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Chapter 3: Problem 11

Prove that every integer is a rational number.

Short Answer

Expert verified
To prove that every integer is a rational number, consider any integer \(n\). We can represent \(n\) as a ratio of two integers by choosing \(a = n\) and \(b = 1\), with \(b 鈮� 0\). Thus, \(n = \frac{n}{1}\), which is a rational number. Therefore, every integer is a rational number.

Step by step solution

01

Definition of Integers and Rational Numbers

Integers are whole numbers that can be positive, negative, or zero. Some examples of integers are -3, -2, -1, 0, 1, 2, 3. Rational numbers are numbers that can be written as a ratio (or fraction) of two integers, where the denominator (the integer in the bottom) is not equal to zero. Rational numbers can be represented as \(\frac{a}{b}\), where \(a\) and \(b\) are both integers and \(b鈮�0\).
02

Represent an Integer as a Rational Number

Now, consider any integer \(n\). To represent n as a rational number, we need to find integers \(a\) and \(b\) (with \(b鈮�0\)) such that \(n = \frac{a}{b}\). Let's use the integer \(n\) itself as the numerator, \(a = n\), and choose the denominator to be 1, \(b = 1\). Now, we can represent \(n\) as a ratio of two integers: \(\frac{a}{b} = \frac{n}{1}\) Since the denominator is not equal to zero (\(b = 1 鈮� 0\)), \(\frac{n}{1}\) is a rational number.
03

Conclusion

As we have just shown that any integer \(n\) can be represented as a rational number in the form \(\frac{n}{1}\), we can conclude that every integer is indeed a rational number.

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Key Concepts

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

Integers
Integers are fundamental numbers in mathematics that consist of whole numbers. These include all positive numbers like 1, 2, 3, etc., negative numbers like -1, -2, -3, and zero. They do not include fractions or decimals.
Integers can be depicted on a number line, which helps visualize them easily. Every integer has a distinct position on this line.

It's important to understand that integers are discrete. This means there are no "in-between" values, unlike fractions that might fall between integers. Furthermore, integers are often used in basic arithmetic operations, such as addition, subtraction, multiplication, and even division when the division does not result in a fraction.
  • Examples of integers: -5, 0, 7.
  • Non-examples of integers: 0.5, -1.25, 3/4.
Understanding integers is essential as they form the building blocks of other types of numbers, like rational numbers.
Fraction Representation
A fraction represents a part of a whole and is used to show numbers between integers. Fractions are composed of two parts: a numerator and a denominator. The numerator, written on top, indicates the number of parts chosen. The denominator, placed below, tells how many equal parts the whole is divided into.
For instance, in the fraction \(\frac{3}{4}\), 3 is the numerator, and 4 is the denominator.

Importantly, a fraction can be seen as a division problem. This is why you often hear phrases like "3 divided by 4," which corresponds to the fraction \(\frac{3}{4}\). Every integer can be represented as a fraction by assigning 1 as its denominator.
  • An integer like 5 can be written as \(\frac{5}{1}\).
  • A negative integer like -7 becomes \(\frac{-7}{1}\).
Thus, any whole number can seamlessly transition into this fraction style, establishing its identity as a rational number.
Mathematical Proof
A mathematical proof is a logical explanation demonstrating that a particular mathematical statement is true. This process is integral to mathematics since it helps affirm the truth of concepts like the idea that every integer is a rational number.
To prove this, we rely on definitions. According to the definition of rational numbers, any number that can be expressed as a fraction with an integer numerator and a non-zero integer denominator is rational.

Here鈥檚 how proof works for integers being rational numbers:
  • Start with an integer, say \( n \).
  • According to our previous explanation, represent \( n \) as a fraction: \( \frac{n}{1} \).
  • The fraction \( \frac{n}{1} \) follows the rule of having an integer numerator and denominator, except the denominator is not zero.
By showing these steps logically, you create a proof. This links back to our fundamental step: every integer can become a fraction with 1 as the denominator, fitting the definition of a rational number. This pattern of thinking and proving is central to mathematics, helping establish truths beyond potential doubt.

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Most popular questions from this chapter

Prove that a necessary and sufficient condition for a nonnegative integer \(n\) to be divisible by a positive integer \(d\) is that \(n\) mod \(d=0\).

"Proof: Suppose \(r\) and \(s\) are rational numbers. Then \(r=a / b\) and \(s=c / d\) for some integers \(a, b, c\), and \(d\) with \(b \neq 0\) and \(d \neq 0\) (by definition of rational). Then \(r+s=\) \(a / b+c / d\). But this is a sum of two fractions, which is a fraction. So \(r+s\) is a rational number since a rational number is a fraction."

Prove that for all positive integers \(a\) and \(b, a \mid b\) if, and only if, \(\operatorname{gcd}(a, b)=a\). (Note that to prove " \(A\) if, and only if, \(B, "\) you need to prove "if \(A\) then \(B\) " and "if \(B\) then \(A . "\) ")

For each integer \(n\) with \(1 \leq n \leq 10, n^{2}-n+11\) is a prime number.

a. Use proof by contradiction to show that for any integer \(n\), it is impossible for \(n\) to equal both \(3 q_{1}+r_{1}\) and \(3 q_{2}+r_{2}\), where \(q_{1}, q_{2}, r_{1}\), and \(r_{2}\), are integers, \(0 \leq r_{1}<\) \(3,0 \leq r_{2}<3\), and \(r_{1} \neq r_{2}\). b. Use proof by contradiction, the quotient-remainder theorem, division into cases, and the result of part (a) to prove that for all integers \(n\), if \(n^{2}\) is divisible by 3 then \(n\) is divisible by 3 . c. Prove that \(\sqrt{3}\) is irrational.
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