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Chapter 4: Problem 59

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

Short Answer

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

Step by step solution

01

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.
02

Identifying the Given Fraction

In this problem, the given fraction is \(\frac{1}{2}\). Here, 1 is the numerator and 2 is the denominator.
03

Calculating the Reciprocal

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Rectangle \(A B C D\) has a length of 5 and a width of \(3 .\) Point \(D\) is the ordered pair \((7,2) .\) Find points \(A, B\) and \(C\). (GRAPH CANT COPY)

Perform the indicated operation. $$-\frac{5}{4}\left(\frac{8}{15}\right)$$

Add or subtract as indicated. $$\frac{1}{3}-\frac{2}{5}$$

Multiply. $$\left(-\frac{4}{3}\right)\left(-\frac{3}{4}\right)$$

Match each statement on the left with the property that justifies it on the right. a. Distributive property b. Associative property c. Commutative property d. Commutative and associative properties $$(a+4)+2=(4+2)+a$$
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