Problem 19 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

True, every integer can be expressed as \(\frac{n}{1}\).

Step 1: Define Integers

First, recall what integers are. Integers include all positive whole numbers, negative whole numbers, and zero. Examples of integers are -3, 0, and 5.

Step 2: Define Rational Numbers

Next, understand what rational numbers are. A rational number is any number that can be expressed as the quotient or fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\).

Step 3: Check if Integers are Rational Numbers

Now, determine if every integer can be written as a rational number. Any integer \(n\) can be written as \(\frac{n}{1}\), which fits the definition of a rational number since both the numerator \(n\) and the denominator \(1\) are integers and \(1 eq 0\).

Step 4: Conclusion

Since we have shown that every integer can be expressed as a fraction (rational number), we can conclude that the statement 'Every integer is a rational number' is true.

Key Concepts

defining integers

An integer is a number that belongs to the set of whole numbers, both positive and negative, including zero. This means every integer can be a whole number like 1 or -1, but it can't have any decimal or fractional part. Examples of integers include:

• -3
• 0
• 5

Integers are fundamental in mathematics as they allow counting, ordering, and operations in arithmetic. Essentially, they form the building blocks for more complex numbers and concepts.

defining rational numbers

A rational number is a number that can be written as a ratio or fraction, specifically in the form \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b eq 0\). This means rational numbers include:

• Fractions like \(\frac{3}{4}\)
• Whole numbers like 2 (which can be written as \(\frac{2}{1}\))
• Decimals that terminate or repeat, such as 0.75 or 0.333...\t

Rational numbers are crucial in mathematics because they help us understand parts of a whole and proportions. They also set the foundation for real numbers, which include both rational and irrational numbers.

integer as rational number

Given the definitions of integers and rational numbers, we need to see if every integer is a rational number. Let's take any integer, say \(n\). By definition, \(n\) can be written as a fraction: \(\frac{n}{1}\). Here, \(n\) is an integer, and \(1\) is also an integer (which importantly is not zero).
So, any integer can be expressed as a fraction of two integers, which fits the definition of rational numbers.
For example:

• 3 can be written as \(\frac{3}{1}\)
• 0 can be written as \(\frac{0}{1}\)
• -7 can be written as \(\frac{-7}{1}\)

This confirms that all integers are rational numbers.
Therefore, the statement 'Every integer is a rational number' is true.