Chapter 2 — Whole Numbers

What are whole numbers? Whole numbers are the set of numbers used for counting and measuring without fractions or negatives. They include 0 and all positive integers: 0, 1, 2, 3, 4, ... . We write this set as W = {0, 1, 2, 3, ...}.

Difference from natural numbers: Natural numbers usually start from 1 (1, 2, 3, ...). Whole numbers include 0 as well. So every natural number is a whole number, but 0 is whole and not a natural number if natural numbers exclude 0.

Basic ideas and operations:

• Successor: the next whole number after n is n + 1. (S(n) = n + 1)
• Predecessor: the previous whole number before n is n - 1, defined only when n > 0.
• Addition and multiplication of whole numbers always give a whole number (closure property).
• Subtraction and division do not always give whole numbers. For subtraction, the result is a whole number only when the minuend is greater than or equal to the subtrahend (for example, 7 - 3 = 4 is whole, but 3 - 7 is not a whole number).

Properties:

• Commutative laws: a + b = b + a, a × b = b × a
• Associative laws: (a + b) + c = a + (b + c), (a × b) × c = a × (b × c)
• Distributive law: a × (b + c) = (a × b) + (a × c)
• Additive identity: a + 0 = a
• Multiplicative identity: a × 1 = a

Even and odd: A whole number is even if it is twice another whole number: n = 2k. It is odd if it is one more than an even number: n = 2k + 1.

Ordering and number line: Whole numbers are ordered on a number line from left to right: 0, 1, 2, 3, ... . A number to the right is greater than a number to the left. Number line models help with visualising successor, predecessor, addition (moving right), and subtraction (moving left).

Use in daily life: Whole numbers are used to count objects, number floors of a building (if ground floor is 0), record scores in games, count people, items, etc. They do not show fractional parts (for that we use fractions or decimals).

Quick tips for students:

• To check if subtraction gives a whole number, compare the two numbers first — subtract only if the first is larger or equal.
• Use a number line for small additions or subtractions; use place value and column addition for larger numbers.

What is a number line? A number line is a straight horizontal line used to represent whole numbers visually. It has equally spaced marks (ticks). The mark for 0 is called the origin. To the right of 0 the numbers increase (1, 2, 3, ...). In Class 6 (Whole Numbers) we work with 0 and positive integers only.

How to draw a number line: draw a horizontal line, mark a point as 0, then make equal spaced ticks to the right and label them 1, 2, 3, ... Place an arrow at the right end to show the line continues.

Key uses and properties:

• Every whole number n is located n units to the right of 0.
• All marks are equally spaced — each gap represents one unit.
• A number to the right of another is greater. Thus, if a is to right of b then a > b.
• Successor and predecessor are easy to see: successor of n is the point one step to the right; predecessor is one step to the left (predecessor exists only for n > 0).

Representing operations:

• Addition: show as hops to the right. To compute a + b, start at a and make b hops right — you land at a + b.
• Subtraction: show as hops to the left. To compute a − b (when a >= b), start at a and make b hops left — you land at a − b.
• Distance between two whole numbers a and b is the number of unit hops between them, equal to |a − b|.

Why it helps: Number lines give a clear visual for comparing numbers, understanding addition/subtraction as movement, and for concepts like successor, predecessor and distance.

What is Face Value? The face value of a digit in a number is the digit itself, irrespective of its position. Example: In 5,482 the face value of 4 is 4.

What is Place Value? The place value of a digit is the value contributed by the digit because of its position in the number. It is found by multiplying the digit by the value of its place (1, 10, 100, 1,000, ...). Example: In 5,482 the place value of 4 (which is in the tens place) is 4 × 10 = 40.

How to find place value (step-by-step): (1) Write the number and identify positions from right to left as units (10^0), tens (10^1), hundreds (10^2), thousands (10^3), and so on. (2) Note the digit at the required position. (3) Multiply the digit by the place value (1, 10, 100, ...). That product is the place value.

Difference in one line: Face value = the digit itself; Place value = digit × value of its position.

Special notes: (a) Face value of 0 is 0. (b) Place value of 0 is also 0, but zeros act as placeholders (e.g., 5,020 keeps the tens place as 2 and hundreds place as 0). (c) Any number can be written in expanded form by writing the sum of the place values of its digits.

What are figures and words? A number written with digits (0–9) is called a number in figures (e.g., 2,15,012). The same number written in English words is called a number in words (e.g., two lakh fifteen thousand twelve).

Place value and the Indian system of commas — In the Indian system, digits are grouped from the right: the first comma comes after three digits (units, tens, hundreds) and then after every two digits (thousands, lakhs, crores). Example: 1,23,45,678 = 1 crore 23 lakh 45 thousand 678. Use a place-value chart to read and write numbers correctly.

Rules for writing numbers in words:

• Separate the number into groups according to commas: crores, lakhs, thousands, hundreds, and the last two digits (tens and units).
• Write the word for each non-zero group followed by its place (e.g., "lakh", "thousand").
• For the last two digits (1–99), use normal English words (e.g., 45 = "forty-five").
• When a group is zero, skip its name (e.g., 4,00,005 = "four lakh five").

Rules for writing numbers in figures from words:

• Identify crores/lakhs/thousands/hundreds and write the numeric value for each group in its correct place-value block.
• Use commas as per the Indian system to separate groups (3,2,2... from the right).

Expanded form — Any number can be written as a sum of each digit times its place value. Example: 5,12,307 = 5 × 1,00,000 + 1 × 10,000 + 2 × 1,000 + 3 × 100 + 0 × 10 + 7 × 1.

These ideas help in reading large numbers correctly, writing monetary amounts, recording population, distances, and many day-to-day uses.

Definition: The successor of a whole number is the number that comes immediately after it when counting. The predecessor of a whole number is the number that comes immediately before it when counting.

In formula form: For any whole number n, Successor = n + 1 and Predecessor = n - 1 (provided n > 0 for predecessor to remain a whole number).

Explanation and method:

• To find the successor: add 1 to the number. Example: successor of 57 is 57 + 1 = 58.
• To find the predecessor: subtract 1 from the number. Example: predecessor of 57 is 57 - 1 = 56.
• Digit rules: If the last digit is not 9, successor is found by increasing only the last digit by 1 (e.g., 243 → 244). If the last digit is 9, adding 1 causes a carry: e.g., 1999 → 2000. Similarly, if the last digit is not 0, predecessor reduces only the last digit by 1 (e.g., 430 → 429). If the last digit is 0, subtracting 1 causes borrowing: e.g., 2000 → 1999.
• Whole-number edge cases: Whole numbers are {0, 1, 2, 3, ...}. Every whole number has a successor. Every whole number except 0 has a predecessor that is also a whole number. The predecessor of 0 is -1, which is not a whole number (it is an integer).

Why this matters: Successor and predecessor are the simplest operations of counting and are used in ordering, sequencing, and understanding place value (how digits change when we add or subtract 1).

What are whole numbers? Whole numbers are non-negative integers: 0, 1, 2, 3, ... . They do not include fractions or negative numbers.

Comparing whole numbers — idea: To compare two whole numbers means to decide which one is greater, which is smaller, or whether they are equal. We use the symbols > (greater than), < (less than) and = (equal to).

Easy rules to compare:

1. If the numbers have different number of digits, the number with more digits is greater. (E.g., 4,321 > 987 because 4,321 has 4 digits and 987 has 3.)
2. If they have the same number of digits, compare digits from the left (highest place value) to the right. The first place where digits differ decides the larger number. (E.g., compare 5,624 and 5,492: at hundreds place 6 > 4 so 5,624 > 5,492.)
3. Use subtraction as a check: if a − b > 0 then a > b; if a − b < 0 then a < b; if a − b = 0 then they are equal.

Ordering whole numbers: Arrange a list of whole numbers in ascending (smallest to largest) or descending (largest to smallest) order. Practical methods:

• Use number of digits first, then left-to-right digit comparison for ties.
• Place the numbers on a number line and read left-to-right for ascending order.
• Use sorting (pairwise compare) or subtraction checks to place each number.

Tips and special points:

• Leading zeros do not change value: 007 = 7.
• Zero is the smallest whole number; no negative whole numbers exist.
• When many numbers are given, grouping by number of digits first speeds up ordering.

Short worked process (example): To decide which is larger: 8,450 or 8,405 — both 4 digits and same thousands and hundreds digits, compare tens place: 5 > 0 so 8,450 > 8,405. You can subtract: 8450 − 8405 = 45 > 0 so 8450 > 8405.

Definition: Whole numbers are classified as even or odd. An even number is a number that can be divided exactly by 2 (no remainder). An odd number is not divisible exactly by 2 (remainder 1).

General forms: Every even whole number can be written as 2n and every odd whole number as 2n+1, where n is a whole number (0, 1, 2, ...).

First few examples: Even numbers: 0, 2, 4, 6, 8, 10...; Odd numbers: 1, 3, 5, 7, 9, 11.... Note: 0 is an even whole number because 0 = 2 × 0.

How to check: Look at the last digit of the number. If it is 0, 2, 4, 6 or 8 the number is even. If it is 1, 3, 5, 7 or 9 the number is odd. Alternatively, use division by 2: if remainder = 0 it is even; remainder = 1 it is odd.

Rules for operations (useful facts):

• Even ± Even = Even
• Odd ± Odd = Even
• Even ± Odd = Odd
• Even × Any whole number = Even
• Odd × Odd = Odd

These rules follow from the general forms 2n and 2n+1. For example, (2a) + (2b) = 2(a+b) which is even; (2a+1)+(2b+1)=2(a+b+1) which is even.

Definition: Whole numbers are the set {0, 1, 2, 3, ...}. Operations on whole numbers are the basic arithmetic actions you can perform: addition, subtraction, multiplication and division. Understanding these operations, their properties and how they relate helps solve numerical and everyday problems.

1. Addition

Addition combines two or more whole numbers to get a sum. Example: 7 + 5 = 12. Key ideas:

• Commutative law: a + b = b + a
• Associative law: (a + b) + c = a + (b + c)
• Additive identity: a + 0 = a
• Closure: sum of whole numbers is a whole number

2. Subtraction

Subtraction finds the difference between two whole numbers: a - b. If a < b, subtracting b from a would give a negative number, which is not a whole number. Key ideas:

• Subtraction is not commutative: a - b ≠ b - a in general
• Subtraction is the inverse of addition
• When subtracting using place value, we use borrowing (regrouping)

3. Multiplication

Multiplication is repeated addition. Example: 4 × 3 = 4 + 4 + 4 = 12. Key ideas:

• Commutative law: a × b = b × a
• Associative law: (a × b) × c = a × (b × c)
• Multiplicative identity: a × 1 = a
• Closure: product of whole numbers is a whole number

4. Division

Division splits a number into equal parts. For whole numbers, dividing a by b (b > 0) gives a quotient q and remainder r such that:

a = b × q + r, where 0 ≤ r < b

Division is not always exact in whole numbers; if the remainder is nonzero, you cannot express the result as a whole number alone. Division by zero is undefined.

5. Relation between operations and order of operations

Use the order: Brackets, Orders (powers), Division and Multiplication (left to right), Addition and Subtraction (left to right). (Often remembered as BODMAS). For whole numbers we generally use brackets, multiplication/division before addition/subtraction.

6. Practical points

• Multiplication as an array or area helps visualize factors and products.
• Subtraction on a number line shows movement left; addition moves right.
• Long division uses successive subtraction/estimation to find quotient and remainder.

Summary: Whole-number arithmetic follows familiar laws (commutative and associative for + and ×, distributive property of × over +). Addition and multiplication are closed in whole numbers; subtraction and division may lead outside whole numbers if results are negative or fractional. Always remember division by zero is not allowed.

Whole numbers are the numbers 0, 1, 2, 3, ... (non‑negative integers). They have several important algebraic and order properties used in arithmetic and problem solving.

• Closure: The sum or product of any two whole numbers is a whole number. (Addition and multiplication keep you inside the set.)
• Commutative Property: Order does not change the result: a + b = b + a and a × b = b × a.
• Associative Property: Grouping does not change the result: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
• Identity Elements: 0 is the additive identity (a + 0 = a). 1 is the multiplicative identity (a × 1 = a).
• Distributive Property: Multiplication distributes over addition: a × (b + c) = a × b + a × c.
• Zero Property of Multiplication: Any whole number multiplied by 0 gives 0: a × 0 = 0.
• Cancellation Law: If a + c = b + c then a = b. If a × c = b × c and c ≠ 0 then a = b.
• No general inverses: Whole numbers do not always have additive inverses (negatives) or multiplicative inverses (reciprocals). So subtraction and division may not yield whole numbers.
• Order and Well‑ordering: For any two whole numbers exactly one of a < b, a = b, or a > b holds. Every nonempty set of whole numbers has a least (smallest) element.
• Successor and No Largest Element: Every whole number n has a successor n + 1. There is no greatest whole number (the set is infinite).

These properties explain why arithmetic with whole numbers behaves predictably and why some operations (like subtraction or division) may lead outside the set.

In the set of whole numbers, 0 and 1 have special and fundamental roles. Zero (0) is the additive identity: adding 0 to any whole number leaves it unchanged. Zero also acts as a placeholder in the place-value system (for example, 205 = 2×100 + 0×10 + 5), and multiplying any number by 0 gives 0. Note that division by 0 is undefined, while 0 divided by a nonzero number equals 0. Zero is neither positive nor negative.

One (1) is the multiplicative identity: multiplying any whole number by 1 leaves it unchanged. Because of this, 1 is called the unit. In the classification of integers, 1 is neither prime nor composite. Together, 0 and 1 determine many fundamental arithmetic rules used in calculations and place-value representation.

• Key properties of 0: additive identity (a + 0 = a), placeholder in place-value, annihilator under multiplication (a × 0 = 0), division by 0 is not allowed.
• Key properties of 1: multiplicative identity (a × 1 = a), unit of counting, 1 is neither prime nor composite.

What is estimation? Estimation is finding an approximate value that is close enough to the exact answer for a given purpose. It gives a quick idea of size, cost, distance or time when exact values are not necessary.

What is rounding? Rounding is a method of replacing a number with another number that has fewer non‑zero digits but is close to the original number. We usually round to a specified place value such as nearest 10, 100, or 1000.

Why use estimation and rounding? They make mental calculation faster, help check answers for reasonableness, and simplify numbers in real life (shopping totals, travel distances, population figures, etc.).

Rounding rules (simple):

• Look at the digit immediately to the right of the place you are rounding to (the "next" digit).
• If that digit is 0,1,2,3 or 4 → keep the rounding digit the same and change all digits to its right to 0 (round down).
• If that digit is 5,6,7,8 or 9 → increase the rounding digit by 1 and change all digits to its right to 0 (round up).

Steps to round a whole number to nearest 10, 100 or 1000

1. Identify the place value you want to round to (10, 100, 1000 ...).
2. Look at the digit to the right of that place.
3. Apply the rounding rule (0–4 keep, 5–9 increase by 1).
4. Replace every digit to the right of the chosen place by 0.

Example (conceptual): Round 3,476 to nearest 100. The hundreds digit is 4 (in 400), the next digit (tens) is 7 → 7 ≥ 5 so increase hundreds digit to 5 → becomes 3,500.

Estimation techniques:

• Rounding method: Round each number to a convenient place and then perform the operation (useful for addition, subtraction, multiplication and division).
• Front‑end (leading digits) method: Use the most significant digits for a quick estimate (e.g., use first 2 digits of large numbers).
• Compatible numbers: Replace numbers with nearby values that are easy to compute mentally (e.g., replace 497 by 500 when multiplying by 4).

How to check an estimate: The rounded number is within half the rounding base of the original number. For example, rounding to nearest 100 changes the number by at most 50.

Use visuals such as number lines, place‑value charts and bar comparisons to build intuition about how close the rounded value is to the original.

What are word problems? Word problems are mathematical questions expressed in everyday language. In the topic of Whole Numbers (Class 6), they require translating a real-life situation into numbers and operations (addition, subtraction, multiplication, division) and then solving.

How to approach a word problem (step-by-step)

• Read the problem carefully at least twice to understand the situation.
• Identify what is asked (the unknown) and the data given (numbers, units).
• Decide which operation(s) to use. Look for keywords: sum/altogether/total → add; left/remaining/less → subtract; each/altogether/times → multiply; equally/share/every → divide.
• Translate the words into a mathematical expression or sequence of expressions.
• Solve step by step, keeping units consistent. For multi-step problems, work in the correct order and write intermediate answers.
• Check the result: substitute into the story to see if it makes sense and roughly estimate to verify magnitude.

Common types of word problems with whole numbers

• Single-operation problems (one addition/subtraction/multiplication/division).
• Multi-step problems (a sequence of operations).
• Grouping and repeated addition (solved by multiplication).
• Sharing and equal distribution (solved by division; may have remainder).
• Problems using simple formulas like average or formation of totals from parts.

Tips

• Always note units (kg, rupees, students, books). Convert to the same unit if needed before computing.
• Estimate before and after solving to catch obvious mistakes.
• Write the answer with appropriate units and, when needed, state remainders or interpret them in context.