Problem 65 - Algebra A Combined Function

Question

Find each multiplicative inverse or reciprocal. $$ 2 $$

Step-by-Step Solution

Answer

The multiplicative inverse of 2 is \( \frac{1}{2} \).

Step 1: Understand the concept of a multiplicative inverse

The multiplicative inverse (or reciprocal) of a number is a value that, when multiplied by the original number, results in the identity element for multiplication, which is 1. For any non-zero number \( a \), the multiplicative inverse is \( \frac{1}{a} \).

Step 2: Apply the concept to the given number

Given the number is 2. We need to find its multiplicative inverse or reciprocal. As per the concept, the reciprocal of any number \( a \) is \( \frac{1}{a} \).

Step 3: Calculate the reciprocal of 2

Since the given number is 2, its multiplicative inverse is \( \frac{1}{2} \).Check: \( 2 \times \frac{1}{2} = 1 \). This confirms that \( \frac{1}{2} \) is indeed the correct reciprocal of 2.

Key Concepts

Reciprocal

In mathematics, the term "reciprocal" is often used interchangeably with "multiplicative inverse." This term is essential when you're working on understanding how numbers interact under multiplication. The reciprocal of a number is very simple to find. For a given non-zero number \( a \), its reciprocal is expressed as \( \frac{1}{a} \).
When you multiply a number by its reciprocal, the result is always 1. This makes reciprocals crucial in simplifying fractions and solving equations.

• If you take the number 2, its reciprocal is \( \frac{1}{2} \).
• This is because \( 2 \times \frac{1}{2} = 1 \).

It's important to note that every non-zero real number has a reciprocal, but zero does not because there is no number which, when multiplied by zero, gives 1.

Identity Element

The idea of an identity element is a fundamental concept in mathematics, particularly in the operation of multiplication. An identity element is a number that, when used in an operation with any other number, leaves that number unchanged.
For multiplication, the identity element is 1. This means that any number multiplied by 1 remains the same. In other words, for any number \( a \), \( a \times 1 = a \).
This concept helps in understanding reciprocals and why multiplying a number by its reciprocal equals the identity element (1).

• For example, the reciprocal of 2 is \( \frac{1}{2} \), and indeed \( 2 \times \frac{1}{2} = 1 \).
• This proves that 1 functions as the identity element in multiplication.

Recognizing the identity element is crucial in solving algebraic equations and working with various mathematical operations.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In many algebraic expressions and equations, understanding inverses, including reciprocals, is essential.
Using the rules of algebra, you can solve equations by isolating variables, often by multiplying both sides by a reciprocal to eliminate coefficients or factors. Consider the equation \( 2x = 1 \).

• To solve, you would multiply both sides by \( \frac{1}{2} \) to isolate \( x \).
• This results in \( x = \frac{1}{2} \).

Understanding reciprocals and identity elements supports solving more complex algebraic problems, as it allows you to simplify and balance equations effectively.
The interconnectedness of algebra and these concepts makes learning them fundamental to progressing in any mathematical journey.