Chapter 3 — Integers

Integers are whole numbers that include positive numbers, negative numbers and zero. They do not include fractions or decimals. We write them as ... , -4, -3, -2, -1, 0, 1, 2, 3, 4, ... showing that integers go on forever in both directions. Integers help us describe situations with two opposite directions: for example, gaining or losing money, walking forwards or backwards, floors above or below the ground, and temperatures above or below zero.

In everyday writing a positive integer is usually written without a + sign, for example 5, but we may write +5 to stress its positivity. Negative integers always have a minus sign, as in -5. Zero is special — it is neither positive nor negative and it separates positive and negative numbers on the number line. Every nonzero integer has an opposite (or additive inverse): the opposite of 7 is -7 and the opposite of -7 is 7. When a number and its opposite are added, they cancel to give zero.

We use concrete examples to understand integers. If the temperature is -2°C, this means 2 degrees below zero. If a wallet shows +300, you have 300 rupees; if it shows -300, you owe 300 rupees. Using integers allows us to record both directions of change in one single system. When we work with integers we must pay attention to signs because they tell us direction. This basic idea — what integers are and why signs matter — prepares learners to work with addition, subtraction, multiplication and division of integers using clear sign rules. Practise naming integers, writing a few on paper and pointing them out on a number line to become comfortable with the set of integers.

A number line is the best visual tool to understand integers. Draw a straight horizontal line, mark a point in the middle as 0. To the right of 0 mark 1, 2, 3,… at equal spaces; to the left mark -1, -2, -3,… at the same equal spaces. Put arrows at both ends of the line to show that numbers continue forever. This picture shows direction: right is positive and left is negative. The number line makes ordering, distance and arithmetic actions easy to see.

Using the number line to compare numbers is simple: the number to the right is always greater. So 2 is to the right of -1 and therefore 2 > -1. The distance between two integers on the number line is the number of unit steps between them; this is useful for absolute value and for solving real problems like the difference in temperature between two days. For example the distance between -3 and 4 is 7 units.

We can also use the number line to perform addition and subtraction. To add a positive integer, start at the first number and move right by that many steps; to add a negative integer move left. For subtraction, convert subtraction into adding the opposite and then use the same moves. For example, to compute 1 + (-4) start at 1 and move 4 steps left to land at -3. To compute 5 - 8 start at 5 and move 8 steps left to reach -3. Number line movements help students see why the sign rules for integers work.

Practice drawing number lines and using them for simple sums and differences. Use real-life situations: if sea level is at 0, a hill at +50 m and a valley at -30 m are easy to represent. This builds strong intuition about integers before doing many written calculations.

Integers are divided into three classes: positive integers (+1, +2, +3, ...), negative integers (-1, -2, -3, ...) and zero (0). A positive integer is any number greater than zero and is usually written without the plus sign in everyday use. A negative integer is any number less than zero and is always written with a minus sign. Zero is neither positive nor negative and acts as the centre of the number line. Knowing these differences helps when comparing numbers and performing arithmetic.

Every nonzero integer has an opposite, also called the additive inverse. The opposite of a number a is written as -a; for example the opposite of +8 is -8 and the opposite of -8 is +8. On the number line opposites are equally distant from zero but on opposite sides, so they have the same absolute value. If you add a number and its opposite you always get zero: a + (-a) = 0. This is an important idea used to simplify expressions and to solve simple equations like x + 5 = 0, where x = -5.

Zero is special in more ways: adding zero to any integer leaves it unchanged (a + 0 = a). Zero is also a boundary: all positive integers lie to the right, all negative integers lie to the left. When reasoning with word problems, translate words into signs: ‘gain’, ‘above’, ‘deposit’ often mean positive; ‘loss’, ‘below’, ‘withdrawal’, ‘owe’ often mean negative. By practicing many examples — such as opposite temperatures, bank balances, and floors of a building — students strengthen their ability to choose signs correctly and to find opposites quickly.

The absolute value of an integer measures how far it is from zero on the number line, ignoring the sign. We write the absolute value of a as |a|. For any integer a, |a| is always non-negative. For example |5| = 5 and |-5| = 5. Absolute value is useful when we want the size or magnitude of a quantity without caring about its direction.

To find the absolute value: if the integer is positive or zero, |a| = a; if the integer is negative, |a| = -a (because -a is positive). Thus |0| = 0, |7| = 7, |-7| = 7. Absolute value helps calculate distance between two integers: distance between a and b equals |a - b|. For example distance between -3 and 4 is | -3 - 4 | = | -7 | = 7 steps on the number line.

Use absolute value in practical contexts. If a mountain peak is +300 m and a valley is -200 m relative to sea level, the vertical difference is |300 - (-200)| = |500| = 500 m. In banking, if a balance is -150, the absolute value | -150 | = 150 is the amount owed. Teach students to use absolute value when asked for 'how far' or 'how much' without direction. Also practise simple properties: |a| = |-a| and |a| ≥ 0 for all integers a. Absolute values can simplify comparisons when direction is not important, and they will be useful later in algebra and geometry for distance calculations.

Comparing integers means deciding which is larger or smaller. Use the number line idea: a number to the right is greater than a number to the left. So all positive integers are greater than zero and all negative integers are less than zero. For two positive integers compare their usual values; for two negative integers the one with smaller absolute value is greater because it is closer to zero. For example -2 > -7 because -2 is to the right of -7 on the number line.

To order a list of integers from smallest to largest, place them according to their positions on the number line. When signs differ, any positive number is greater than any negative number. Zero lies between negatives and positives. Practise ordering mixed lists that include zero and both signs. Use simple rules: if sign differ, choose the positive; if both negative, the larger absolute value means the number is smaller (more negative).

Teach strategies: convert words into comparison symbols (<, >, =) and explain reasoning. For example to compare -3 and 2, note 2 is right of -3 so 2 > -3. To compare -5 and -2, note -2 is closer to zero so -2 > -5. Also practise placing numbers on a blank number line as a check. Ordering helps find minimum and maximum values, solve simple inequalities, and answer questions like which day was coldest or which bank balance was smallest. Frequent short exercises increase speed and confidence in working with mixed integers.

Adding integers depends on their signs. There are two main cases. Case 1: If both numbers have the same sign (both positive or both negative), add their absolute values and keep the common sign. For example (+6) + (+3) = +9 and (-6) + (-3) = -9. Case 2: If the numbers have different signs, subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value. For example (+7) + (-4) = +3 because 7 − 4 = 3 and the positive had larger absolute value. Similarly (-7) + (+4) = -3.

It helps to think in terms of gains and losses: + means gain, − means loss. If a gain and a loss occur, the net change is the difference. Using a number line: to add a positive number move to the right; to add a negative number move to the left. This makes addition of integers concrete and reduces mistakes with signs. Also use cancellation idea: a + (−a) = 0. When adding several integers, group positives together and negatives together first to simplify, then combine the two results. This reduces errors and is especially useful when dealing with long sums.

Practice many short examples and apply addition in word problems. Encourage checks: after getting a result, see if reversing an operation leads back to the original number (using inverse operations). Also illustrate with counters: use red counters for negative and green for positive, pair them to cancel and count what remains. Mastering these addition rules is essential for later algebra and for working with mixed-sign expressions in higher classes.

Subtraction with integers is easiest when we convert it into addition of the opposite. The rule is a − b = a + (−b). This means subtracting a positive number is the same as adding its negative, and subtracting a negative number is the same as adding its positive. For example 7 − 10 = 7 + (−10) = −3, and −3 − (−6) = −3 + 6 = 3. This single method removes the need for many special-case rules and keeps thinking consistent.

On the number line, subtraction means moving opposite to the number you remove. If you subtract a positive, move left; if you subtract a negative, move right because subtracting a negative equals adding a positive. For students, draw the number line and show examples step by step; this visual method clarifies sign changes. When subtracting multiple numbers, convert each subtraction into addition of opposite and then apply addition rules.

Practice with zero and opposites: a − 0 = a, 0 − a = −a. Also show that subtraction is not commutative: a − b ≠ b − a in general. Teach short checks: after computing a − b = c, check by c + b = a. Use real-life examples like temperature change: if temperature was +2°C and it falls by 6°C, final = 2 − 6 = −4°C. Converting problems into addition of opposites helps students avoid confusion with signs and solve subtraction problems reliably.

Multiplication of integers extends the idea of repeated addition and follows a simple sign rule. If two integers have the same sign (both positive or both negative) their product is positive. If they have different signs (one positive and one negative) the product is negative. For example, (+4) × (+3) = +12, (−4) × (−3) = +12, and (+4) × (−3) = −12. This rule comes from the fact that multiplying by −1 reverses direction: (−1) × a = −a, and (−1) × (−1) = +1.

Use repeated addition to understand multiplication with positive factors: 3 × 5 = 5 + 5 + 5 = 15. For negative factors, think of direction or use the rule with (−1). Multiplication by zero gives zero: a × 0 = 0 for any integer a. Teach that multiplication is commutative and associative for integers: a × b = b × a and (a × b) × c = a × (b × c). This helps to rearrange factors to make calculation easier; for instance combine two numbers to make 10 first, then multiply.

For classroom practice use multiplication tables for absolute values and then apply the sign rule. Also show using area model or grids to visualise small cases. Explain real-life examples: if you lose 3 rupees every day for 4 days the total change is (−3) × 4 = −12. Encourage checking by dividing product by one factor to recover the other, keeping careful watch on sign. Mastery of multiplication rules prepares students for division and algebraic manipulations later on.

Division of integers is the inverse process of multiplication and follows the same sign rules we use for multiplication. If the dividend and the divisor have the same sign (both positive or both negative) the quotient is positive. If they have different signs (one positive, one negative) the quotient is negative. For example, (+24) ÷ (+6) = +4, (−24) ÷ (−6) = +4, and (+24) ÷ (−6) = −4. Always remember the important restriction: division by zero is not defined; you cannot divide any number by 0.

Teach division as either sharing into equal groups or as the inverse of multiplication. If 18 apples are shared equally among 3 children, each child gets 6; similarly if -18 rupees debt is shared over 3 days equally, each day is -6 rupees. Use the multiplication check: if a ÷ b = c then c × b should return a. This check is useful for verifying answers and for fixing sign errors. Practise small exact divisions first and learn to confirm results by multiplication.

Discuss quotients with remainders only briefly at this stage: if a does not divide evenly by b, write quotient and remainder (for example 20 ÷ 6 = 3 remainder 2), but focus mainly on sign understanding. When negative numbers are involved, be careful with remainders—class 6 work keeps to simple examples where division is exact or remainder is handled without sign complications. Also show why division by zero is undefined: there is no number c such that c × 0 = a when a ≠ 0, so a ÷ 0 cannot be defined.

Give practice examples in real contexts such as equal sharing of costs, splitting debts across days, speeds and distances (when both may be negative directions), and checks using multiplication. Reinforce the sign rule with many pairs of examples and remind students to always check whether the divisor is zero before attempting division.

Many familiar properties of arithmetic remain true when we work with integers. These properties make calculation easier and help rearrange expressions. Key properties include:

• Commutative property: For addition and multiplication the order of numbers does not matter: a + b = b + a and a × b = b × a. This helps to group numbers conveniently.
• Associative property: How we group numbers in addition or multiplication does not change the result: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
• Distributive property: Multiplication distributes over addition: a × (b + c) = a × b + a × c. This is especially useful when expanding brackets or simplifying expressions with integers of different signs.

Also note identities: zero is the additive identity since a + 0 = a; one is the multiplicative identity since a × 1 = a. Multiplication by zero always gives zero: a × 0 = 0. Subtraction and division are not commutative: a − b ≠ b − a generally, and a ÷ b ≠ b ÷ a. When using these properties with negatives, apply correct sign rules at each step. For example, using distributive law with negative numbers: 2 × (−3 + 4) = 2×(1) = 2, and separately 2×(−3) + 2×4 = −6 + 8 = 2.

Teach students to use these properties to simplify calculations: reorder terms to add numbers making tens or zeros first, factor common numbers, and check results by reverse operations. Understanding and practising these properties will make integer arithmetic faster and more reliable.

When an expression contains more than one operation (addition, subtraction, multiplication, division, and brackets), follow the order of operations to get the correct answer. The commonly used rule is BODMAS or BIDMAS: Brackets first, Orders (powers; not used in depth in Class 6), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). This order applies to integers as well and helps avoid mistakes with signs.

Apply the rule step by step. First evaluate expressions inside brackets completely. Next perform all multiplication and division in the order they appear from left to right, taking care to apply sign rules for negative numbers. Finally carry out all additions and subtractions from left to right; for subtraction convert to addition of the opposite to keep calculations consistent. For example 5 − 3 × (−2) = 5 − (−6) = 5 + 6 = 11. If the expression has nested brackets evaluate inner brackets before outer ones.

Teach students to rewrite subtractions as addition of negatives and to check each step by a quick reverse operation. When many terms are added, grouping positives and negatives separately can simplify mental calculation. When multiplying or dividing negative numbers, apply sign rules immediately to avoid sign errors. Practise varied examples combining brackets and mixed signs so pupils become comfortable applying order of operations and sign rules together. This will prepare them for algebraic expressions later on.

Word problems show how integers describe real-life situations. Common contexts: temperature changes, bank deposits and withdrawals, floors above/below ground, gains and losses, and elevations above/below sea level. Read carefully to decide which quantities are positive and which are negative: words like 'rise', 'gain', 'above', 'deposit' indicate positive; 'fall', 'loss', 'below', 'withdraw' indicate negative. Convert these statements into integer expressions and then calculate using the rules learned earlier.

For example, if temperature in the morning is −3°C and it rises by 7°C, compute −3 + 7 = 4°C. For financial problems, if a balance is +600 and there is withdrawal of 750, result = 600 + (−750) = −150, meaning a debt of 150. Use number line or simple addition/subtraction rules to check answers and ensure sign correctness. Also check results against the story: a final temperature above zero should be positive, a final bank balance that is owed should be negative.

Patterns with integers help predict future terms and understand regular change. An arithmetic sequence has a common difference d which may be positive or negative. Example: 10, 7, 4, 1, −2,... has common difference −3. The nth term formula a_n = a_1 + (n−1)d works with integers too. Alternating patterns like 3, −3, 3, −3 repeat signs; use these patterns to create quick sums or to identify symmetry about zero. Encourage students to draw simple number-line patterns and timelines for successive changes to visualise net effects. Practise both word problems and pattern questions together to strengthen translating words into integer operations and recognising regular number behaviour.