Introduction to the Number System

Mathematics is built on the foundation of numbers. From counting objects to measuring distances, numbers help us understand and describe the world around us. The number system is a way of organizing and classifying numbers into different types based on their properties. Understanding these types helps us solve problems more easily and prepares us for exams like the Assam Direct Recruitment Examination (ADRE) for Grade 4 posts.

In this chapter, we will explore the main types of numbers: natural numbers, whole numbers, integers, rational numbers, and irrational numbers. We will learn what each type means, how they relate to each other, and why they are important in everyday life and exams.

Natural Numbers

Natural numbers are the numbers we use to count objects. They start from 1 and go on infinitely: 1, 2, 3, 4, 5, and so on. These are sometimes called counting numbers.

Natural numbers do not include zero or any negative numbers.

For example, if you have 3 apples, you can count them as 1, 2, 3 - these are natural numbers.

Properties of Natural Numbers

• They start from 1 and go on infinitely.
• Used for counting and ordering.
• Always positive.

Whole Numbers

Whole numbers are like natural numbers but include zero. So, whole numbers start from 0 and continue as 0, 1, 2, 3, 4, and so on.

Zero is important because it represents the absence of quantity - for example, if you have zero mangoes, it means you have none.

Difference Between Natural and Whole Numbers

The only difference is that whole numbers include zero, while natural numbers start from one.

Integers

Integers include all whole numbers and their negative counterparts. This means integers are {..., -3, -2, -1, 0, 1, 2, 3, ...}.

Integers are useful when we talk about temperatures below zero, money owed (debt), or levels below sea level.

Real-Life Examples of Integers

• Temperature: -5°C means 5 degrees below zero.
• Bank balance: -Rs.200 means you owe Rs.200.
• Floor levels: Basement floors can be represented by negative integers.

Rational Numbers

Rational numbers are numbers that can be written as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\). This means rational numbers include integers, fractions, and decimals that either terminate or repeat.

For example:

Decimal Representation of Rational Numbers

Rational numbers can have decimals that:

• Terminate: End after a finite number of digits (e.g., 0.75)
• Repeat: Have a repeating pattern of digits (e.g., 0.666...)

Irrational Numbers

Irrational numbers cannot be written as fractions of integers. Their decimal forms neither terminate nor repeat. These numbers go on forever without any pattern.

Common examples include:

Why Are Irrational Numbers Important?

Irrational numbers appear in many real-world situations like geometry (circle measurements), physics, and engineering. Knowing their properties helps us understand the limits of fractions and decimals.

Worked Example 1: Classifying Numbers on a Number Line Easy

Problem: Classify the following numbers as natural, whole, integer, rational, or irrational: 3, 0, -2, 0.5, \(\sqrt{2}\).

Solution:

Step 1: Identify each number's type.

• 3: Natural (counting number), whole, integer, rational (3 = \(\frac{3}{1}\))
• 0: Whole number, integer, rational (0 = \(\frac{0}{1}\)), but not natural
• -2: Integer, rational (=\(\frac{-2}{1}\)), not natural or whole
• 0.5: Rational (=\(\frac{1}{2}\)), not integer, whole, or natural
• \(\sqrt{2}\): Irrational, not rational, integer, whole, or natural

Answer: The classifications are as above.

Worked Example 2: Identifying Rational and Irrational Numbers Medium

Problem: Determine whether the numbers 0.333... and \(\sqrt{3}\) are rational or irrational.

Solution:

Step 1: Look at 0.333... (repeating decimal)

Since it is a repeating decimal, it can be expressed as a fraction. For example, 0.333... = \(\frac{1}{3}\).

Step 2: Look at \(\sqrt{3}\)

The decimal form of \(\sqrt{3}\) is approximately 1.7320508..., which neither terminates nor repeats.

Therefore, \(\sqrt{3}\) is irrational.

Answer: 0.333... is rational; \(\sqrt{3}\) is irrational.

Worked Example 3: Expressing Rational Numbers as Fractions Medium

Problem: Convert the decimals 0.75 and 0.2 (recurring) into fractions and simplify.

Solution:

Step 1: Convert 0.75 (terminating decimal)

0.75 = \(\frac{75}{100}\) = \(\frac{3}{4}\) after dividing numerator and denominator by 25.

Step 2: Convert 0.2 recurring (0.222...)

Let \(x = 0.222...\)

Multiply both sides by 10: \(10x = 2.222...\)

Subtract original equation: \(10x - x = 2.222... - 0.222...\)

\(9x = 2\)

\(x = \frac{2}{9}\)

Answer: 0.75 = \(\frac{3}{4}\), 0.2 recurring = \(\frac{2}{9}\)

Worked Example 4: Real-life Application: Budgeting with Integers and Rational Numbers Hard

Problem: Raju has Rs.500 in his wallet. He spends Rs.150.75 on groceries and owes Rs.200 to a friend. What is his current balance?

Solution:

Step 1: Initial amount = Rs.500

Step 2: Amount spent = Rs.150.75

Step 3: Amount owed = Rs.200 (negative balance)

Step 4: Calculate remaining money after groceries:

Rs.500 - Rs.150.75 = Rs.349.25

Step 5: Deduct the amount owed:

Rs.349.25 - Rs.200 = Rs.149.25

Answer: Raju has Rs.149.25 left after expenses and debt.

Worked Example 5: Number System Puzzle for Exam Practice Hard

Problem: Which of the following numbers are integers? 0, -7, 3.5, \(\frac{4}{2}\), \(\sqrt{16}\), \(\pi\)

Solution:

• 0: Integer
• -7: Integer
• 3.5: Not an integer (decimal number)
• \(\frac{4}{2} = 2\): Integer
• \(\sqrt{16} = 4\): Integer
• \(\pi \approx 3.14159\): Not an integer (irrational number)

Answer: Integers are 0, -7, 2, and 4.

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