Question

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

Short answer

In set \(E\), all numbers are irrational and real. There are no natural, integer, or rational numbers.

Step-by-step solution

Identify Natural Numbers

Natural numbers are positive integers starting from 1. In set \(E = \{\sqrt{2}, \pi, \sqrt{2}+1, \pi+\frac{1}{2} \}\), none of the numbers are natural numbers.

Identify Integers

Integers include all whole numbers and their opposites (negative whole numbers). In set \(E\), none of the numbers are integers.

Identify Rational Numbers

Rational numbers can be expressed as the quotient of two integers (i.e., fractions). In set \(E\), none of the numbers can be expressed in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q eq 0\).

Identify Irrational Numbers

Irrational numbers cannot be expressed as a simple fraction; they have non-terminating, non-repeating decimal parts. In set \(E\), all the numbers \(\sqrt{2}\), \(\pi\), \(\sqrt{2}+1\), and \(\pi+\frac{1}{2}\) are irrational numbers.

Identify Real Numbers

Real numbers include all rational and irrational numbers. Therefore, all elements in set \(E\), \(\sqrt{2}\), \(\pi\), \(\sqrt{2}+1\), and \(\pi+\frac{1}{2}\), are real numbers.

Key concepts

Natural Numbers

Natural numbers are the numbers we naturally count with. They start from 1 and go upwards. For example, 1, 2, 3, and so on are all natural numbers. Natural numbers do not include zero, negative numbers, fractions, or decimals. In the given set, which is \[E = \{ \sqrt{2}, \pi, \sqrt{2} + 1, \pi + \frac{1}{2} \ \} \], none of these numbers are natural numbers. This is because:

• \( \sqrt{2} \) is an irrational number
• \( \pi \) is also an irrational number
• \( \sqrt{2} + 1 \) and \( \pi + \frac{1}{2} \) are sums of irrational numbers, making them irrational

Integers

Integers are whole numbers and their negative counterparts, such as -3, -2, -1, 0, 1, 2, 3, and so on. They do not include fractions or decimals. In our set \[ E = \{ \sqrt{2}, \pi, \sqrt{2} + 1, \pi + \frac{1}{2} \ \} \], none of these elements qualify as integers. Specifically:

• \( \sqrt{2} \) lies between 1 and 2, and is not a whole number
• \( \pi \) is approximately 3.14, also not a whole number
• \( \sqrt{2} + 1 \) and \( \pi + \frac{1}{2} \) both result in non-integer values

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q e 0 \). Rational numbers have finite or repeating decimal expansions. None of the numbers in our set \[ E = \{ \sqrt{2}, \pi, \sqrt{2} + 1, \pi + \frac{1}{2} \ \} \] are rational. Here’s why:

• \( \sqrt{2} \) has a non-repeating, non-terminating decimal expansion
• \( \pi \) is a well-known irrational number
• \( \sqrt{2} + 1 \) and \( \pi + \frac{1}{2} \) are combinations involving irrational numbers, thus irrational

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a fraction of two integers. Their decimal expansions are non-terminating and non-repeating. The set \[ E = \{ \sqrt{2}, \pi, \sqrt{2} + 1, \pi + \frac{1}{2} \ \} \] is composed entirely of irrational numbers because:

• \( \sqrt{2} \) is an irrational number
• \( \pi \) is an irrational number
• \( \sqrt{2} + 1 \) combines an irrational number (\( \sqrt{2} \)) with an integer
• \( \pi + \frac{1}{2} \) combines an irrational number (\( \pi \)) with a fraction

Real Numbers

Real numbers include all rational and irrational numbers. This means every number that can be located on the number line is a real number. Therefore, the set \[ E = \{ \sqrt{2}, \pi, \sqrt{2} + 1, \pi + \frac{1}{2} \ \} \] consists of real numbers. This is because:

• \( \sqrt{2} \) is irrational, thus a real number
• \( \pi \) is irrational, thus a real number
• \( \sqrt{2} + 1 \) and \( \pi + \frac{1}{2} \) are sums involving irrational numbers, making them irrational, and hence real

Real numbers encompass every other number type, including natural numbers, integers, rational numbers, and irrational numbers.