Problem 49 - 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. $$\left\\{1.8, \frac{1}{10}, 7,-2.75,1,-3\right\\}$$

Step-by-Step Solution

Answer

The set contains the following types of numbers: \natural numbers: \(1, 7\), \integers: \(1, 7, -3\), \rational numbers: \(1.8, \frac{1}{10}, 7, -2.75, 1, -3\), And there are no irrational numbers in the set.

Step 1 Identify Natural Numbers

Identify the numbers in the set without fractional parts and are positive. Natural numbers in this set are: \(1, 7\).

Step 2 Identify Integers

Identify the numbers in the set without fractional parts. These include both positive and negative numbers. The integers in the set are: \(1, 7, -3\).

Step 3 Identify Rational Numbers

Identify the numbers in the set that could be expressed as a fraction. Rational numbers in this set can be: \(1.8, \frac{1}{10}, 7, -2.75, 1, -3\).

Step 4 Identify Irrational Numbers

Identify the numbers in the set that could not be expressed as a fraction. None of the numbers in the set is an irrational number as all can be expressed as fractions.

Key Concepts

Natural Numbers

Diving into the realm of mathematics begins with understanding the basics, such as natural numbers. These are the numbers we first learn as children, and they serve as the foundation for more complex math concepts. Natural numbers, also known as counting numbers, encompass all positive integers starting from 1 and moving upwards indefinitely — 1, 2, 3, and so forth. They are used to count objects and measure quantity.

When looking at a set of numbers, like \(1.8, \frac{1}{10}, 7, -2.75, 1, -3\), to distinguish natural numbers, we look for whole, positive figures. Here, only \(1\) and \(7\) fit the bill. It's essential for students to understand that natural numbers don't include fractions, decimals, or negative values. They're the simplest building blocks within the number system.

Integers

Graduating from natural numbers, we encounter integers. These are similar to natural numbers but they introduce the concept of zero and negatives. Essentially, integers are all the whole numbers, both positive and negative, including zero: ..., -3, -2, -1, 0, 1, 2, 3, ...

From our example, the integers can be found by locating the non-fractional numbers. The identified integers are \(1\), \(7\), and \( -3\). Unlike natural numbers, integers allow for the expression of deficits or values below zero. Decimals even if they are negative or positives, like -2.75, don't qualify as integers because they are not whole numbers.

Rational Numbers

As we delve deeper, rational numbers emerge with a bit more complexity. These numbers can be expressed as fractions, where both the numerator (top number) and the denominator (bottom number) are integers. The denominator, importantly, must never be zero. Rational numbers can take many forms — they can be whole numbers, proper fractions, improper fractions, or even repeating or terminating decimals.

In terms of our example set \(1.8, \frac{1}{10}, 7, -2.75, 1, -3\), we can observe that each number can indeed be written as a fraction. For instance, \(1.8\) is the same as \(\frac{9}{5}\), and \( -2.75\) is \(\frac{-11}{4}\), making all of these numbers rational. It's vital for learners to grasp that decimals, if they can be converted into such fractions, are part of the rational number family.

Irrational Numbers

Venturing into the mysterious world of irrational numbers, we find numbers that cannot be tidily expressed as fractions or ratios of integers. These are the numbers that keep going on forever without repeating — and they can't be captured with simple fractions. Classic examples of irrational numbers include the square root of 2 (\sqrt{2}), the mystical \(\pi\), and the natural logarithm base \(e\).

Back to our number set, you’ll notice we didn't identify any irrational numbers. Each number given could be turned into a fraction of integers. Thus, irrational numbers are missing from this particular set. When students learn about irrational numbers, it's important to highlight that these numbers are not 'impossible' or 'incomprehensible.' They have distinct properties and play a crucial role in advanced mathematics, such as in geometry and calculus.