Question

List the numbers in each set that are (a) Natural numbers, (b) Integers, (c) Rational numbers, (d) Irrational numbers, (e) Real numbers. $$ F=\left\\{-\sqrt{2}, \pi+\sqrt{2}, \frac{1}{2}+10.3\right\\} $$

Short answer

No natural or integer numbers. Rational: \(\frac{1}{2}+10.3\). Irrational: \(-\sqrt{2}, \pi + \sqrt{2}\). All real.

Step-by-step solution

Understanding the Problem

Identify and classify each number from the set \( F=\left\{-\sqrt{2}, \pi+\sqrt{2}, \frac{1}{2}+10.3\right\} \). Determine which of these numbers are Natural numbers, Integers, Rational numbers, Irrational numbers, and Real numbers.

- Identify Natural Numbers

Natural numbers are positive whole numbers including zero. Check each element in the set to see if it meets this criterion. \(-\sqrt{2}\) is not a natural number. \(\pi + \sqrt{2}\) is not a natural number. \(\frac{1}{2}+10.3\) is not a natural number.

- Identify Integers

Integers are whole numbers which can be positive, negative, or zero. Check each element in the set: \(-\sqrt{2}\) is not an integer. \(\pi + \sqrt{2}\) is not an integer. \(\frac{1}{2}+10.3\) is not an integer.

- Identify Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers (a fraction). Check each element: \(-\sqrt{2}\) is an irrational number because \sqrt{2}\ is irrational. \(\pi + \sqrt{2}\) is an irrational number as both \(\pi\) and \(\sqrt{2}\) are irrational. \(\frac{1}{2} + 10.3\) is a rational number because it can be expressed as a fraction.

- Identify Irrational Numbers

Irrational numbers cannot be expressed as a fraction. Check each element in the set: \(-\sqrt{2}\) is an irrational number. \(\pi + \sqrt{2}\) is an irrational number. \(\frac{1}{2}+10.3\) is a rational number.

- Identify Real Numbers

Real numbers include all rational and irrational numbers. Every element in the given set is a real number.

Key concepts

Natural Numbers

Natural numbers, also known as counting numbers, are the first set of numbers we learn as children. These include numbers like 1, 2, 3, and so on. They are the simplest form of numbers used for counting. One key point to note is that natural numbers do not include decimals, fractions, or negative numbers.
In the given set \( F = \left\{ -\sqrt{2}, \pi + \sqrt{2}, \frac{1}{2} + 10.3 \right\} \), none of the elements are natural numbers. This is because:

• \( -\sqrt{2} \) is a negative number.
• \( \pi + \sqrt{2} \) involves the irrational number \( \pi \).
• \( \frac{1}{2} + 10.3\) includes a fraction and a decimal.

Integers

Integers extend the set of natural numbers to include negative numbers and zero. Therefore, integers can be positive, negative, or zero. This set includes numbers like -3, 0, 4, etc. They still exclude numbers with fractions or decimals.
When we analyze the set \( F = \left\{ -\sqrt{2}, \pi + \sqrt{2}, \frac{1}{2} + 10.3 \right\} \), we find that none of the elements qualify as integers because:

• \( -\sqrt{2} \) is not a whole number.
• \( \pi + \sqrt{2} \) is a sum of two irrational numbers.
• \( \frac{1}{2} + 10.3 \) involves both a fraction and a decimal.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b eq 0 \). This implies that any number that can be written as a simple fraction is rational.
Looking at our set \( F = \left\{ -\sqrt{2}, \pi + \sqrt{2}, \frac{1}{2} + 10.3 \right\} \), we see that:

• \( -\sqrt{2} \) is not a rational number because \( \sqrt{2} \) is irrational.
• \( \pi + \sqrt{2} \) combines two irrational numbers, making it irrational.
• \( \frac{1}{2} + 10.3 \) is rational because it can be expressed as the fraction \( \frac{53}{5} \).

Irrational Numbers

Irrational numbers cannot be expressed as simple fractions. They have non-repeating and non-terminating decimal expansions. Examples include \( \sqrt{2} \), \( \pi \), and \( e \).
Observing our set \( F = \left\{ -\sqrt{2}, \pi + \sqrt{2}, \frac{1}{2} + 10.3 \right\} \), we identify these as irrational:

• \( -\sqrt{2} \): The square root of 2 is irrational, hence its negative is also irrational.
• \( \pi + \sqrt{2} \): Both \( \pi \) and \( \sqrt{2} \) are irrational, so their sum remains irrational.

The other number, \( \frac{1}{2} + 10.3 \), is rational.

Real Numbers

Real numbers include all numbers that can be found on the number line. They encompass both rational and irrational numbers. So, the set of real numbers consists of natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
For the set \( F = \left\{ -\sqrt{2}, \pi + \sqrt{2}, \frac{1}{2} + 10.3 \right\} \):

• \( -\sqrt{2} \) is a real number.
• \( \pi + \sqrt{2} \) is a real number.
• \( \frac{1}{2} + 10.3 \) is a real number.

Every element in this set is a real number, as they all exist on the number line.