Problem 123 - Beginning and Intermediate Algebra

Question

Find any real numbers that are their own reciprocal.

Step-by-Step Solution

Answer

The real numbers that are their own reciprocal are 1 and -1.

Step 1: Define the Concept of Reciprocal

In mathematics, a reciprocal of a number is its multiplicative inverse. This means that if you multiply a number by its reciprocal, the result is 1. For a real number \( x \), its reciprocal is \( \frac{1}{x} \).

Step 2: Set Up the Equation

We need to find real numbers \( x \) such that \( x \) is equal to its own reciprocal. This gives us the equation: \[ x = \frac{1}{x} \] We need to solve this equation to find the values of \( x \).

Step 3: Solve the Equation

To solve \( x = \frac{1}{x} \), first multiply both sides by \( x \) to eliminate the fraction: \[ x^2 = 1 \] This is because multiplying \( x \) by \( \frac{1}{x} \) cancels out the \( x \) in the denominator on the right-hand side.

Step 4: Find Solutions for the Quadratic Equation

Now, solve the equation \( x^2 = 1 \). We can do this by taking the square root of both sides: \[ x = \pm 1 \] This indicates that there are two solutions: \( x = 1 \) and \( x = -1 \).

Step 5: Verify the Solutions

Check whether \( x = 1 \) and \( x = -1 \) satisfy the original condition, \( x = \frac{1}{x} \). For \( x = 1 \), \( \frac{1}{1} = 1 \) is true. For \( x = -1 \), \( \frac{1}{-1} = -1 \) is also true. Both values satisfy the condition.

Key Concepts

Real Numbers

Real numbers are a fundamental concept in mathematics. They include all the numbers that you might typically encounter in both everyday life and advanced mathematics. This set comprises:

• Positive numbers (e.g., 1, 2, 3, 4...)
• Negative numbers (e.g., -1, -2, -3...)
• Zero (0)
• Fractions (e.g., 1/2, -3/4)
• Decimals (e.g., 0.75, -8.1)
• Irrational numbers (e.g., π, √2)

Real numbers can be visualized on the number line. This line extends infinitely in both directions, encapsulating all possible real numbers.

The main property of real numbers is that any two real numbers can be compared in size by being either equal or one being larger than the other. Understanding real numbers is crucial as they form the basis for more complex mathematical constructs, such as quadratic equations, which we will explore next.

Quadratic Equation

A quadratic equation is a polynomial equation of degree 2. The standard form is:\[ ax^2 + bx + c = 0 \]where a, b, and c are constants, and \( a eq 0 \).

In our exercise, we encountered a quadratic equation in the form:\[ x^2 = 1 \]This is a simplified quadratic equation where:

• a = 1
• b = 0
• c = -1

To solve a quadratic equation, one common technique is to factor the equation, set each factor equal to zero, and solve for x. However, in the simple equation \( x^2 = 1 \), we can directly take the square root of both sides:

• Solutions: \( x = 1 \) and \( x = -1 \)

These solutions are known as the roots of the equation, and they help us understand the behavior of parabolic functions in graphs. Quadratic equations appear frequently in diverse areas such as physics, engineering, and finance, illustrating their broad applicability.

Multiplicative Inverse

The multiplicative inverse, also known as the reciprocal, is a key concept in mathematics involving the operation of multiplication. For any real number \( x \), the multiplicative inverse is \( \frac{1}{x} \) such that:\[ x \cdot \frac{1}{x} = 1 \]

The important aspect of the inverse is that when a number is multiplied by its inverse, the product is always 1. For example:

• The inverse of 2 is \( \frac{1}{2} \) because \( 2 \times \frac{1}{2} = 1 \)
• The inverse of -3 is \( \frac{-1}{3} \) because \( -3 \times \frac{-1}{3} = 1 \)

In the context of our exercise, finding a real number that is its own reciprocal involves solving the condition \( x = \frac{1}{x} \). Real numbers that are their own reciprocals must satisfy this equation, leading us to the solutions \( x = 1 \) and \( x = -1 \).

The concept of a multiplicative inverse is fundamental in solving equations, particularly in fields such as algebra and calculus, where it assists in simplifying expressions and solving equations efficiently.