Chapter 1 — Number System

Counting and natural numbers
Counting means saying or writing numbers in order: 1, 2, 3, … The numbers we use to count objects are called natural numbers. They start at 1 and continue without end. Natural numbers are used for quantities you can count: apples, students, books.

Writing and reading numbers
Use digits (0–9) to write numbers. Read numbers from left to right using place-value names (units, tens, hundreds...). In the Indian system groups go: ones, tens, hundreds, thousands, ten-thousands, lakhs, ten-lakhs, etc.

Number line
A number line is a straight line on which numbers are marked at equal distances. Mark points for 1, 2, 3 and show arrow at the right end to indicate numbers continue. A number line helps compare sizes and show addition (moving right) and subtraction (moving left).

Properties and uses

• Natural numbers are closed under addition and multiplication: adding or multiplying two natural numbers gives another natural number.
• Subtraction may not always give a natural number (for example 2 − 5 is not a natural number).
• Counting builds the idea of order and size which you will use everywhere in maths.

Learning activities

• Practice counting forwards and backwards in steps of 1, 2, 5 and 10.
• Place given natural numbers on a number line and use the line to show simple additions and subtractions.
• Write numbers in words and figures for practice.

What are whole numbers?
Whole numbers are the set of natural numbers together with zero: 0, 1, 2, 3, … Zero is used to represent none or the absence of any quantity, for example 0 books or 0 pens. Whole numbers are useful when we want to include the idea of nothing in counting.

Zero as a number
Zero has special properties. It is neither positive nor negative. In addition it is the additive identity: adding zero to any whole number leaves it unchanged. In multiplication, any number times zero is zero.

Operations with whole numbers
Adding two whole numbers always gives a whole number. Multiplying two whole numbers gives a whole number. Subtraction may not give a whole number if the smaller number is subtracted from a larger one (for example 3 − 5 leads outside whole numbers). Division of whole numbers can give a whole number or leave a remainder.

Place on a number line
On a number line place zero at a chosen origin. Numbers to the right of zero are positive whole numbers. Using zero makes it easier to show subtraction and later leads to negative numbers.

Using zero in daily life

• Zero in money: ₹0 means no money left in a purse.
• Scores: 0 marks indicate no correct answers.
• Counting empty seats or empty boxes uses zero.

Practice suggestions

• Write whole numbers starting from 0 and count forward and backward.
• Use the number line including 0 to perform simple additions and subtractions and notice where results fall.

Definition and meaning
Integers include whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, …. Negative numbers express loss, debt, temperature below zero or movement in the opposite direction. For example, −5°C means five degrees below zero.

Number line with negatives
Extend the number line left of zero to show negative integers. Numbers increase to the right; every step to the right increases value by 1 and every step to the left decreases by 1. Thus −2 is left of −1, and −1 is left of 0.

Comparing integers
Comparisons follow the number line: larger if to the right. Examples: 3 > −1, −2 < 1, and −3 < −2. Remember zero is greater than any negative integer.

Basic operations with integers

• Addition: adding a positive integer moves right, adding a negative moves left. Example: (−3) + 5 = 2.
• Subtraction: a − b can be thought of as a + (−b). If we subtract a negative, a − (−b) = a + b.
• Multiplication: signs matter. Positive × positive = positive; negative × negative = positive; positive × negative = negative.

Practical examples and rules

• Bank balance: if you owe ₹50: balance −50.
• Use integer rules in problems involving gains and losses or temperatures.

Practice

• Place integers on a line, compare pairs and perform simple operations using sign rules.
• Solve word problems that describe gains and losses in daily contexts.

Understanding place value
Place value is the value a digit has because of where it sits in a number. In the Indian system we use groups such as ones, tens, hundreds, thousands, ten-thousands, lakhs, ten-lakhs. For example in 3,45,678 the digit 3 is in the lakh place so its place value is 3 × 1,00,000 = 3,00,000. The same digit may have different values when placed elsewhere.

Face value
Face value is the digit itself without considering its place. In 3,45,678 the face value of 3 is simply 3 while its place value is 3,00,000. Always distinguish face and place value as both are used in problems.

Expanded form
Any number can be expanded as the sum of each digit multiplied by its place value. For example 45,321 = 4×10000 + 5×1000 + 3×100 + 2×10 + 1. Writing numbers in expanded form helps to understand what the digits represent and is useful for comparing and rounding.

Comparing numbers
To compare two whole numbers, first count digits: the number with more digits is larger. If digits are equal, compare from the leftmost digit using place values. For decimals, make the number of digits after the decimal equal by adding zeros and then compare left to right. For fractions, convert to common denominators or to decimals to compare easily.

Rounding and estimation
Rounding uses place value: to round to nearest ten look at the units digit; to round to nearest hundred look at tens, and so on. Estimation by rounding helps check answers in calculations.

Practice methods

• Identify face and place values of digits in various numbers.
• Convert numbers to expanded form and compare pairs using place values.
• Round numbers to nearest ten, hundred or thousand for quick estimates.

Vertical addition
To add whole numbers, write them one below another aligning digits by place value (units under units, tens under tens). Start adding from the rightmost column. If the sum in a column is 10 or more, write down the units digit and carry the tens to the next column on the left. Continue until all columns are added and include any final carry as the leftmost digit.

Vertical subtraction and borrowing
Write numbers in columns aligned by place value. Subtract each column starting from units. If a digit in the minuend is smaller than the corresponding digit in the subtrahend, borrow 1 (which equals 10 in that place) from the next higher place. Reduce the higher-place digit by 1 and add 10 to the current digit, then subtract. Continue across all columns.

Checking answers
To check subtraction, add the subtrahend and the difference; the sum should equal the minuend. For addition, estimate by rounding the numbers to check the result is reasonable.

Solving word problems
Read the problem carefully and identify the quantities given and what is asked. Translate words like 'total', 'more than', 'left', 'altogether', 'remaining' into mathematical operations. Draw simple diagrams or lists if helpful. Choose the correct operation and compute using vertical methods to reduce mistakes.

Tips for accuracy

• Always align by place value when writing numbers vertically.
• Write carries and borrows clearly above or below the line to avoid errors.
• Use estimation to verify answers quickly.

Long multiplication
Multiplication is repeated addition. For multiplying multi-digit numbers use the long multiplication method. Write the multiplicand (larger number) above and the multiplier (smaller number) below, aligning by place value. Multiply the multiplicand by each digit of the multiplier starting with units; write each partial product shifted left according to the place of the digit of the multiplier. Add all partial products to get the final product. Carry digits are handled as you multiply each column.

Short multiplication
For one-digit multipliers multiply each digit of the multiplicand and carry as necessary. For example 234 × 6: multiply units, tens and hundreds one by one carrying tens forward.

Long division
Division finds how many times a divisor fits into a dividend. Use long division: begin with the leftmost digit or group of digits in the dividend that the divisor can divide into. Write the quotient digit above, multiply divisor by that quotient digit, subtract to get remainder, bring down next digit and repeat. If the divisor does not divide exactly, a remainder is left and you may continue with decimals if needed.

Relationship and checking
Multiplication and division are inverse operations: if a × b = c then c ÷ b = a and c ÷ a = b. Use this fact to check answers. Estimation by rounding numbers helps check if the product or quotient is reasonable.

Tips and practice

• Practice partial products carefully and align them correctly before adding.
• When dividing, choose groups of dividend digits so that the divisor fits at least once into the chosen group.
• Always write remainders clearly and check by multiplying quotient by divisor and adding remainder to verify the dividend.

Multiples
A multiple of a number is obtained by multiplying that number by an integer. For example multiples of 6 are 6, 12, 18, 24, … Multiples are useful when finding common times or synchronising repeating events.

Factors
A factor (or divisor) of a number is a number that divides it exactly with no remainder. For example factors of 12 are 1, 2, 3, 4, 6, 12. To find factors list pairs whose product equals the number, checking up to the square root saves time.

Divisibility rules
Divisibility rules give quick tests to see if a number is divisible by small integers without full division. Common rules used often include:

• Divisible by 2 if the last digit is even (0,2,4,6,8).
• Divisible by 3 if the sum of digits is divisible by 3.
• Divisible by 4 if the last two digits form a number divisible by 4.
• Divisible by 5 if the last digit is 0 or 5.
• Divisible by 9 if the sum of digits is divisible by 9.
• Divisible by 10 if the last digit is 0.

Using factors and multiples
Knowing factors helps to simplify fractions and divide things into equal groups. Multiples help to find LCM or to schedule repeated events. Use prime factorisation to find both factors and multiples systematically: express each number as a product of primes, then combine powers of primes as required for HCF or LCM.

Practice

• List factors of numbers by checking divisibility up to their square roots.
• Use divisibility tests to quickly identify small factors before using prime factorisation.

Definitions
HCF (highest common factor) of two or more numbers is the largest number that divides all of them exactly. LCM (least common multiple) is the smallest positive number that is a multiple of each of the numbers.

Methods to find HCF

• Listing method: list factors of each number and choose the greatest common one. This works well for smaller numbers.
• Prime factorisation method: write each number as a product of prime factors. The HCF is the product of common prime factors taken with the lowest power that appears in each factorisation.

Methods to find LCM

• Listing multiples: list multiples until you find the smallest common one (useful for small numbers).
• Prime factorisation: take each prime that appears in any factorisation with the highest power found among the numbers and multiply them to get the LCM.
• Relation: For two numbers a and b, HCF(a,b) × LCM(a,b) = a × b, which can be used to find one if the other is known.

Examples of use
HCF is used to reduce fractions to simplest form and to divide items into equal groups with no remainder. LCM is used when adding or comparing fractions with different denominators and for finding common repeating intervals, such as cycles aligning every certain number of days.

Practice tips

• Use prime factor trees to find prime factors clearly.
• Always check answers by using the relation HCF × LCM = product for two numbers.

What is a fraction?
A fraction expresses a part of a whole. It is written as a/b where the numerator a counts how many parts we have and the denominator b tells into how many equal parts the whole is divided. For example 3/4 represents three of four equal parts.

Types of fractions

• Proper fraction: numerator < denominator (e.g., 2/5).
• Improper fraction: numerator ≥ denominator (e.g., 7/4).
• Mixed number: a whole number and a proper fraction together (e.g., 1 3/4).

Visual models
Draw shapes like circles, rectangles or use fraction bars divided into equal parts to show fractions. Shading helps see how many parts make up the fraction. Visuals are useful to compare fractions and to understand equivalent fractions.

Equivalent fractions and simplification
Fractions that represent the same amount but have different numerators and denominators are equivalent. Multiply or divide numerator and denominator by the same non-zero number to find equivalents (for example 1/2 = 2/4 = 3/6). To simplify a fraction, divide numerator and denominator by their HCF to get the simplest form.

Converting between improper fractions and mixed numbers
To convert improper to mixed: divide numerator by denominator to get a whole part and a remainder; write remainder over the original denominator. To convert mixed to improper: multiply whole number by denominator, add numerator and place over denominator.

Practice

• Draw diagrams and shade parts to represent given fractions.
• Find equivalent fractions and simplify to lowest terms using HCF.

Adding and subtracting fractions with like denominators
If fractions have the same denominator, add or subtract their numerators and keep the denominator unchanged: a/b + c/b = (a + c)/b. After computation simplify the result by dividing numerator and denominator by their HCF if possible. For example 3/8 + 2/8 = 5/8.

Adding and subtracting with unlike denominators
When denominators differ, convert fractions to equivalent fractions having a common denominator before adding or subtracting. The LCM of denominators is the usual choice for the common denominator. Change each fraction by multiplying numerator and denominator by the factor needed to reach the LCM, then add or subtract numerators and simplify the result.

Multiplication of fractions
Multiply numerators to get the new numerator and denominators to get the new denominator: (a/b) × (c/d) = (a×c)/(b×d). To make calculations easier, cancel any common factors between numerator of one fraction and denominator of the other before multiplying.

Division of fractions
Dividing by a fraction uses the reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c). For dividing by a whole number treat it as that number over 1. Always simplify the result.

Mixed numbers
To operate on mixed numbers, either convert them to improper fractions and perform the operation, or add/subtract whole parts and fractional parts after making fractional parts have like denominators. For multiplication and division it is easier to convert to improper fractions first.

Practice tips

• Always simplify answers and convert improper fractions to mixed numbers for easier interpretation.
• Use diagrams for addition and subtraction with like denominators to visualise the result.

Decimals and their place value
Decimals represent numbers that are not whole using a decimal point. Places to the right of the decimal point are tenths (1/10), hundredths (1/100), thousandths (1/1000) and so on. For example 4.36 means 4 units + 3 tenths + 6 hundredths which equals 4 + 0.3 + 0.06.

Reading and writing decimals
Read 0.7 as 'zero point seven' or 'seven tenths'. Write decimals in expanded form to show value: 6.304 = 6 + 3/10 + 0/100 + 4/1000. When comparing decimals make equal digits after the point by adding zeros: 3.5 → 3.50 to compare with 3.45.

Addition and subtraction of decimals
Align numbers by the decimal point before adding or subtracting. If necessary add zeros to equalise the number of decimal places. Perform column-wise addition or subtraction as for whole numbers and place the decimal point directly below the other decimal points in the result.

Multiplication of decimals
Multiply as whole numbers ignoring the decimal points. After multiplying, count the total number of decimal places in the multiplicands and place that many digits from the right in the product as decimal places. For example 2.5 × 0.4 → 25 × 4 = 100, total decimal places 1 + 1 = 2 → 1.00 = 1.

Division of decimals
To divide by a decimal move the decimal point in the divisor to the right to make it a whole number and move the decimal point in the dividend the same number of places. Then divide as whole numbers and place the decimal point above accordingly in the quotient. Use rounding where necessary for decimal answers.

Conversion between fractions and decimals
Fractions with denominators that are powers of 10 convert easily to decimals: 75/100 = 0.75. To convert other fractions divide numerator by denominator. Practice conversions both ways to become fluent.