Problem 48 - Elementary Algebra Within Reach

Question

Determine which of the numbers in the set are (a) natural numbers, (b) integers, (c) rational numbers, and (d) irrational numbers. $$\left\\{-6, \frac{7}{4}, 2,1, \sqrt{49},-8, \frac{4}{3}\right\\}$$

Step-by-Step Solution

Answer

The natural numbers in the set are 2, 1, and 7 (\(\sqrt{49}\)). The integers in the set are -6, 2, 1, 7, and -8. The rational numbers are -6, \(\frac{7}{4}\), 2, 1, 7, -8, and \(\frac{4}{3}\). The set does not include any irrational numbers.

Step 1: Identifying Natural Numbers

Natural numbers are counting numbers starting from 1. In the given set, the natural numbers are \(2\) and \(1\). The number \(\sqrt{49}\) is also a natural number since square root of \(49\) is \(7\), which is a natural number.

Step 2: Identifying Integers

Integers include all natural numbers, zero, and their negatives. From the set, therefore, the integers are \(-6\), \(2\), \(1\), \(\sqrt{49}\) (which is \(7\)), and \(-8\). Note that the fractional numbers, namely \(\frac{7}{4}\) and \(\frac{4}{3}\), are not integers since they are not whole numbers.

Step 3: Identifying Rational Numbers

Rational numbers are numbers that can be written as a fraction where the denominator is not zero. This means all integers are rational (because they can be written as the integer divided by \(1\)), and any fraction with non-zero denominator is also a rational number. Therefore, all numbers in the set except for one are rational: \(-6\), \(\frac{7}{4}\), \(2\), \(1\), \(\sqrt{49}\), \(-8\), and \(\frac{4}{3}\).

Step 4: Identifying Irrational Numbers

Irrational numbers cannot be expressed as fractions and their decimal expansion neither terminates nor become periodic. In the given set, however, all numbers can be written as a fraction or terminate as decimal; hence, there are no irrational numbers.

Key Concepts

Natural Numbers

Natural numbers are the simplest and most straightforward type of numbers. They are the numbers you use for counting. Think of how you count your fingers or toys - these are the natural numbers. This means they start at 1 and go up like 2, 3, 4, and so on. Importantly, they do not include zero or negative numbers.
In the given exercise set, let's identify the natural numbers:

• The number 2 is a natural number, simply because you count it.
• The number 1 is also a natural number for the same reason.
• Then there's \( \sqrt{49} \). When you solve this, you get 7, which is undoubtedly a natural number!

So, whenever you count things, remember you are using natural numbers.

Integers

Integers expand on the idea of natural numbers by including zero and negative numbers. This inclusion makes integers a broader category. To grasp integers, imagine a number line that stretches infinitely in both directions and includes not just positive numbers, but also negative numbers and zero.
From the given set of numbers, these are the integers:

• -6 is an integer because it continues on this number line below zero.
• 2 and 1 remain integers, as they are already natural numbers and thus fit into integers, too.
• \( \sqrt{49} \) is 7, and 7 is an integer as well.
• -8 is, like -6, an integer, falling on the negative side of the number line.

However, numbers like fractions \( \frac{7}{4} \) and \( \frac{4}{3} \) are not integers. They are parts of whole numbers, not the whole thing itself.

Rational Numbers

Rational numbers are where the fraction fun begins! They're considered rational because they are ratios (or fractions) of integers. Another important fact about rational numbers is that their decimal expansions either end or repeat.
For example:

• A simple integer like -6 can be considered a rational number since you can express it as \( \frac{-6}{1} \), with 1 as the denominator.
• Fractions like \( \frac{7}{4} \) and \( \frac{4}{3} \) are already in quintessential rational form (number over a number, denominator isn't zero).
• The numbers 2 (\( \frac{2}{1} \)), 1 (\( \frac{1}{1} \)), and 7 (since it's \( \sqrt{49} \); \( \frac{7}{1} \)) fit as rational numbers too.
• Finally, -8 can slide right into the rational zone as \( \frac{-8}{1} \).

Ultimately, the clarity of rational numbers lies in their ability to be written as a fraction with integers. In this set, every single number can satisfy that, leaving it purely rational.

Irrational Numbers

Irrational numbers live on a number line too, but they come with a tricky twist of being impossible to fully write down as fractions of integers. Their decimal versions aren't neat or repeating.
Explore examples to clarify:

• A number like \( \pi \) or \( \sqrt{2} \) would be deemed irrational for its non-repeating, non-ending decimal form.
• In contrast, our set lacks such endlessly diverse numbers, indicating no entries are irrational.

These special numbers add a fascinating complexity to the number world as we need them for certain calculations that need precision to the nth degree. However, in our example exercise, none of the given numbers are Irrational since each one can nest comfortably into a rational form.