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Chapter 1: Problem 83

Writing Explain why \(\frac{8}{4}\) is a natural number, but \(\frac{7}{4}\) is not.

Short Answer

Expert verified
The fraction \(\frac{8}{4}\) is a natural number because the quotient of the numerator (8) divided by the denominator (4) gives a natural number (2). However, \(\frac{7}{4}\) is not a natural number because the quotient of the numerator (7) divided by the denominator (4) gives 1.75, a number that does not fall in the set of natural numbers.

Step by step solution

01

Understanding Natural Numbers

A natural number is a number that is used for counting. It is a positive integer starting from 1 and goes up to infinity. They are denoted by the set \(\{1,2,3,4,....\}\). These numbers are both the numerator and the denominator of a fraction. If the numerator is divisible by the denominator and gives a quotient which is a positive integer, then the fraction can be considered a natural number.
02

Checking for \(\frac{8}{4}\)

In the fraction \(\frac{8}{4}\), both 8 and 4 are natural numbers. When 8 (the numerator) is divided by 4 (the denominator), the result is 2, a natural number. Therefore, \(\frac{8}{4}\) falls under the category of natural numbers.
03

Checking for \(\frac{7}{4}\)

In the fraction \(\frac{7}{4}\), 7 and 4 are both natural numbers. However, when 7 (the numerator) is divided by 4 (the denominator), the result is 1.75, which is not a natural number. Therefore, \(\frac{7}{4}\) does not fall under the category of natural numbers.

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Key Concepts

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

Fractions
Fractions are a way to represent parts of a whole, using a combination of two natural numbers. A fraction is written with two numbers separated by a line. The upper number is called the numerator, and the lower number is the denominator.
Fractions can be:
  • Proper Fraction: Where the numerator is smaller than the denominator, e.g., \(\frac{3}{4}\).
  • Improper Fraction: Where the numerator is equal to or greater than the denominator, e.g., \(\frac{5}{3}\).
  • Mixed Number: This is a whole number combined with a proper fraction, e.g., 2\(\frac{1}{4}\).
Fractions are useful in situations where a whole cannot be expressed entirely in full numbers, like slices of a pizza or portions of a pie. Understanding how to interpret fractions is crucial because they can also be simplified or converted into decimals or percentages.
Positive Integers
Positive integers are a subset of integers that are greater than zero. In mathematics, these numbers form the set of natural numbers since they start from 1 and increase infinitely.
  • Examples of positive integers include 1, 2, 3, 4, and beyond.
  • They do not include zero or negative numbers.
  • Positive integers are used for counting and ordering.
When a fraction like \(\frac{8}{4}\) results in a positive integer (like 2), it can be considered a natural number. However, if the division results in a non-whole number, like 1.75 in \(\frac{7}{4}\), it is not a positive integer and thus not a natural number.
Numerator and Denominator
The terms numerator and denominator are crucial in understanding fractions. The numerator, located at the top of the fraction, indicates how many parts of the whole are being considered. The denominator, at the bottom, indicates into how many parts that whole is divided.
  • Numerator: In \(\frac{8}{4}\), 8 is the numerator, representing the number of parts taken.
  • Denominator: In \(\frac{8}{4}\), 4 is the denominator, representing the total number of equal parts.
  • A fraction is a division of the numerator by the denominator.
The relationship between the numerator and the denominator is what determines whether the fraction can be reduced to a natural number. For instance, when the numerator is exactly divisible by the denominator, like in \(\frac{8}{4}\), it simplifies to a natural number, which is 2, unlike \(\frac{7}{4}\), which cannot be simplified to a natural number.

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Most popular questions from this chapter

\(\frac{5^{2}+12^{2}}{13}\)

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