Problem 38 - Algebra and Trigonometry

Question

List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. $$\\{-5,-0 . \overline{3}, 0, \sqrt{2}, \sqrt{4}\\}$$

Step-by-Step Solution

Answer

Natural Numbers: None. Whole Numbers: 0. Integers: -5 and 0. Rational Numbers: -5, -0.3 recurring, 0. Irrational Numbers: √2. Real Numbers: -5, -0.3 recurring, 0, √2, √4.

Step 1: Identify Natural Numbers

Natural numbers are all positive numbers without any fractional or decimal component. Hence, in the given set, there are no natural numbers since none of the given numbers are positive and without decimal or fractional component.

Step 2: Identify Whole Numbers

Whole numbers are all positive numbers including zero, without any fractional or decimal components. So, in the given set, only 0 is a whole number.

Step 3: Identify Integers

Integers are all whole numbers including their negatives. However, they do not have any decimal or fractional component. Therefore, from the given set, -5 and 0 are integers.

Step 4: Identify Rational Numbers

Rational numbers are those numbers which can be expressed in the form of p/q, where p and q are integers and q is not equal to 0. Rational numbers can have decimal and fractional components. Hence in the given set, -5, -0.3 (recurring) and 0 are rational numbers. Rational numbers can either be expressed as decimals or fractions.

Step 5: Identify Irrational Numbers

An irrational number cannot be expressed as a ratio of two integers. In other words, they cannot be expressed in the form p/q. Furthermore, their decimal expansion is neither terminating nor recurring. From the given set, only √2 which equals 1.41421356... (non-terminating and non-repeating decimal) is irrational.

Step 6: Identify Real Numbers

Real numbers include all rational and irrational numbers. Essentially, they include every number that can be placed on the number line. Therefore, all numbers in the set (-5, -0.3 recurring, 0, √2, √4) are real numbers.