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Chapter 1: Problem 17

Decide whether each statement is true or false. Every integer is a rational number.

Short Answer

Expert verified
True. Every integer can be expressed as a fraction, making it a rational number.

Step by step solution

01

Understand the Definition of an Integer

An integer is any whole number, whether positive, negative, or zero. Examples of integers include -3, 0, and 5.
02

Understand the Definition of a Rational Number

A rational number is any number that can be expressed as the quotient or fraction \(\frac{p}{q}\), where p and q are integers and q 鈮� 0. Examples of rational numbers include \(\frac{1}{2}\), -5 (which can be written as \(\frac{-5}{1}\)), and 0 (which can be written as \(\frac{0}{1}\)).
03

Determine if Every Integer is a Rational Number

Since every integer can be written as a fraction with the integer itself as the numerator and 1 as the denominator (e.g., \(-3 = \frac{-3}{1}\)), it follows that every integer is a rational number.
04

Conclusion

Based on the above steps, the statement 'Every integer is a rational number' is true.

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Key Concepts

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

integer
An integer is a type of number without any fractional or decimal part. Integers can be positive, negative, or zero.
Examples of integers include:
  • -5
  • 0
  • 23


Unlike fractions or decimals, integers are standalone whole numbers. For instance, -3 is an integer, while -3.5 is not.

Understanding integers is crucial in many areas of mathematics and number theory. They are the building blocks for more complex concepts and calculations. It鈥檚 helpful to remember the properties of integers when exploring rational numbers.
rational number definition
A rational number is a number that can be expressed as the fraction \(\frac{p}{q}\), where both p and q are integers, and q is not zero.

Here's a breakdown:
  • **Fraction Form:** Rational numbers are expressed as \(\frac{p}{q}\), such as \(\frac{3}{4}\).
  • **Integers:** Both the numerator (p) and the denominator (q) must be integers.
  • **Non-Zero Denominator:** The denominator, q, should never be zero because division by zero is undefined.

Some examples of rational numbers are:
  • \(\frac{1}{2}\)
  • \(\frac{-5}{1}\)
  • \(\frac{0}{1}\)

It鈥檚 interesting to note that integers are also rational numbers. For example, the integer -3 can be written as \(\frac{-3}{1}\). Rational numbers play an important role in understanding ratios, proportions, and algebra.
number theory
Number theory is a branch of mathematics dedicated to studying the properties and relationships of numbers, particularly integers.

Key areas of number theory include:
  • **Prime Numbers:** Numbers greater than 1 with no positive divisors other than 1 and themselves.
  • **Divisibility:** Exploring how integers divide each other.
  • **Greatest Common Divisors (GCD):** The largest integer that divides two numbers without leaving a remainder.
  • **Least Common Multiples (LCM):** The smallest integer that is a multiple of two numbers.

Understanding number theory aids in solving a variety of mathematical problems and developing critical thinking. It鈥檚 foundational for advanced subjects, from algebra to cryptography. By mastering the basic principles in number theory, one builds a strong base for all future mathematical learning.

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Most popular questions from this chapter

The operation of division is used in divisibility tests. A divisibility test allows us to determine whether a given number is divisible (without remainder) by another number. An integer is divisible by 9 if the sum of its digits is divisible by \(9,\) and not otherwise. Show that (a) \(4,114,107\) is divisible by \(9 \quad\) and \(\quad\) (b) \(2,287,321\) is not divisible by \(9 .\)

Solve each problem. Linda Des Jardines owes \(\$ 870.00\) on her MasterCard account. She returns two items costing \(\$ 35.90\) and \(\$ 150.00\) and receives credit for these on the account. Next, she makes a purchase of \(\$ 82.50\) and then two more purchases of \(\$ 10.00\) each. She makes a payment of \(\$ 500.00 .\) She then incurs a finance charge of \(\$ 37.23 .\) How much does she still owe?

In September, Jeffery Cooper began with a checking account balance of 904.89 Dollar. His checks and deposits for September are as follows: $$ \begin{array}{|l|c|} \hline \text { Checks } & \text { Deposits } \\ \$ 41.29 & \$ 80.59 \\ \$ 13.66 & \$ 276.13 \\ \$ 84.40 & \\ \hline \end{array} $$ Assuming no other transactions, what was his account balance at the end of September?

Decide whether each statement is true or false. \(|13-8| \leq|7-4|\)

Solve each problem. The top of Mt. Whitney, visible from Death Valley, has an altitude of \(14,494\) ft above sea level. The bottom of Death Valley is 282 ft below sea level. Using 0 as sea level, find the difference between these two elevations.
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