Problem 59 - Prealgebra

Question

Find the reciprocal of each number. $$\frac{1}{2}$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Reciprocal

The reciprocal of a number is simply 1 divided by that number. For a fraction, the reciprocal is found by flipping the numerator and the denominator.

Step 2: Identifying the Given Fraction

In this problem, the given fraction is \(\frac{1}{2}\). Here, 1 is the numerator and 2 is the denominator.

Step 3: Calculating the Reciprocal

To find the reciprocal of \(\frac{1}{2}\), swap the numerator and the denominator. This results in\(\frac{2}{1}\).

Key Concepts

Fraction

A fraction is a mathematical way to represent a part of a whole. It is expressed as two numbers, one on top of the other, separated by a line called a fraction bar. This line is more than just a separator; it indicates division. For instance, in the fraction \(\frac{1}{2}\), you are essentially dividing 1 by 2.

Fractions can represent different things:

• Parts of a whole, like half a pizza represented as \(\frac{1}{2}\).
• Divisions, like dividing 1 liter into 4 parts, or \(\frac{1}{4}\) of a liter.
• Ratios, showing that two out of every ten apples are rotten as \(\frac{2}{10}\).

When you become comfortable with fractions, you can start to see the world in a more detailed way, from slicing up desserts to understanding data in statistics.

Numerator

The numerator is the top number in a fraction, and it tells you how many parts of a whole you have. In the fraction \(\frac{1}{2}\), the numerator is 1. This means you possess one part out of the two equal parts that make up a whole.

The role of the numerator is crucial:

• It indicates the quantity of the parts you are considering. In \(\frac{3}{4}\), the numerator 3 means you have 3 parts out of 4.
• The numerator can be any integer, including 0, which would mean having no parts at all, like \(\frac{0}{5}\).
• In a mixed number, it refers to the fractional part. For example, in 2\(\frac{1}{3}\), 1 is the numerator.

Understanding the numerator's function helps in grasping how much of a fraction you actually have.

Denominator

The denominator is the bottom number in a fraction and it indicates into how many equal parts the whole is divided. In the fraction \(\frac{1}{2}\), the denominator is 2, meaning the whole is divided into 2 equal parts.

The denominator provides essential information:

• It shows the size of each part. For \(\frac{1}{4}\), the whole is divided into 4 smaller parts, making each part smaller than if it were \(\frac{1}{2}\).
• The denominator is never zero, as dividing by zero is undefined in mathematics.
• In simplifying fractions, a common factor cancels out, potentially changing the denominator, as in \(\frac{2}{4}\) simplifying to \(\frac{1}{2}\).

Recognizing how the denominator operates allows you to see how each fraction piece fits into the larger structure.