Problem 37 - 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. $$\left\\{-11,-\frac{5}{6}, 0,0.75, \sqrt{5}, \pi, \sqrt{64}\right\\}$$

Step-by-Step Solution

Answer

a. Natural Numbers: \(\sqrt{64}\) (equivalent to 8) \n b. Whole Numbers: 0, \(\sqrt{64}\) \n c. Integers: -11, 0, \(\sqrt{64}\) \n d. Rational Numbers: -11, -\(\frac{5}{6}\), 0, 0.75, \(\sqrt{64}\) \n e. Irrational Numbers: \(\sqrt{5}\), \(\pi\) \n f. Real Numbers: All of the numbers in the given set are real numbers.

Step 1: Identify Natural Numbers

Natural Numbers are the positive numbers without fractions or decimals, starting from 1. In the given set (\(-11,-\frac{5}{6}, 0,0.75, \sqrt{5}, \pi, \sqrt{64}\)), \(\sqrt{64}\) which is equal to 8 is the natural number.

Step 2: Identify Whole Numbers

Whole Numbers are similar to natural numbers but they also include 0. In the given set, 0 and \(\sqrt{64}\) (which equals 8) are the whole numbers.

Step 3: Identify Integers

Integers include all whole numbers and their negatives. In the given set, -11, 0 and \(\sqrt{64}\) (which equals 8) are considered as integers.

Step 4: Identify Rational Numbers

Rational Numbers are numbers which can be represented as a fraction of two integers. In the given set, -11, -\(\frac{5}{6}\), 0, 0.75 (which can be written as \(\frac{3}{4}\)) and \(\sqrt{64}\) (which equals 8), are the rational numbers.

Step 5: Identify Irrational Numbers

Irrational numbers cannot be represented as a fraction of two integers and they are not ending or repeating decimals. In the given set, \(\sqrt{5}\) and \(\pi\) are irrational numbers.

Step 6: Identify Real Numbers

Real numbers include all rational and irrational numbers, basically all numbers that can be placed on the number line. Hence, all numbers in the given set are real numbers.