Chapter 2 — Whole Numbers

Whole numbers are numbers used for counting and measuring whole objects; they include 0 and the positive integers 1, 2, 3, ... with no fractions or decimals. Whole numbers answer questions like 'how many apples?' or 'how many students?' They are the simplest number type and the base of arithmetic. When you count from 0 on, you go through all whole numbers: 0, 1, 2, 3, 4, and so on.

Zero is special: it means none or nothing. It is an important whole number used as a placeholder in writing numbers (place value) and in arithmetic rules (for example, adding zero leaves a number unchanged). Whole numbers are used in everyday life: counting money (whole rupees), objects, steps, and items that cannot be split in the given context.

We can show whole numbers using objects (five stones = 5), tallies (|||| = 4), or numerals on paper. Understanding whole numbers helps you learn how to add, subtract, multiply and divide quantities. For example, putting two groups of 3 objects together gives total 6, which is simple addition. Repeated addition leads to multiplication, and sharing equal parts leads to division. Learning whole numbers also prepares students for negative numbers, fractions and decimals later, because the same ideas of operations and place value extend to other number types.

In class you will practise writing whole numbers, comparing sizes (which is bigger?), and using number sentences such as 7 + 3 = 10. You will use methods like counting on a number line, grouping objects, and simple algorithms for larger numbers. Good practice with whole numbers builds accuracy, speed and confidence in mathematics and daily tasks.

Place value tells us the value of a digit depending on its position in a number. For whole numbers we have units (ones), tens, hundreds, thousands, ten-thousands, lakhs etc. Each place value is ten times the value of the place to its right. For example, in 52,814 the digit 5 is in ten-thousands place and stands for 50,000; 2 stands for 2,000; 8 stands for 800; 1 stands for 10 and 4 stands for 4 units.

There are three common ways to write a number. The standard form uses digits as usual: 1,23,456. The word form writes the number in words: one lakh twenty-three thousand four hundred fifty-six. The expanded form breaks the number into the sum of each place value: 1,23,456 = 1,00,000 + 20,000 + 3,000 + 400 + 50 + 6. Expanded form helps understand the contribution of each digit to the whole number.

In the Indian system commas are placed after the first three digits from the right and then after every two digits: for example, 12,34,567. Practice reading and writing numbers using this system. Use a place value table to write the digits in separate columns (Lakh | Ten-thousand | Thousand | Hundred | Ten | Unit). This makes addition and subtraction easier because digits align under their place values.

While learning, practise converting between forms: given the word form write the numeral, given the numeral write the expanded form and words. This strengthens number sense. Also learn to compare digits by their place values: the digit in a higher place has more value than any digit in lower places, which helps in ordering and rounding numbers correctly. To develop fluency, do exercises that ask you to break numbers into parts, form numbers from parts and spot errors when digits are placed in the wrong column.

A number line is a straight line where whole numbers are placed at equal intervals. We usually begin the line at 0 and mark positive whole numbers to the right. The number line helps visualise position, distance and direction: moving right increases value (addition); moving left decreases value (subtraction). Number line also helps compare numbers: a number to the right is larger.

To compare two whole numbers, first look at the number of digits: the number with more digits is larger (for positive numbers). If digits count is equal, compare the leftmost digit; the larger leftmost digit makes the number larger. For example, to compare 7,852 and 7,629, examine thousands (both 7), then hundreds: 8 > 6 so 7,852 > 7,629. Use symbols: > (greater than), < (less than), = (equal to).

Ordering a group of numbers means arranging them from smallest to largest (ascending) or largest to smallest (descending). Use number of digits or place-value comparison, or mark them on number line to order visually. When numbers are close, lining up digits in a place value table helps avoid mistakes.

For practice, place numbers 0 to 20 on a line, then use jumps to show addition and subtraction. To compare larger numbers quickly, use place value: e.g., 54,321 > 5,432 because five-digit numbers are larger than four-digit numbers. Ordering and comparing are needed when solving word problems, planning events or arranging objects by size. Regular practice using both number line and place-value checks makes answers fast and accurate. Try exercises where you are given a mixed list of two-, three-, four- and five-digit numbers and must quickly arrange them in ascending or descending order; practice will improve speed and reduce errors in exams.

Estimation gives a quick approximate value that is close to the exact answer. Estimation is useful to check whether a calculated result is reasonable. One common method is rounding. To round a number to the nearest ten, look at the units digit: if it is 5 or more, increase the tens digit by one and write 0 in units; if it is 4 or less, keep the tens digit and make units 0. Similar rules apply for rounding to the nearest hundred, thousand etc.

Example: 4,376 rounded to nearest ten is 4,380 because units 6 > 5. Rounded to nearest hundred it is 4,400 because tens digit 7 > 5. Rounding can simplify mental calculations: to add 3,998 + 2,347, round 3,998 to 4,000 and 2,347 to 2,300 to get an estimated sum 6,300. After exact calculation we can see whether the result is close to this estimate.

Mental maths strategies include partitioning numbers (breaking a number into parts), using compatible numbers (numbers easy to work with), and using distributive property. For example, to calculate 25 × 16 mentally: 25 × (10 + 6) = 250 + 150 = 400. Or to add 399 + 247, do 400 + 247 = 647 then subtract 1 to get 646. Such techniques speed up calculations in exams and daily life.

Practice mental calculation of tables, small multiplications, and quick rounding checks. Use estimation at each step of long procedures to avoid big errors. Estimation does not replace exact calculation but helps detect slips and improves number sense. Learners should practice a mix of exact and approximate problems to become confident in deciding which method is suitable for time and accuracy required.

Addition combines two or more whole numbers to obtain their sum. For small numbers we can add using objects or mentally. For larger numbers we use the column (vertical) method. In the column method write the numbers one below the other with digits aligned by place value (units under units, tens under tens etc.). Start adding from the units column. If the sum of a column is 10 or more, write down the units digit and carry the tens digit to the next left column. Carrying must be tracked carefully for multi-digit sums.

It helps to know addition facts (tables of single-digit sums) so carrying becomes quick. Use place-value partitioning as a mental technique: break numbers into hundreds, tens and units and add corresponding parts. For example, 247 + 386 = (200+300) + (40+80) + (7+6) = 500 + 120 + 13 = 633. After summing parts, combine them again, taking care to add any carries generated when parts exceed place limits.

Another useful method is column addition with more than two numbers: add all units first, write the units digit of the result and carry the extra tens to the tens column. Continue column by column until all numbers are added. For very large lists of numbers, use grouping to make friendly pairs such as numbers that add to 10, 100 or other round values. Always align numbers correctly; misalignment is a common source of error in exams.

To check addition, use the inverse operation: subtract one addend from the sum and verify you get the other addend. Alternatively use estimation by rounding each addend to check the sum is in the right range. Practice word problems that combine addition with other operations and use clear steps: identify addends, choose method, compute, and verify. With practice, column addition becomes fast and reliable for sums of many whole numbers.

Subtraction finds the difference between two whole numbers or removes one quantity from another. In column subtraction, write the minuend (the larger number) on top and the subtrahend below, digits aligned by place value. Start subtracting from the units column. If the digit in the top row is smaller than the one below, borrow (or regroup) from the next left column. Borrowing reduces the left digit by one and adds ten to the current column.

Example: 7,103 − 2,478. Units: 3 − 8 cannot, so borrow from tens. But tens digit is 0 so we borrow from hundreds: hundreds 1 becomes 0, tens becomes 10, then borrow 1 from tens so units become 13 and tens become 9. Now units 13 − 8 = 5; tens 9 − 7 = 2; hundreds 0 − 4 cannot, borrow from thousands: thousands 7 becomes 6, hundreds become 10, so hundreds 10 − 4 = 6; thousands 6 − 2 = 4. Result: 4,625. Such multi-step borrowing needs careful tracking of each change.

Subtraction is checked by inverse addition: difference + subtrahend = minuend. Special cases: subtracting zero leaves the number unchanged; a number minus itself gives zero. Use number line for simple subtractions by moving left. Practice subtraction with and without borrowing to gain speed and accuracy. Learn tricks for subtracting from round numbers: subtracting from 10, 100 or 1000 can be done by complement methods for quick mental subtraction.

Work on word problems where subtraction appears as 'left', 'remaining', 'difference' and 'how many more'. Translate phrases into subtraction sentences, set up columns when numbers are large and check answers with reverse addition. With practice, students will become confident in handling borrow across zeros and multi-digit subtractions required in everyday calculations and examination problems.

Multiplication is repeated addition. When we write 4 × 3 it means 4 added three times (4 + 4 + 4). The first number is multiplicand and the second is multiplier. Multiplication tables (times tables) up to 10 or 12 help perform multiplication quickly. Key properties include commutative and associative laws which let us change order and grouping: a × b = b × a and (a × b) × c = a × (b × c). Knowing these properties helps in mental calculation and simplifies problem solving.

There are several methods to multiply: repeated addition for small numbers; grid or box method for two-digit by two-digit; and long (column) multiplication for larger numbers. In long multiplication multiply the multiplicand by each digit of the multiplier, starting from units. Write each partial product shifting one place to the left for each next digit of the multiplier, then add the partial products. Keep track of carries within each partial multiplication step.

Example: multiply 234 by 12. Multiply 234 by 2 (units) to get 468. Then multiply 234 by 1 (tens) meaning 234 × 10 = 2,340. Add partial products: 468 + 2,340 = 2,808. Use distributive property for mental ease: 234 × 12 = 234 × (10 + 2) = 2,340 + 468. For two-digit by two-digit, grid method splits numbers into tens and units and multiplies the parts, then sums the four partial products.

Special rules: any number multiplied by 0 gives 0; multiplied by 1 gives the number itself. Use estimation to check results — round multiplicand and multiplier to easy numbers and see if product is close. Multiplication is used in finding area (later), calculating total cost of items, and problems with repeated equal groups. Regular practice of tables and multiplication methods increases speed and accuracy for exams and real life.

Division splits a whole number into equal parts or finds how many groups of a given size fit into a number. In the expression 20 ÷ 5, 20 is the dividend, 5 is the divisor and the result 4 is the quotient. If the number cannot be divided exactly, the leftover is the remainder. The relation is: Dividend = Divisor × Quotient + Remainder, where remainder < divisor. Remember division by zero is not allowed.

For small integers, division can be done by repeated subtraction or using multiplication facts: find a number which multiplied by divisor gives the dividend. For larger numbers use the long division method. In long division start from the highest place of the dividend. Decide how many times the divisor fits into that part, write that digit in the quotient, subtract the product, bring down the next digit and repeat until all digits are processed. If, after bringing down all digits, there is still a number smaller than divisor, it is the remainder.

Example: 1,536 ÷ 8. Take 15 (first two digits) because 1 < 8: 15 ÷ 8 = 1 (write 1 in quotient), remainder 7 (15 − 8 = 7). Bring down 3 to get 73. 73 ÷ 8 = 9, remainder 1 (9×8=72). Bring down 6 to get 16. 16 ÷ 8 = 2 remainder 0. Quotient = 192. Use the relation to check: 8×192 + 0 = 1,536. If a remainder exists and we need a more precise answer, express the remainder as a fraction (remainder/divisor) or convert to decimals (later topics).

Division problems appear as sharing equally or grouping items. Word problems may ask how many complete groups can be formed and how many are left over. Practice dividing by one- and two-digit divisors and interpreting remainders. Use estimation before dividing to check that the quotient is in a reasonable range. Long division skills are valuable for larger calculations and for checking arithmetic in examinations.

Understanding properties of operations helps simplify calculations and reason correctly. The main properties for addition and multiplication are commutative and associative. The commutative property means the order does not matter: a + b = b + a and a × b = b × a. This means 7 + 3 = 3 + 7 and 4 × 6 = 6 × 4. The associative property says grouping does not matter: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c). This helps when adding many numbers or multiplying many factors.

The distributive property links multiplication and addition: a × (b + c) = a×b + a×c. Use distributive property to break difficult multiplications into easier parts, for example 23 × 7 = (20 + 3) × 7 = 140 + 21 = 161. The identity elements include 0 for addition (a + 0 = a) and 1 for multiplication (a × 1 = a). Knowing identities helps when simplifying expressions.

Some operations do not have these properties: subtraction and division are not commutative or associative in general. For example, 8 − 3 ≠ 3 − 8 and (8 − 3) − 2 ≠ 8 − (3 − 2). Use properties to rearrange sums and products to perform mental maths and checks. In problem solving, rearrange terms to form friendly numbers, use distributive law for grouping, and always verify using inverse operations to avoid mistakes.

Factors of a number are integers that divide it exactly with no remainder. For example factors of 12 are 1, 2, 3, 4, 6, 12. To find factors, test divisibility by integers up to the square root and pair them. Multiples of a number are the results of multiplying that number by whole numbers: multiples of 4 are 4, 8, 12, 16, ... Every number is a multiple of itself and of 1.

Prime numbers are numbers greater than 1 that have exactly two positive divisors: 1 and itself. Examples: 2, 3, 5, 7, 11. Note 2 is the only even prime. Composite numbers have more than two factors, for example 4, 6, 8, 9. The number 1 is neither prime nor composite because it has only one divisor.

Prime factorization writes a number as a product of prime factors, for example 84 = 2 × 2 × 3 × 7 = 2^2 × 3 × 7. Prime factorization helps find the Highest Common Factor (HCF) and Least Common Multiple (LCM). HCF is the greatest number that divides two or more numbers; LCM is the smallest positive number that is a multiple of all numbers. Use prime factorization or listing methods to find HCF and LCM. For two positive integers a and b, HCF(a,b) × LCM(a,b) = a × b when using their full prime factor forms. Practice with factor trees and Venn diagrams to visualise common and uncommon prime factors.

Divisibility tests are quick rules to decide whether a number can be divided by another without performing full division. They help find factors and simplify calculations. Common tests are easy to remember and apply to many examination problems.

Test for 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8). Test for 3: A number is divisible by 3 if the sum of its digits is divisible by 3. Test for 5: Last digit 0 or 5 means divisible by 5. Test for 9: If the sum of digits is divisible by 9 then the number is divisible by 9. Test for 10: Last digit 0 means divisible by 10. There are also tests for 4 (last two digits divisible by 4), for 8 (last three digits divisible by 8) and for 6 (number divisible by both 2 and 3).

Examples: To test 5,462 for divisibility by 3, add digits 5+4+6+2 = 17; 17 is not divisible by 3, so 5,462 is not divisible by 3. To test 1,350 for 5, last digit 0 so divisible by 5. Use these tests in factor finding and when simplifying calculations or checking answers. When using tests based on digit sums, you can repeat the process: e.g., for 18,234 sum digits 1+8+2+3+4 = 18 and since 18 is divisible by 9, the original number is divisible by 9. Practise applying rules quickly to improve speed in exams.

This topic shows how whole-number concepts apply to everyday situations and how to solve word problems step by step. Word problems test both reading and arithmetic skills. They commonly involve finding totals (use addition), differences (use subtraction), equal sharing or grouping (use division), and repeated equal groups (use multiplication). Some problems need HCF or LCM, for arranging objects in rows or finding common meeting times.

Follow a clear method when solving: (1) Read the problem carefully at least twice. (2) Identify the data (numbers given) and variables (what is asked). (3) Decide which operations to use and write a number sentence or equation. (4) Carry out calculations step by step and show working. (5) Check the final answer using inverse operations or estimation to ensure it is reasonable. Draw a diagram, make a table or use a bar model when the situation is complex—this helps visualise quantities and relationships.

Example problem types: sharing items into equal boxes, total cost of several items, how many rows of chairs for students given per row count, and combining groups with different sizes. Use HCF to arrange items into the largest equal groups without leftover; use LCM to find when repeating cycles coincide. For multi-step problems, solve each step in sequence and label intermediate results. Write the final answer with correct unit (students, boxes, rupees) and, if required, give remainder interpretation (e.g., leftover sweets). Practise many problems that mix two or more operations to build confidence in translating words into correct calculations and to improve speed in exams.