Overview
This chapter introduces integers — the set of whole numbers and their negatives including zero — and shows how to represent and use them on a number line. It explains why integers matter by linking them to real-life situations (temperature, bank balance, elevations, gains and losses) and by preparing students for algebra and higher arithmetic. Key themes include classification (positive, negative, zero), absolute value, comparison and ordering, representation on the number line, rules for addition, subtraction, multiplication and division of integers, and simple applications and word problems. By the end of the chapter students will be able to identify and represent integers, compare and order them, apply sign rules to perform operations, solve routine problems using integers, and build a foundation for algebraic thinking.
Learning Objectives
- Define integers and give examples of positive integers, negative integers and zero.
- Represent integers on a number line and locate their relative positions.
- Identify the opposite (additive inverse) and determine the absolute value of an integer.
- Compare and order integers using >, < and = symbols and justify the comparison.
- Explain the rules for addition of integers with like and unlike signs.
- Apply addition rules to compute sums of integers mentally and on paper.
- Explain the rules for subtraction of integers and relate subtraction to adding the additive inverse.
- Apply subtraction rules to solve arithmetic problems and simple word problems involving integers.
Topics in this chapter
14 topics · tap a topic title to jump straight to it.
Introduction to Integers
What are Integers?
Integers are whole numbers that can be positive, negative or zero. They do not have fractions or decimals. Examples: ...,-3, -2, -1, 0, 1, 2, 3,...
Types of Integers
- Positive integers: 1, 2, 3, ... (written with a + sign or no sign)
- Negative integers: -1, -2, -3, ... (written with a - sign)
- Zero: 0 (neither positive nor negative)
Number Line
Integers are shown on a number line. Zero is in the middle, positives to the right, negatives to the left. The number line helps compare and operate on integers.
Key ideas
- Opposite (Additive inverse): Every integer a has an opposite −a such that a + (−a) = 0. Example: opposite of 5 is −5.
- Absolute value: |a| is the distance of a from 0 on the number line (always non-negative). Example: |−4| = 4, |4| = 4.
- Comparing integers: On the number line, a number to the right is greater. So 2 > −1 and −2 < −1.
Basic rules for operations (intuitive)
- Addition: Move right for positive addends, left for negative addends on the number line. Example: 3 + (−5) = −2.
- Subtraction: Subtracting a number means adding its opposite: a − b = a + (−b).
- Multiplication/Division sign rules: Same sign gives positive, different signs give negative. Example: (−2) × (−3) = 6, (−2) × 3 = −6.
These ideas form the foundation for working with integers in higher arithmetic and algebra.
- Temperature: If the temperature is −5°C and it rises by 7°C, new temperature = −5 + 7 = 2°C.
- Bank balance: If you owe ₹200 (represented as −200) and you pay ₹300, new balance = −200 + 300 = ₹100 (positive means you have money).
- Elevations: Sea level = 0, a mountain 150 m above = +150, a valley 20 m below = −20.
- Floors in a building: Ground floor = 0, two floors above = +2, three basements = −3.
- Gain/Loss: Profit of 50 is +50, loss of 30 is −30; net = +50 + (−30) = +20.
- Sports: If a team has −2 goal difference and wins by 3, new difference = −2 + 3 = +1.
- |a| = absolute value of a (distance from 0). Example: |−7| = 7.
- Opposite (additive inverse): a + (−a) = 0.
- Addition identity: a + 0 = a.
- Zero multiplication: a × 0 = 0.
- Division by zero is undefined: a ÷ 0 is not defined (for any a).
- Sign rules for multiplication/division: (+)×(+) = +, (−)×(−) = +, (+)×(−) = − (same for ÷).
Number Line Representation
What is a number line? A number line is a straight horizontal line used to represent integers as points placed at equal distances. The point marked 0 is called the origin. Integers to the right of 0 are positive (1, 2, 3, ...); integers to the left are negative (−1, −2, −3, ...).
Key ideas:
- Placement: Every integer n corresponds to a point n units from 0. If n > 0 place it n units to the right; if n < 0 place it |n| units to the left.
- Direction and sign: Rightward movement = positive direction; leftward movement = negative direction.
- Equality and uniqueness: Each integer has a unique position on the line.
- Ordering: A number that lies to the right of another on the number line is greater. Example: 2 is to the right of −1, so 2 > −1.
- Absolute value: The absolute value |a| is the distance of a from 0 on the number line (always non‑negative).
- Addition and subtraction as jumps: To add a positive number, move right; to add a negative number, move left. Subtraction a − b is equivalent to adding the opposite: a + (−b).
- Distance between two integers a and b: It is |a − b|, the number of equal intervals between their points.
How to use the number line for operations (steps):
- Draw a horizontal line and mark 0 at the center. Choose equal spacing and mark integers left and right (… −3, −2, −1, 0, 1, 2, 3 …).
- To represent an integer n, count |n| marks from 0 in the correct direction and place a dot.
- To compute a + b: start at a, then make b jumps (right if b > 0, left if b < 0); the landing point is the sum.
- To compute a − b: start at a and add −b (i.e., jump left if b > 0, right if b < 0).
Why it helps: The number line gives a visual understanding of sign, magnitude, ordering, absolute value and the effect of adding or subtracting positive and negative quantities.
- Place integers −4, −1, 0, 3 on a number line: mark 0, then mark 1,2,3 to the right and 1,2,3,4 to the left; locate −4 four steps left of 0 and 3 three steps right of 0.
- Addition: −2 + 5. Start at −2, make 5 jumps to the right → land at 3. So −2 + 5 = 3.
- Addition of negatives: (−3) + (−4). Start at −3, make 4 jumps left → land at −7. So −3 + (−4) = −7.
- Subtraction using opposite: 4 − (−2). Start at 4, add opposite of (−2) which is +2 → move 2 right → land at 6. So 4 − (−2) = 6.
- Distance between integers: distance between −5 and 2 is |−5 − 2| = |−7| = 7. On the line there are 7 equal intervals between them.
- Compare integers: Which is greater, −1 or −4? On the number line −1 is to the right of −4, so −1 > −4.
- Absolute value: |a| = distance of a from 0 (e.g., |−3| = 3, |5| = 5).
- Distance between two integers a and b: distance = |a − b| = |b − a|.
- Subtraction as addition: a − b = a + (−b).
- Ordering: if a is to the right of b on the number line, then a > b; if a is to the left, then a < b.
- Addition rules on number line: moving right adds a positive number; moving left adds a negative number.
Successor and Predecessor
Definition: For any integer n, the successor of n is the integer that comes immediately after n when integers are arranged in increasing order; it is n + 1. The predecessor of n is the integer that comes immediately before n; it is n − 1.
Key points:
- Every integer has both a successor and a predecessor because integers extend infinitely in both directions.
- Successor means one step to the right on the number line; predecessor means one step to the left.
- The same rule applies to positive numbers, zero and negative numbers. Example: successor of −5 is −4; predecessor of −5 is −6.
How to find them (step-by-step):
- To find the successor of n, add 1 to n. (Successor = n + 1)
- To find the predecessor of n, subtract 1 from n. (Predecessor = n − 1)
Short examples in words: The successor of a day (Tuesday) is the next day (Wednesday). The predecessor of seat number 10 is seat number 9.
Why this matters: Understanding successor and predecessor builds the idea of 'one more' and 'one less' which is fundamental for counting, addition, subtraction and working with number lines and integers.
- Successor of 7 is 7 + 1 = 8; Predecessor of 7 is 7 − 1 = 6.
- Successor of 0 is 0 + 1 = 1; Predecessor of 0 is 0 − 1 = −1.
- Successor of −5 is −5 + 1 = −4; Predecessor of −5 is −5 − 1 = −6.
- Find successor and predecessor of 1999: successor = 2000, predecessor = 1998.
- Real-life: If today is 14th, its successor (next day) is 15th and its predecessor (previous day) is 13th.
- Successor(n) = n + 1
- Predecessor(n) = n − 1
- On number line: successor = one step to the right; predecessor = one step to the left
Opposite (Additive Inverse)
Definition: The opposite (or additive inverse) of an integer a is the integer which, when added to a, gives 0. It is written as −a. So a + (−a) = 0.
For any integer:
- If a is positive, its opposite is the same number with a negative sign. Example: opposite of 7 is −7.
- If a is negative, its opposite is the same number without the negative sign (i.e. positive). Example: opposite of −4 is 4.
- The opposite of 0 is 0, because 0 + 0 = 0.
How to find the opposite: Change the sign. If there is no minus sign, put one; if there is a minus sign, remove it.
Properties: The opposite of an integer is unique. Taking the opposite twice returns the original number: −(−a) = a. Opposites are symmetrically placed about 0 on the number line and are equidistant from 0.
- Opposite of 12 is −12 because 12 + (−12) = 0.
- Opposite of −9 is 9 because −9 + 9 = 0.
- Opposite of 0 is 0 because 0 + 0 = 0.
- Bank example: If your balance is +150 (₹150 in hand), the opposite is −150 (₹150 owed). Together they cancel to zero.
- Temperature example: +5°C and −5°C are opposites; they are equally distant from 0°C but on different sides.
- a + (−a) = 0 (definition of additive inverse)
- −(−a) = a (double opposite returns the original)
- Opposite of 0 = 0
- If a > 0 then opposite is −a; if a < 0 then opposite is |a|
Absolute Value
What is Absolute Value? The absolute value of an integer is its distance from 0 on the number line, without considering direction. We write it as |a| and read it as "the absolute value of a." Because distance cannot be negative, |a| is always non-negative.
How to find it: If a is non-negative (a ≥ 0), then |a| = a. If a is negative (a < 0), then |a| = −a (which makes it positive). Examples: |5| = 5, |−5| = 5, |0| = 0.
Key ideas and properties:
- |a| ≥ 0 for every a.
- |a| = |−a| (absolute value ignores sign).
- The distance between two numbers a and b on the number line is |a − b|.
- Multiplication and division: |ab| = |a|·|b| and |a/b| = |a|/|b| (b ≠ 0).
Absolute value helps us measure how far numbers are from zero or from each other, which is useful in everyday contexts like temperature changes, depths, or score differences.
- Example 1: |7| = 7 because 7 is 7 units to the right of 0 on the number line.
- Example 2: |−7| = −(−7) = 7 because −7 is 7 units to the left of 0.
- Example 3: Distance between 3 and −4 is |3 − (−4)| = |3 + 4| = |7| = 7 units.
- Example 4: If temperature drops from 5°C to −3°C, the change in temperature = |−3 − 5| = |−8| = 8°C.
- Example 5: |0| = 0 because 0 is at the origin; there is no distance from 0 to itself.
- |a| = a, if a ≥ 0
- |a| = −a, if a < 0
- |−a| = |a|
- Distance between a and b: |a − b|
- |ab| = |a|·|b|
- |a/b| = |a|/|b| (b ≠ 0)
Comparing and Ordering Integers
What are integers? Integers are whole numbers that can be positive, negative, or zero: ...,-3,-2,-1,0,1,2,3,...
What does comparing integers mean? To compare two integers means to decide which is greater, which is smaller, or if they are equal. We write a>b to mean a is greater than b, a<b to mean a is less than b, and a=b if they are equal.
Use of number line: Integers increase from left to right on a number line. Any point to the right is greater than a point to the left. This is the most reliable visual tool for comparing and ordering integers.
Rules for comparing integers (quick summary):
- If one integer is positive and the other is negative, the positive integer is always greater. (e.g., 4 > -2)
- If both integers are positive, the one with larger absolute value is greater. (e.g., 7 > 3)
- If both integers are negative, the one with smaller absolute value is greater. For negatives, larger absolute value means smaller integer. (e.g., -2 > -5 because 2 < 5)
- Zero is greater than every negative integer and less than every positive integer. (e.g., 0 > -1 and 0 < 1)
- Transitive property: if a > b and b > c, then a > c. Useful when ordering several numbers.
Ordering integers means arranging them either in ascending order (from smallest to largest; left to right on number line) or descending order (from largest to smallest; right to left). Use comparisons or plot them on a number line to order several integers.
Why absolute value matters: The absolute value |x| is the distance of x from 0. For positive numbers |x| = x; for negative numbers |x| = -x. When both numbers are negative, compare |x| values in reverse: the integer with the smaller |x| is the greater integer.
- Compare 2 and -3: 2 is positive and -3 is negative → 2 > -3. On a number line 2 is to the right of -3.
- Compare -5 and -2: both negative. |-5| = 5 and |-2| = 2. Since 5 > 2, -5 < -2 → so -2 is greater than -5.
- Compare 0 and -1: zero is greater than any negative number → 0 > -1.
- Order the list: -3, 1, -7, 0, 5. Ascending (smallest to largest): -7, -3, 0, 1, 5. Descending: 5, 1, 0, -3, -7.
- Real-life: Temperatures: comparing -4°C, 2°C and 0°C. 2°C > 0°C > -4°C. On a thermometer, -4°C is below 0, so it is colder.
- Real-life: Bank balances: ₹200 (credit) vs ₹-150 (debt). ₹200 > ₹-150 because a positive balance is greater than a negative debt.
- a > b means a is to the right of b on number line; a < b means a is to the left of b.
- If sign(a) = + and sign(b) = -, then a > b (any positive > any negative).
- If a, b > 0 then a > b ⇔ |a| > |b| (compare absolute values directly).
- If a, b < 0 then a > b ⇔ |a| < |b| (for negatives compare absolute values in reverse).
- |x| = distance of x from 0 (absolute value).
- Transitive property: if a > b and b > c, then a > c.
Addition of Integers
What are we adding? Integers are whole numbers that can be positive, negative or zero. Addition of integers means combining two or more directed quantities (with signs) to get a result called the sum.
Rules (short):
- Same sign: Add the absolute values and keep the common sign. Example: (+6)+(+3)=+(6+3)=+9; (−6)+(−3)=−(6+3)=−9.
- Different signs: Subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value. Example: (+7)+(−4)=+(7−4)=+3; (−7)+(+4)=−(7−4)=−3.
- Equal and opposite: If the absolute values are equal and signs are opposite, the sum is 0. Example: (+5)+(−5)=0.
- Zero: Adding 0 changes nothing: a+0=a.
Number-line method: To add a positive integer, move right by that many units; to add a negative integer, move left. For a+b, start at a and then move according to b. For more than two integers, add sequentially.
Why it works (intuitive): Think of positives as steps to the right (gain, rise, deposit) and negatives as steps to the left (loss, fall, withdrawal). Combining steps is either reinforcing (same direction) or cancelling (opposite directions).
- 1) (+4) + (+6) = +(4+6) = +10. (Same sign → add absolute values; keep +.)
- 2) (−5) + (−2) = −(5+2) = −7. (Same sign → add; keep −.)
- 3) (+8) + (−3) = +(8−3) = +5. (Different signs → subtract; sign of larger absolute value: +8.)
- 4) (−4) + (+9) = +(9−4) = +5. (Different signs → subtract; sign of +9.)
- 5) (+6) + (−6) = 0. (Equal and opposite → cancel out.)
- 6) Word problem — Temperature: If the temperature is −3°C at night and it rises by 7°C, new temperature = (−3)+(+7)=+4°C.
- Same sign: a + b = sign(a)·(|a| + |b|). Example: (−a) + (−b) = −(|a|+|b|).
- Different signs: a + b = sign_of_larger(|a|,|b|) · (|larger| − |smaller|). Example: (+7) + (−4) = + (7−4) = +3.
- Opposite equal: a + (−a) = 0.
- Identity: a + 0 = a.
Subtraction of Integers
Subtraction of integers means finding how much one integer differs from another on the number line. The easiest and most reliable way to handle subtraction of integers is to convert the subtraction into addition: for any two integers a and b,
a − b = a + (−b)
So subtracting b is the same as adding the opposite of b. After this conversion, use the rules for addition of integers:
- If the two numbers you add have the same sign, add their absolute values and keep the common sign.
- If they have opposite signs, subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value.
Practical interpretation on a number line:
- To compute a − b, start at point a. Then move to the right if −b is positive (i.e., when b is negative), or move to the left if −b is negative (i.e., when b is positive).
- Equivalently: subtracting a positive number moves left; subtracting a negative number moves right.
Steps to solve a subtraction of integers:
- Rewrite a − b as a + (−b).
- Decide the signs of a and −b.
- Apply addition rules: either add absolute values (same sign) or subtract them (different signs) and assign the correct sign.
Common mistakes to avoid: do not treat subtraction as simply taking the smaller number from the larger without considering signs; always convert to addition of the opposite and use integer addition rules.
- 5 − 3 = 5 + (−3) → same signs? No. 5 and −3 have opposite signs → 5 − 3 = 2.
- 3 − 5 = 3 + (−5) → opposite signs → 5 − 3 = 2, larger absolute is 5 with sign − → result = −2.
- −2 − 4 = −2 + (−4) → same sign (both −) → add absolutes 2 + 4 = 6 and keep − → −6.
- −2 − (−5) = −2 + 5 → opposite signs → 5 − 2 = 3, sign of larger absolute (5) is + → 3.
- 7 − (−2) = 7 + 2 = 9 (subtracting a negative = adding the positive).
- 0 − 3 = 0 + (−3) = −3.
- a − b = a + (−b) (convert subtraction into addition)
- If x and y have the same sign: x + y = sign(x) · (|x| + |y|)
- If x and y have opposite signs: x + y = sign(larger absolute) · (|larger| − |smaller|)
- Subtracting positive: move left on number line; subtracting negative: move right on number line.
Multiplication of Integers
What it means: Multiplication of integers is repeated addition or grouping. a × b means 'a groups of b' (or 'b added a times'). For integers, we also consider direction (sign): positive means right/up, negative means left/down on a number line.
Sign rules (short): positive × positive = positive; positive × negative = negative; negative × positive = negative; negative × negative = positive. Intuitively, 'a negative times a negative gives a positive' because two reversals cancel out.
Why the sign rules work: Use repeated addition for positive multipliers (3 × 4 = 4 + 4 + 4 = 12). For negative multiplicands, think of adding negative numbers (-4 + -4 + -4 = -12). For a negative number of groups (like -3 × 4) interpret as reversing direction: -3 × 4 = -(3 × 4) = -12. Two reversals (−3 × −4) bring you back to positive: −(3) × −(4) = +12.
Key properties that still hold:
- Commutative: a × b = b × a
- Associative: (a × b) × c = a × (b × c)
- Distributive over addition: a × (b + c) = a × b + a × c
- Multiplicative identity: 1 × a = a
- Multiplying by 0: 0 × a = 0
- Negation rule: (−1) × a = −a and |a × b| = |a| × |b|
Tips for understanding: Use a number line for visualizing repeated jumps. Use the idea of direction (right = +, left = −). For area/array models, treat one side as negative to represent a negative area (below axis) which helps visualize sign of product.
- 4 × 3 = 3 + 3 + 3 + 3 = 12 (positive × positive = positive).
- -5 × 2 = (-5) + (-5) = -10 (negative × positive: add negative twice → negative).
- 5 × (-2) = -(5 × 2) = -10 (positive × negative = negative; same magnitude, negative sign).
- -3 × -4 = 3 × 4 = 12 (negative × negative = positive because two reversals cancel).
- 0 × 7 = 0 (anything multiplied by 0 is 0).
- Using distributive property: 6 × (2 + -3) = 6×2 + 6×(-3) = 12 + (-18) = -6.
- (+a) × (+b) = + (a × b)
- (+a) × (−b) = − (a × b)
- (−a) × (+b) = − (a × b)
- (−a) × (−b) = + (a × b)
- a × b = b × a (commutative)
- (a × b) × c = a × (b × c) (associative)
Division of Integers
What is division of integers?
Division of integers is the operation of splitting an integer (called the dividend) into equal parts specified by another integer (the divisor) to get the quotient. Division is the inverse operation of multiplication: a ÷ b = c exactly when a = b × c.
Key points:
- If dividend = 0 and divisor ≠ 0, then quotient = 0. Example: 0 ÷ 5 = 0.
- Division by 0 is undefined: a ÷ 0 is not defined for any integer a.
- The quotient may be an integer if the divisor divides the dividend exactly; otherwise it may be a fraction/decimal.
Rules for signs (how signs affect the result)
| (+) | ÷ | (+) | = | (+) |
| (-) | ÷ | (-) | = | (+) |
| (+) | ÷ | (-) | = | (-) |
| (-) | ÷ | (+) | = | (-) |
In words: dividing two integers with the same sign gives a positive quotient; with opposite signs gives a negative quotient.
Division with remainder and divisibility
When a does not divide b exactly, we write b ÷ a = q remainder r, where 0 ≤ r < |a| and b = a×q + r. If remainder r = 0, b is divisible by a.
- 12 ÷ 3 = 4 because 3 × 4 = 12 (positive ÷ positive = positive).
- -12 ÷ 3 = -4 because 3 × (-4) = -12 (negative ÷ positive = negative).
- -12 ÷ -3 = 4 because (-3) × 4 = -12 (negative ÷ negative = positive).
- 0 ÷ 5 = 0 (zero divided by nonzero is zero).
- 5 ÷ 0 is undefined (division by zero is not allowed).
- 15 ÷ 4 = 3 remainder 3 since 15 = 4×3 + 3 (shows remainder and divisibility).
- a ÷ b = c ⇔ a = b × c (division is the inverse of multiplication)
- (+ ÷ +) = + ; (- ÷ -) = + ; (+ ÷ -) = - ; (- ÷ +) = - (sign rules)
- 0 ÷ a = 0 for a ≠ 0
- a ÷ 0 is undefined
- If b = a×q + r with 0 ≤ r < |a|, then q is the integer quotient and r is the remainder
- b is divisible by a ⇔ remainder r = 0
Properties of Operations on Integers
Introduction: Integers are whole numbers and their negatives (..., -3, -2, -1, 0, 1, 2, 3, ...). When we add, subtract, multiply or divide integers, certain rules (properties) help us compute and understand results quickly. Below are the main properties with short explanations and examples.
- Closure: The set of integers is closed under addition and multiplication. This means if a and b are integers, then a + b and a × b are also integers. (Not always closed under division: a ÷ b may not be an integer.)
- Commutative Property:
- Addition: a + b = b + a. Example: 3 + (-5) = -2 and (-5) + 3 = -2.
- Multiplication: a × b = b × a. Example: (-2) × 4 = 4 × (-2) = -8.
- Associative Property:
- Addition: (a + b) + c = a + (b + c). Example: (2 + (-3)) + 4 = 3 and 2 + ((-3) + 4) = 3.
- Multiplication: (a × b) × c = a × (b × c). Example: ((-1) × 2) × 3 = -6 and (-1) × (2 × 3) = -6.
- Distributive Property: Multiplication distributes over addition/subtraction: a × (b + c) = a × b + a × c, and a × (b - c) = a × b - a × c. Example: 3 × (2 + (-5)) = 3×2 + 3×(-5) = 6 - 15 = -9.
- Additive Identity: 0 is the additive identity because a + 0 = a for any integer a. Example: -7 + 0 = -7.
- Multiplicative Identity: 1 is the multiplicative identity because a × 1 = a for any integer a. Example: 5 × 1 = 5.
- Additive Inverse (Negative): For each integer a there exists −a such that a + (−a) = 0. Example: 8 + (−8) = 0. Note: multiplicative inverses (reciprocals) of integers are not always integers (only ±1 have integer reciprocals).
- Zero Property of Multiplication: a × 0 = 0 for any integer a.
- Subtraction as Addition of Negative: a − b = a + (−b). This allows subtraction to be handled using addition rules. Example: 4 − (−3) = 4 + 3 = 7.
- Sign Rules (useful when adding or multiplying):
- Addition: If signs are same, add magnitudes and keep the sign. If different, subtract smaller magnitude from larger and keep sign of larger magnitude. Example: (−7) + (−2) = −9; 7 + (−12) = −5.
- Multiplication: (+)×(+) = +, (−)×(−) = +, (+)×(−) = −, (−)×(+) = −. Example: (−3)×(−4)=12; (−3)×4=−12.
- Division: Division of integers may not give an integer (not closed). Example: 5 ÷ 2 = 2.5 (not an integer). Division by 0 is undefined.
Summary: For integers, addition and multiplication are closed, commutative, associative; multiplication distributes over addition; 0 is additive identity and 1 is multiplicative identity; every integer has an additive inverse; zero times any integer is zero; subtraction is addition of the negative; division is not always an integer and division by zero is not allowed.
- Temperature: If morning is 5°C and temperature drops by 8°C, final = 5 + (−8) = −3°C (addition with negative).
- Bank balance: If balance is Rs. 200 and you withdraw Rs. 350, then 200 + (−350) = −150 (overdraft).
- Elevator floors: If ground floor is 0, going down 3 floors and then up 5 floors = (0 − 3) + 5 = 2 (use associative and subtraction as addition of negative).
- Repeated addition (multiplication): 4 × (−3) means add (−3) four times: (−3) + (−3) + (−3) + (−3) = −12.
- Distributive in shopping: 3 × (10 + (−2)) = 3×10 + 3×(−2) = 30 − 6 = 24 (three packs each priced at ₹10 with ₹2 discount each).
- Closure (addition/multiplication): If a, b ∈ Z then a + b ∈ Z and a × b ∈ Z.
- Commutative: a + b = b + a, a × b = b × a
- Associative: (a + b) + c = a + (b + c), (a × b) × c = a × (b × c)
- Distributive: a × (b + c) = a × b + a × c and a × (b − c) = a × b − a × c
- Additive identity: a + 0 = a
- Multiplicative identity: a × 1 = a
Mixed Operations and Order of Operations
What are mixed operations? Mixed operations are arithmetic expressions that contain more than one operation (addition, subtraction, multiplication, division) and sometimes brackets. When integers (positive and negative whole numbers) are involved we must apply the rules for integer arithmetic together with the correct order of operations.
Order of operations (BODMAS/BIDMAS):
- Brackets: evaluate expressions inside ( ) or [ ] first.
- Orders (or Indices): powers or roots — usually introduced later, but follow them if present.
- Division and Multiplication: do from left to right.
- Addition and Subtraction: do from left to right.
When integers are involved, always handle signs carefully and apply integer rules at each step. A helpful technique is to remove subtraction by rewriting it as addition of the opposite: a - b = a + (−b).
Rules for integers (quick reference):
- Addition: same signs → add absolute values and keep the sign; different signs → subtract smaller absolute value from larger and take the sign of the larger absolute value.
- Subtraction: a - b = a + (−b). Change the sign of b then add.
- Multiplication & Division: same signs → positive; different signs → negative. Multiply/divide absolute values as usual.
- Distributive law: a(b + c) = ab + ac, works with integers (and with negative numbers too).
How to evaluate a mixed expression with integers — stepwise method:
- Remove spaces and identify brackets. Work from innermost brackets outward.
- Within a bracket, follow BODMAS: do multiplication/division left to right, then addition/subtraction left to right.
- When you see subtraction, consider rewriting as addition of a negative if it helps clarity.
- Apply integer sign rules at each arithmetic step.
- After simplifying inside all brackets, perform remaining division/multiplication, then addition/subtraction.
Tips for students:
- Work neatly line by line and show intermediate answers with signs.
- Use a number line for visualizing addition/subtraction of negatives and positives.
- Check each step by estimating the sign and size of the result to catch mistakes.
- 1) Evaluate: 7 − (−3) + 2 × (−4) Solution: 7 − (−3) = 7 + 3 = 10. Then 2 × (−4) = −8. So 10 + (−8) = 2.
- 2) Evaluate: (−5) + 12 ÷ 3 − 4 Solution: Do division first: 12 ÷ 3 = 4. Expression becomes (−5) + 4 − 4. Now left to right: (−5) + 4 = −1. Then −1 − 4 = −5.
- 3) Evaluate: 3 × [2 − (−1 + 4)] Solution: Innermost bracket: (−1 + 4) = 3. Then [2 − 3] = −1. Now 3 × (−1) = −3.
- 4) Evaluate: (−6) ÷ 2 × (−3) + 5 Solution: Division and multiplication left to right: (−6) ÷ 2 = −3. Then (−3) × (−3) = +9. Finally 9 + 5 = 14.
- 5) Evaluate: 8 − [3 × (−2) − 4] Solution: Inner: 3 × (−2) = −6. Then −6 − 4 = −10. So 8 − (−10) = 8 + 10 = 18.
- a − b = a + (−b) (rewrite subtraction as addition of the opposite)
- Addition rules: if signs are same: |a| + |b| with that sign; if signs differ: |larger| − |smaller| with sign of larger absolute value
- Multiplication: (−a) × (−b) = +ab ; (−a) × b = −ab ; a × (−b) = −ab
- Division: (−a) ÷ (−b) = a ÷ b ; (−a) ÷ b = −(a ÷ b), same sign → positive, different → negative
- Order of operations (BODMAS): Brackets → Orders/Indices → Division & Multiplication (left to right) → Addition & Subtraction (left to right)
- Distributive law: a(b + c) = ab + ac (works with negative integers as well)
Word Problems and Applications
What this topic means
Word problems and applications in the chapter on Integers teach how to translate real-life situations into integer statements, perform integer operations, and interpret the result. Integers are used to represent quantities with direction or opposite nature — for example, gains and losses, temperatures above/below zero, heights above/below sea level, profit/loss, deposits/withdrawals.
General approach to solve word problems with integers
- Read the problem carefully and identify quantities that can be positive or negative (e.g., gain = positive, loss = negative).
- Assign integers (positive or negative) to each quantity and write the mathematical expression.
- Use integer operation rules (addition, subtraction, multiplication, division) to compute the result.
- Interpret the numerical answer back in the context of the problem (what the sign and magnitude mean).
- Use a number line or quick checks (absolute values, signs) to verify your result.
Common real-life situations modelled by integers
- Temperature changes (°C or °F): rise (positive), fall (negative).
- Bank balances: deposit (positive), withdrawal (negative).
- Elevation: above sea level (positive), below sea level (negative).
- Profit and loss: profit (positive), loss (negative).
- Game scores: points gained (positive), points lost (negative).
Tips for students
- Translate words: "rise/increase/gain" → add (positive); "fall/decrease/loss" → subtract (negative).
- When subtracting an integer, convert to addition of its opposite: a − b = a + (−b).
- Use the number line to show step-by-step changes — it reduces errors.
- 1) Temperature change: "Morning temperature is −5°C. By noon it rises by 8°C. What is the temperature at noon?" Solution: −5 + 8 = 3°C. (Interpretation: 3°C above zero.)
- 2) Elevation: "A submarine is 30 m below sea level. It rises 18 m and then sinks 5 m. What is its final position?" Solution: Start −30 m, rise +18 → −12 m; sink −5 → −12 + (−5) = −17 m (17 m below sea level).
- 3) Bank balance: "Riya's account was ₹2500. She withdrew ₹3200 and then deposited ₹1500. What is her final balance?" Solution: 2500 + (−3200) + 1500 = 2500 − 3200 + 1500 = 800. Final balance = ₹800.
- 4) Distance above/below ground: "A tunnel goes 12 m below ground, then a passage goes up 20 m to a chamber. What is the chamber's height relative to ground?" Solution: −12 + 20 = +8 m (8 m above ground).
- 5) Repeated losses (multiplication): "A player loses 4 points each for 5 wrong answers. What is the total change in score?" Solution: Total = 5 × (−4) = −20 (score decreased by 20 points).
- 6) Division of negative value: "A debt of ₹600 is shared equally among 3 people. What is each person's share (as a negative value)?" Solution: (−600) ÷ 3 = −200 (each owes ₹200).
- Addition rules: If signs are same → add absolute values and keep the common sign. Example: (+a)+(+b)=+(a+b), (−a)+(−b)=−(a+b).
- Addition with different signs: Subtract smaller absolute value from larger absolute value; keep sign of the larger magnitude. Example: (+a)+(−b)=sign_of_larger(|a|,|b|) × (|a|−|b|).
- Subtraction rule: a − b = a + (−b). Convert subtraction into addition of the opposite.
- Multiplication sign rules: Same sign → positive, different signs → negative. Example: (−a)×(−b)=+(ab), (−a)×b=−(ab).
- Division sign rules: Same sign → positive, different signs → negative. Example: (−a)÷(−b)=+(a÷b), (−a)÷b=−(a÷b).
- Absolute value: |a| = distance of a from 0 on number line; always non-negative. Useful to compare magnitudes.
Problem-solving Strategies and Practice
Overview: Problem-solving with integers means translating everyday situations into signed numbers (positive and negative) and using rules for addition, subtraction, multiplication and division. Use clear steps: understand the problem, choose a strategy (number line, inverse operations, grouping), execute carefully, and check the result.
Step-by-step strategy (Polya adapted):
- Understand: Read the problem, identify quantities and whether they are gains (+) or losses (−), above/below, deposits/withdrawals.
- Plan: Convert words to integers (e.g., "3 below zero" → −3). Decide whether to use a number line, algebraic rules, or model drawing.
- Execute: Apply integer rules (use number line jumps for addition/subtraction). For subtraction, rewrite a − b as a + (−b) and then add.
- Check: Use inverse operations (e.g., check addition with subtraction) and estimate whether the answer's sign and size make sense.
Helpful concrete strategies:
- Use a number line to visualise moves right (positive) and left (negative).
- Group operations using commutative and associative properties for addition (reorder to make easy pairs).
- Turn subtraction into addition of the additive inverse: a − b = a + (−b).
- Always track the sign: for mixed signs subtract magnitudes and keep the sign of the larger magnitude.
- Use word-to-symbol translation templates: "loss/withdrawal/below" → negative, "gain/deposit/above" → positive.
Why these help: These strategies reduce mistakes with signs, provide visual confirmation (number line), and let you break difficult expressions into simpler parts that are easy to compute and check.
- 1) -3 + 7: On the number line start at -3, move 7 steps right → land at 4. Answer: 4.
- 2) -8 + (-5): Both negative, add magnitudes 8 + 5 = 13 and keep negative sign → -13.
- 3) 5 - (-2): Rewrite as 5 + 2 = 7. Answer: 7.
- 4) -6 - 4: Rewrite as -6 + (-4). Both negative, add magnitudes 6 + 4 = 10 → -10.
- 5) (-3) × 4: One negative, one positive → product negative. 3×4 = 12 → -12.
- 6) (-2) × (-3): Both negative → product positive. 2×3 = 6 → 6.
- Addition rules: If signs are same, add magnitudes and keep the sign (e.g., (-a)+(-b)=-(a+b); a+b positive if both positive). If signs differ, subtract smaller magnitude from larger and take the sign of the larger magnitude (e.g., a+(-b)=±( |a|-|b| ) ).
- Subtraction: a - b = a + (−b). Always convert subtraction to addition of the opposite before applying addition rules.
- Multiplication signs: positive × positive = positive; negative × negative = positive; positive × negative = negative. (Sign same → positive; sign different → negative.)
- Division signs: same as multiplication for signs. (e.g., (-a)/(-b) = a/b; (-a)/b = -(a/b)).
- Zero rules: a + 0 = a; a × 0 = 0; 0 divided by non-zero a is 0; division by 0 is undefined.
- Absolute value: |a| = distance of a from 0. For integers, |a| ≥ 0. Example: |−5| = 5.
Key Concepts
- Integer
- A whole number that can be positive, negative or zero; no fractions or decimals.
- Positive Integer
- An integer greater than zero (numbers to the right of 0 on the number line).
- Negative Integer
- An integer less than zero (numbers to the left of 0 on the number line).
- Zero
- The integer that is neither positive nor negative; it is the additive identity.
- Number Line
- A straight line with equally spaced points representing integers, 0 at the origin; positives to the right, negatives to the left.
- Successor
- The next integer greater than a given integer (add 1).
- Predecessor
- The previous integer less than a given integer (subtract 1).
- Absolute Value
- The distance of an integer from 0 on the number line; always non‑negative.
- Additive Inverse (Opposite)
- For any integer a, the number which when added to a gives 0; the opposite of a is -a.
- Additive Identity
- The integer 0, because adding 0 to any integer leaves it unchanged.
- Multiplicative Identity
- The integer 1, because multiplying any integer by 1 leaves it unchanged.
- Zero Pair
- A pair of integers whose sum is zero, typically a and -a; they cancel each other.
- Comparison of Integers
- Using symbols <, >, = to show which integer is smaller, larger or equal; integers to the right on the number line are greater.
- Ordering of Integers
- Arranging integers in ascending (smallest to largest) or descending (largest to smallest) order.
- Addition of Integers
- Combine integers using rules: same sign — add magnitudes and keep sign; different signs — subtract smaller magnitude from larger and take sign of larger magnitude.
- Subtraction of Integers
- Subtracting b from a is same as adding the additive inverse: a - b = a + (-b).
- Multiplication of Integers
- Multiply magnitudes; the product is positive if signs are the same, negative if signs are different.
- Division of Integers
- Divide magnitudes; quotient is positive if signs are the same, negative if different; division by zero is undefined.
- Rules of Signs
- Rules to determine sign in multiplication/division: + × + = +, - × - = +, + × - = -, and similarly for division; for addition/subtraction use sign and magnitudes.
- Even and Odd Integers
- Even integers are divisible by 2 (…,-4, -2, 0, 2,4,…). Odd integers are not divisible by 2 (…,-3,-1,1,3,…).
End-of-Chapter Trial Paper & Test Questions
Topic-wise questions to test your understanding of every concept in this chapter.
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Which of the following is NOT an integer? / निम्न में से कौन-सा पूर्णांक नहीं है? (a) -7 / -7 (b) 0 / 0 (c) 3/4 / 3/4 (d) 100 / 100
Show answer
(c) 3/4 / 3/4 — Integers are whole numbers (positive, negative, or zero) with no fractions or decimals; 3/4 is a fraction, not an integer. / पूर्णांक भिन्न या दशमलव रहित पूर्ण संख्याएँ होती हैं; 3/4 एक भिन्न है, पूर्णांक नहीं।
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What is the sum of (-8) + (+5)? / (-8) + (+5) का योग क्या है? (a) 13 / 13 (b) -3 / -3 (c) 3 / 3 (d) -13 / -13
Show answer
(b) -3 / -3 — Different signs: subtract smaller absolute value (5) from larger (8) and keep the sign of the number with larger absolute value (negative). So -8 + 5 = -3. / भिन्न चिह्न: बड़े निरपेक्ष मान (8) में से छोटा (5) घटाएँ और बड़े निरपेक्ष मान वाले अंक (ऋणात्मक) का चिह्न रखें। अतः -8 + 5 = -3।
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The product (-3) × (-4) equals: / (-3) × (-4) का गुणनफल है: (a) -12 / -12 (b) 7 / 7 (c) 12 / 12 (d) -7 / -7
Show answer
(c) 12 / 12 — Negative × Negative = Positive; multiply the absolute values: 3 × 4 = 12, and the result is positive. / ऋणात्मक × ऋणात्मक = धनात्मक; निरपेक्ष मान गुणा करें: 3 × 4 = 12, परिणाम धनात्मक होगा।
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The absolute value of -15 is ______. / -15 का निरपेक्ष मान ______ है।
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15 / 15 — Absolute value is the distance of a number from 0 on the number line, always non-negative. |-15| = 15. / निरपेक्ष मान संख्या की 0 से दूरी है, जो सदा अऋणात्मक होती है। |-15| = 15।
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Subtracting -6 from 10 gives ______. / 10 में से -6 घटाने पर ______ मिलता है।
Show answer
16 / 16 — Subtracting a negative is the same as adding its positive: 10 − (−6) = 10 + 6 = 16. / ऋणात्मक संख्या घटाना उसे जोड़ने के बराबर है: 10 − (−6) = 10 + 6 = 16।
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True or False: On the number line, -3 is greater than -1. / सत्य या असत्य: संख्या रेखा पर -3, -1 से बड़ा है।
Show answer
False / असत्य — On the number line, numbers increase from left to right. -1 is to the right of -3, so -1 > -3 (not the other way around). / संख्या रेखा पर बाएँ से दाएँ संख्याएँ बढ़ती हैं। -1, -3 के दाईं ओर है, अतः -1 > -3।
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A submarine is at -30 m below sea level. It rises by 18 m. What is its new position? / एक पनडुब्बी समुद्र तल से -30 मीटर नीचे है। वह 18 मीटर ऊपर उठती है। उसकी नई स्थिति क्या है?
Show answer
-12 m / -12 मीटर — Start at -30, add +18: (-30) + 18 = -12. The submarine is now 12 m below sea level. / -30 से शुरू करके +18 जोड़ें: (-30) + 18 = -12। पनडुब्बी अब समुद्र तल से 12 मीटर नीचे है।
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Evaluate: 6 − (−4) + (−3) × 2 / निकालें: 6 − (−4) + (−3) × 2
Show answer
4 / 4 — Step 1: (-3) × 2 = -6. Step 2: 6 − (−4) = 6 + 4 = 10. Step 3: 10 + (−6) = 4. Follow BODMAS — multiplication before addition/subtraction. / चरण 1: (-3) × 2 = -6। चरण 2: 6 − (−4) = 6 + 4 = 10। चरण 3: 10 + (−6) = 4। BODMAS नियम पालन करें — गुणा पहले, फिर जोड़/घटाव।
Related Laws & Principles
Explore allFoundational laws & principles behind this chapter. Each one opens a full page — what it says, why it matters, five practice questions and the mistakes to avoid.