Chapter 1: The Number System - The Foundation of Mathematics
🎯 Objective: This chapter aims to build a strong conceptual understanding of the number system, from the simplest counting numbers to the complexities of fractions and decimals. We will explore their properties, operations, and interconnections, which are crucial for teaching at the primary and upper-primary levels .
🧮 Section 1.1: Knowing Our Numbers
This section is all about understanding the language of numbers. We learn to read them, write them, compare them, and give them meaning.
🔍 Comparing and Ordering Numbers
How to Compare: Numbers are compared based on the number of digits.
Rule 1: A number with more digits is always greater. (e.g., 8,500 > 999).
Rule 2: If two numbers have the same number of digits, we compare the digits at the highest place value, then the next, and so on. (e.g., To compare 5,678 and 5,689, we start with the thousands place (both 5), then hundreds (both 6), then tens (7 vs. 8). Since 7 < 8, we get 5,678 < 5,689).
📍 Place Value and Face Value
This is a fundamental concept. Every digit in a number has two values:
Face Value: The value of the digit itself. The face value of 7 in 7,56,432 is always 7.
Place Value: The value of the digit based on its position in the number. The place value of 7 in 7,56,432 is 7,00,000 (Seven Lakh).
| Concept | Definition | Example: Number 8,36,425 |
|---|---|---|
| Face Value | The intrinsic value of the digit. | Face value of 3 is 3. |
| Place Value | The value of the position it holds. | Place value of 3 is 30,000. |
🔢 Formation of Numbers with Given Digits
To form the smallest number:
Arrange the given digits in ascending order (from smallest to largest).
Important: If '0' is one of the digits, place it in the second position from the left to give it value.
Example: Digits: 5, 0, 2, 9. Ascending order: 0, 2, 5, 9. Smallest number: 2059.
To form the largest number:
Arrange the given digits in descending order (from largest to smallest).
Example: Digits: 5, 0, 2, 9. Descending order: 9, 5, 2, 0. Largest number: 9520.
⬆️⬇️ Ascending and Descending Order
Ascending Order: Arranging numbers from the smallest to the largest (increasing order). Example: 15, 27, 39, 42.
Descending Order: Arranging numbers from the largest to the smallest (decreasing order). Example: 98, 76, 50, 21.
🌍 Indian and International System of Numeration
Knowing how to read and write large numbers in both systems is essential.
| System | Places/Crores | How to Write | How to Read | ||||
|---|---|---|---|---|---|---|---|
| Indian | Crore (Cr) | Lakh (L) | Thousand (Th) | Ones | 5,42,68,123 | Five crore forty-two lakh sixty-eight thousand one hundred twenty-three. | |
| 10,00,000 | 1,00,000 | 1,000 | 1 | ||||
| International | Millions (M) | Hundred-Thousands (HTh) | Ten-Thousands (TTh) | Thousands (Th) | Ones | 54,268,123 | Fifty-four million two hundred sixty-eight thousand one hundred twenty-three. |
| 1,000,000 | 100,000 | 10,000 | 1,000 | 1 |
📏 Estimation and Rounding Off Numbers
In real life, we often don't need exact numbers. Estimation helps us get a rough idea.
Rounding Off Rules:
To the nearest 10: Look at the ones digit. If it is 0, 1, 2, 3, or 4, keep the tens digit the same. If it is 5, 6, 7, 8, or 9, increase the tens digit by 1.
To the nearest 100: Look at the tens digit. If it is 0, 1, 2, 3, or 4, keep the hundreds digit the same. If it is 5, 6, 7, 8, or 9, increase the hundreds digit by 1.
To the nearest 1000: Look at the hundreds digit. Apply the same rule.
Example: Round off 7,453 to the nearest:
10 → 7,450 (Ones digit 3 < 5)
100 → 7,500 (Tens digit 5 ≥ 5)
1000 → 7,000 (Hundreds digit 4 < 5)
🎲 Section 1.2: Playing with Numbers
Numbers are not just for counting; they have fascinating properties and relationships. This section is all about exploring these patterns.
🔗 Factors and Multiples
Factors: A factor of a number is an exact divisor of that number. Example: Factors of 12 are 1, 2, 3, 4, 6, and 12.
Multiples: A multiple of a number is the product of that number and any natural number. Example: Multiples of 5 are 5, 10, 15, 20,...
Key Fact: Every number is a factor of itself and the smallest factor is 1. The smallest multiple of a number is the number itself.
🔢 Prime and Composite Numbers
Prime Numbers: Numbers greater than 1 that have exactly two distinct factors: 1 and the number itself. Examples: 2, 3, 5, 7, 11. (Note: 2 is the smallest and the only even prime number).
Composite Numbers: Numbers greater than 1 that have more than two factors. Examples: 4, 6, 8, 9, 10.
Trick: The number 1 is neither prime nor composite. It's called a unit.
⚫⚪ Even and Odd Numbers
Even Numbers: Numbers that are divisible by 2. They end in 0, 2, 4, 6, or 8. Example: 2, 14, 36, 58.
Odd Numbers: Numbers that are not divisible by 2. They end in 1, 3, 5, 7, or 9. Example: 3, 17, 25, 49.
✅ Tests of Divisibility
These are shortcuts to check if a number is divisible by another without performing long division.
| Divisible by | Test | Example |
|---|---|---|
| 2 🔴 | The last digit is even (0, 2, 4, 6, 8). | 1,238 → Yes |
| 3 🔵 | The sum of all digits is divisible by 3. | 561: 5+6+1=12 → Yes |
| 4 🟢 | The last two digits form a number divisible by 4. | 5324: 24÷4=6 → Yes |
| 5 🟡 | The last digit is 0 or 5. | 325 → Yes |
| 6 🟠 | The number is divisible by both 2 and 3. | 234: Even (Yes), Sum=9 (Yes) → Yes |
| 8 🟣 | The last three digits form a number divisible by 8. | 9128: 128÷8=16 → Yes |
| 9 🔴 | The sum of all digits is divisible by 9. | 873: 8+7+3=18 → Yes |
| 10 🟢 | The last digit is 0. | 450 → Yes |
| 11 🔵 | The difference between the sum of digits in odd places and the sum of digits in even places is either 0 or a multiple of 11. | 9152: (9+5) - (1+2) = 14 - 3 = 11 → Yes |
📈 Highest Common Factor (HCF)
Definition: The largest number that divides two or more given numbers exactly.
Methods to Find HCF:
Prime Factorization Method: Find the prime factors of each number. Identify the common prime factors and multiply them (with the smallest power).
Example: Find HCF of 24 and 36.
24 = 2³ × 3¹
36 = 2² × 3²
Common factors: 2² and 3¹. HCF = 2² × 3¹ = 4 × 3 = 12.
Long Division Method:
Divide the larger number by the smaller number.
Then, divide the divisor by the remainder.
Repeat this process until the remainder becomes 0. The last divisor is the HCF.
📉 Lowest Common Multiple (LCM)
Definition: The smallest positive number that is a multiple of two or more given numbers.
Methods to Find LCM:
Prime Factorization Method: Find the prime factors of each number. Multiply the highest powers of all the prime factors that appear in any of the numbers.
Example: Find LCM of 24 and 36.
24 = 2³ × 3¹
36 = 2² × 3²
Highest powers: 2³ and 3². LCM = 8 × 9 = 72.
Common Division Method (Ladder Method): Arrange the numbers in a row. Divide by a common prime number. Continue until no two numbers have a common factor. The product of all divisors is the LCM.
🤝 Relationship between HCF and LCM
For any two positive integers, there is a beautiful relationship:
HCF (a, b) × LCM (a, b) = a × b
Application: This formula is extremely useful in solving word problems where one of the values (HCF or LCM) is unknown.
🌱 Section 1.3: Whole Numbers
This section introduces us to the concept of 'zero' and the properties that govern arithmetic operations.
0️⃣1️⃣ Natural Numbers and Whole Numbers
Natural Numbers (N): The counting numbers. N = {1, 2, 3, 4, 5, ...} 🌻
Whole Numbers (W): All natural numbers including zero. W = {0, 1, 2, 3, 4, 5, ...} 🍀
⚙️ Properties of Whole Numbers
These properties make calculations easier and are the backbone of arithmetic.
| Property | Addition | Multiplication |
|---|---|---|
| Closure | Yes (Sum of two whole numbers is always a whole number). *Ex: 5 + 8 = 13* | Yes (Product of two whole numbers is always a whole number). *Ex: 4 × 9 = 36* |
| Commutative | Yes (a + b = b + a). *Ex: 3 + 7 = 7 + 3* | Yes (a × b = b × a). *Ex: 5 × 6 = 6 × 5* |
| Associative | Yes (a + (b + c) = (a + b) + c). *Ex: 2 + (3 + 4) = (2 + 3) + 4* | Yes (a × (b × c) = (a × b) × c). *Ex: 2 × (3 × 4) = (2 × 3) × 4* |
| Identity | Yes (Zero - Additive Identity). *a + 0 = a*. *Ex: 9 + 0 = 9* | Yes (One - Multiplicative Identity). *a × 1 = a*. *Ex: 7 × 1 = 7* |
| Distributive | Multiplication distributes over Addition: a × (b + c) = (a × b) + (a × c). This is a key property for mental math. *Ex: 7 × 105 = 7 × (100 + 5) = (7 × 100) + (7 × 5) = 700 + 35 = 735.* |
📏 Number Line Representation and Operations
The number line is a powerful visual tool.
Addition: Move to the right on the number line. 🚶♂️➡️
Example: 4 + 3. Start at 4, move 3 steps right. You reach 7.
Subtraction: Move to the left on the number line. 🚶♂️⬅️
Example: 7 - 2. Start at 7, move 2 steps left. You reach 5.
➖ Section 1.4: Negative Numbers and Integers
The number system expands to include numbers less than zero, which are essential for describing real-world situations like debt or temperature.
❄️ Concept of Negative Numbers
Integers (Z): The collection of all whole numbers and their negatives.
Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
Positive integers are on the right side of zero, and negative integers are on the left side of zero.
📏 Representation on Number Line
The number line now extends infinitely in both directions.<---|-----|-----|-----|-----|-----|-----|-----|---> -3 -2 -1 0 1 2 3 4
📈 Ordering of Integers
On the number line, numbers increase as we move to the right.
Every positive integer is greater than 0 and every negative integer.
Every negative integer is less than 0.
Example: -5 < -2 (because -5 is to the left of -2 on the number line).
🧮 Addition and Subtraction of Integers
Rules of Integer Operations:
| Operation | Rule | Example |
|---|---|---|
| Addition | Same Sign: Add the numbers and keep the common sign. | (+5) + (+3) = +8 (-5) + (-3) = -8 |
| Different Signs: Find the difference (larger absolute value minus the smaller). The result takes the sign of the number with the larger absolute value. | (+5) + (-3) = +2 (-5) + (+3) = -2 | |
| Subtraction | Convert to Addition: Change the subtraction sign to addition and change the sign of the number that follows. Then, follow the rules of addition. | (+5) - (+3) = (+5) + (-3) = +2 (+5) - (-3) = (+5) + (+3) = +8 |
💰 Real-Life Applications of Integers
Banking: Deposits (+), Withdrawals (-).
Weather: Temperature above zero (+), below zero (-).
Floors of a Building: Above ground (+), basement/parking (-).
Geography: Altitude above sea level (+), depth below sea level (-).
🥧 Section 1.5: Fractions
Fractions represent a part of a whole or a part of a collection. They are a critical bridge to more advanced math.
🍕 Concept of Fractions – Proper, Improper, Mixed Fractions
Proper Fraction: Numerator < Denominator. Its value is less than 1. *Example: 3/5, 7/8.*
Improper Fraction: Numerator ≥ Denominator. Its value is 1 or more. *Example: 7/5, 9/4.*
Mixed Fraction: A combination of a whole number and a proper fraction. *Example: 2 1/3 (Two and one-third).* It represents the sum of the whole number and the fraction (2 + 1/3).
🔄 Equivalent Fractions
Fractions that represent the same part of a whole are called equivalent fractions.
To get an equivalent fraction, multiply or divide the numerator and denominator by the same non-zero number.
Example: 1/2 = 2/4 = 3/6 = 4/8
⚖️ Comparing Fractions
Like Fractions (Same Denominator): The fraction with the larger numerator is greater.
Example: 5/7 > 3/7.
Unlike Fractions (Different Denominators): Find the LCM of the denominators to make them like fractions (find a common denominator), then compare the numerators.
Example: Compare 3/4 and 5/6. LCM of 4 and 6 is 12.
3/4 = (3×3)/(4×3) = 9/12
5/6 = (5×2)/(6×2) = 10/12
Since 10/12 > 9/12, we have 5/6 > 3/4.
➕➖✖️➗ Operations on Fractions
Addition/Subtraction:
For like fractions, simply add/subtract the numerators.
For unlike fractions, first convert them into like fractions (using LCM), then add/subtract the numerators.
Example (Addition): 1/4 + 3/8 = (2/8) + (3/8) = 5/8.
Multiplication:
Multiply the numerators and multiply the denominators. Simplify if possible.
Example: 2/3 × 4/5 = (2×4)/(3×5) = 8/15.
Division:
To divide one fraction by another, multiply the first fraction (dividend) by the reciprocal of the second fraction (divisor).
Example: 2/3 ÷ 4/5 = 2/3 × 5/4 = (2×5)/(3×4) = 10/12 = 5/6.
📏 Fractions on a Number Line
Just like whole numbers, fractions can also be placed on a number line. The space between 0 and 1 is divided into equal parts based on the denominator. For example, for 3/4, divide the space between 0 and 1 into 4 equal parts and count 3 parts from 0.
📝 Word Problems Involving Fractions
Strategy: Read carefully, identify the 'whole', and determine the operation needed.
Example: Riya ate 1/4 of a pizza and her brother ate 2/5 of the same pizza. What fraction of the pizza did they eat together? (Answer: 1/4 + 2/5 = 5/20 + 8/20 = 13/20).
💰 Section 1.6: Decimals (Integration with Fractions)
Decimals are another way of representing fractions, especially those with denominators like 10, 100, 1000, etc.
🔢 Decimal Representation
A decimal number has a whole number part and a fractional part, separated by a decimal point (.).
Example: In 23.45, 23 is the whole number part, and 45 is the fractional part.
📍 Place Value in Decimals
The place value chart extends to the right of the decimal point as well.
| Thousands (1000) | Hundreds (100) | Tens (10) | Ones (1) | Decimal Point (.) | Tenths (1/10) | Hundredths (1/100) | Thousandths (1/1000) |
|---|---|---|---|---|---|---|---|
| 2 | 3 | . | 4 | 5 | 6 |
In the number 23.456, the place value of 4 is 4/10, of 5 is 5/100, and of 6 is 6/1000.
🔄 Converting Fractions to Decimals and Vice Versa
Fraction → Decimal: Divide the numerator by the denominator.
Example: 3/4 = 3 ÷ 4 = 0.75.
Decimal → Fraction: Write the decimal without the decimal point as the numerator. For the denominator, write 1 followed by as many zeros as there are digits after the decimal point. Then simplify.
Example: 0.75 = 75/100 = 3/4.
➕➖ Addition and Subtraction of Decimals
Crucial Step: Always line up the decimal points vertically before adding or subtracting. This ensures that you are adding tenths to tenths, hundredths to hundredths, etc.
Example: Add 12.34 and 5.6.
text12.34 + 5.60 (You can add a zero to 5.6 to make it 5.60 for clarity) ------- 17.94
💵 Money as Decimals
Money is the most common real-life application of decimals.
₹1 = 100 paise.
So, 75 paise = ₹ 75/100 = ₹ 0.75.
This perfectly demonstrates the connection between fractions, decimals, and our daily lives.
📝 Chapter Summary: Quick Revision Table for PSTET
I hope this detailed chapter serves as a valuable resource in your PSTET preparation journey. Remember, a strong foundation in the number system is the key to unlocking the rest of mathematics. Happy teaching and best of luck! 👍
