Types of Numbers
Mathematics for Hearing Impaired Students
Understanding Different Types of Numbers
In mathematics, numbers are classified into different types based on their properties. Understanding these categories helps us work with numbers more effectively.
Natural Numbers
Natural numbers are the counting numbers starting from 1. They are positive integers and do not include zero. Example: 1, 2, 3, 4, 5, …
Whole Numbers
Whole numbers include all natural numbers plus zero. Example: 0, 1, 2, 3, 4, …
Integers
Integers include all whole numbers and their negative counterparts. Example: …, -3, -2, -1, 0, 1, 2, 3, …
Rational Numbers
Rational numbers are numbers that can be expressed as a fraction where both numerator and denominator are integers, and the denominator is not zero. Example: 1/2, 0.75, -4/5, 2 (which is 2/1).
Irrational Numbers
Irrational numbers cannot be expressed as a simple fraction. Their decimal representation goes on forever without repeating. Example: √2, π (pi), e.
Real Numbers
Real numbers include all rational and irrational numbers. They can be represented on a number line. Example: -5, 0, 1/2, √3, π.
Number Relationships
Visual Guide to Number Types
Natural Numbers
1, 2, 3, 4, 5, …
Whole Numbers
0, 1, 2, 3, 4, …
Integers
…, -2, -1, 0, 1, 2, …
Rational Numbers
½, 0.75, -4/5
Irrational Numbers
√2, π, e
Real Numbers
All numbers on the number line
Number Line Representation
Integers are shown as solid points, while rational numbers can be between integers.
Venn Diagram
Number System Venn Diagram
The diagram below shows how different types of numbers are related:
Explanation: This Venn diagram shows how number types relate:
- Natural numbers are inside whole numbers
- Whole numbers are inside integers
- Integers are inside rational numbers
- Rational and irrational numbers together make real numbers
Solved Examples
Example 1: Identifying Natural Numbers
Question: Which of these are natural numbers: -3, 0, 5, 12, ½?
Solution: Natural numbers are positive integers starting from 1. So, 5 and 12 are natural numbers. -3 is negative, 0 is not included, and ½ is a fraction.
Answer: 5 and 12
Example 2: Classifying Whole Numbers
Question: Identify whole numbers from: -5, 0, ¼, 7, 10.5
Solution: Whole numbers include 0 and all positive integers without fractions or decimals. So, 0 and 7 are whole numbers.
Answer: 0 and 7
Example 3: Recognizing Integers
Question: Which of these are integers: -10, 0, 3.5, 7, √4?
Solution: Integers include negative numbers, zero, and positive numbers without fractions. √4 = 2, which is an integer. So, -10, 0, 7, and √4 are integers.
Answer: -10, 0, 7, and √4
Example 4: Finding Rational Numbers
Question: Identify rational numbers from: π, 0.75, -8, √9, 1/3
Solution: Rational numbers can be expressed as fractions. 0.75 = 3/4, -8 = -8/1, √9 = 3 = 3/1, 1/3 is already a fraction. π is irrational.
Answer: 0.75, -8, √9, 1/3
Example 5: Distinguishing Irrational Numbers
Question: Which of these are irrational: √2, 0.5, π, 5, √9?
Solution: Irrational numbers cannot be expressed as fractions. √2 ≈ 1.414… (non-repeating, non-terminating), π ≈ 3.14159… (non-repeating, non-terminating). 0.5 = 1/2, 5 = 5/1, √9 = 3 = 3/1 are all rational.
Answer: √2 and π
Practice Exercises
1. Which of these are natural numbers: -2, 0, 4, 8, 5.5?
2. Identify whole numbers from: -1, 0, ½, 3, 10.2
3. Which of these are integers: -15, 0, 2.25, 9, √16?
4. Identify rational numbers from: √3, 0.6, -5, √25, 2/7
5. Which of these are irrational: √5, 0.25, π/2, 4, √36?
6. Classify -7 as natural, whole, integer, rational, or irrational.
7. Is 0 a natural number? Explain.
8. Which of these is not an integer: -10, 0, ¾, 12?
9. Is 1.333… a rational number? Why?
10. Identify all real numbers from: -2, 0, √2, 3/4, π
Answers
1. 4, 8
2. 0, 3
3. -15, 0, 9, √16
4. 0.6, -5, √25, 2/7
5. √5, π/2
6. Integer and Rational
7. No, natural numbers start from 1
8. ¾
9. Yes, it can be expressed as 4/3
10. All are real numbers: -2, 0, √2, 3/4, π