is vital in solving equations because it helps in finding symmetrical relationships within numbers.
In the original exercise, we're asked which real numbers are their own reciprocals. To solve this, think about how multiplying these numbers by themselves results in 1, written mathematically as \( x^2 = 1 \). Solving this gives us two special numbers: 1 and -1, which are their own reciprocals.
These particular numbers are unique because they represent balance in multiplication and division, showing how understanding the reciprocal property can lead to surprising and elegant solutions.