Operations with Integers
Discover how positive and negative numbers behave when we add, subtract, multiply and divide them.
Use number lines, integer tokens, patterns, puzzles and real-world situations to understand the rules instead of simply memorising them. Explore Rakesh's number game, carrom coin movement, the 4 sign rules, the 4×4 Magic Grid, function machines, and the strategic game of Terhüchü.
What You Will Discover in This Chapter
A Quick Recap of Integers
Rakesh's Number Game, Carrom Coin Movement, Magnitude vs Direction & Token Model
Rakesh's Puzzle: A Number Game
Rakesh gives you a challenge: “I have thought of two numbers. Their sum is 25, and their difference is 11. Can you tell me the two numbers?” (Remember: difference means first number − second number).
| First Number | Second Number | Sum (a + b) | Difference (a − b) | Check |
|---|---|---|---|---|
| 10 | 15 | 25 | -5 | Too low |
| 20 | 5 | 25 | 15 | Too high |
| 19 | 6 | 25 | 13 | Close! |
| 18 | 7 | 25 | 11 | ✓ Exact Match! (18 and 7) |
Notice what happens when you swap the numbers: 7 − 18 = −11. Swapping the pair reverses the sign of the difference!
Carrom Coin Integers: Directed Movement
A carrom coin begins at 0. Striking it rightwards is defined as positive (+), while striking it leftwards is negative (−). The final position is simply P = a + b.
Textbook Puzzle: Sequence of Multiple Strikes
If the coin is struck in order: 1, −2, 3, −4, 5, −6, 7, −8, 9, −10, what is the final position?
Magnitude vs Direction: An Integer Captures Both
Token Model Recall & The Additive Inverse
We represent positive 1 with a green token (+) and negative 1 with a red token (−). Together, one green and one red form a zero pair and cancel out.
Step 1: Start with 7 positive tokens (+++++++). We need to remove 18 positives, but we only have 7!
Multiplication of Integers: The Token Model
Placing & Removing Tokens for All 4 Sign Cases: (+)(+), (+)(−), (−)(+), (−)(−)
The Empty Bag Token Laboratory
In this model, start with an empty bag (value = 0).
• Positive Multiplier: Place groups of tokens INTO the bag.
• Negative Multiplier: REMOVE groups of tokens FROM the bag (by inserting zero pairs first).
Place 2 negative tokens 3 times = 6 negatives = -6.
Remove 2 negative tokens 5 times from zero pairs = leaves 10 positive tokens = +10.
Remove 1 negative token 4 times = leaves 4 positive tokens = +4.
Remove 3 positive tokens 7 times = leaves 21 negative tokens = -21.
Multiplication Patterns & Sign Rules
Decreasing Multipliers, Times Tables, Commutativity & Brahmagupta (628 CE)
Pattern A: Multiplicand is Positive (+3)
For every unit decrease in the multiplier, the product decreases by 3:
Pattern B: Multiplicand is Negative (−3)
For every unit decrease in the multiplier, the product increases by 3:
Commutativity of Integer Multiplication
Does the product change if we swap the multiplier and multiplicand? Test below with our interactive swap animator:
For every integer a, multiplying by 1 leaves the integer unchanged: 1 × 5 = 5 and 1 × (−5) = −5.
Multiplying any integer by −1 flips its sign while preserving its magnitude: (−1) × 5 = −5 and (−1) × (−5) = +5 (the additive inverse!).
Brahmagupta's Rules (628 CE)
In his monumental treatise Brāhmasphuṭasiddhānta (chapter 18, verses 30–32), Indian astronomer and mathematician Brahmagupta formulated the world's first recorded rules for operations on positive and negative numbers, using the terms fortune (dhana) for positive and debt (ṛṇa) for negative:
• “The product or quotient of two debts is a fortune.” (− × − = +)
• “The product or quotient of a fortune and a debt is a debt.” (+ × − = −)
Real-Life Applications of Integers
Exam Scoring with Penalties, Mining Shaft Elevators & Cement Company Profit/Loss
Example 1: Exam Scoring (50 MCQs, +5 Correct, −2 Wrong)
Mala answered 30 questions correctly and 20 incorrectly in a 50-question test. What were her total marks?
Example 2: Mining Shaft Elevator (Descending at 3 m/min)
Method 1 (Distance Subtraction): 0 − (60 × 3) = 0 − 180 = −180 m.
Method 2 (Speed as Directed Velocity −3 m/min): 60 × (−3) = −180 m.
Final Position: 180 metres below ground level (−180 m).
A Magic Grid of Integers
Circle 4 Numbers (No Two in Same Row or Column) & Discover the Secret Constant Product!
Interactive 4×4 Magic Grid
Rules: Circle any number. Strike out its row and column. Circle another unstruck number, strike out its row and column, until 4 numbers are circled. Then multiply them!
Division of Integers
Division as Inverse of Multiplication & Why Division by Zero is Undefined
Division Reframed as Multiplication
We convert integer division into multiplication:
• (−100) ÷ 25 asks: “What number multiplied by 25 gives −100?” → 25 × (−4) = −100, so (−100) ÷ 25 = −4.
• (−100) ÷ (−4) asks: “What number multiplied by −4 gives −100?” → (−4) × 25 = −100, so (−100) ÷ (−4) = 25.
• 50 ÷ (−25) asks: “What number multiplied by −25 gives 50?” → (−25) × (−2) = 50, so 50 ÷ (−25) = −2.
Asking for 15 ÷ 0 means finding a number such that 0 × ? = 15. But 0 multiplied by anything is 0, so no solution exists! On the other hand, 0 ÷ 15 = 0 because 15 × 0 = 0.
Expressions, Associativity & Distributivity
Grouping Orders, Negative Sign Counting & Two-Color Rectangular Token Model
Associative Property: a × (b × c) = (a × b) × c
Consider 5 × (−3) × 4. Does it matter how we group the factors?
Rule for the Sign of Many Negative Factors
Look at powers of (−1):
• Even number of negative factors → POSITIVE product.
• Odd number of negative factors → NEGATIVE product.
Distributive Property: a × (b + c) = (a × b) + (a × c)
In the textbook model for 4 × (2 + (−3)), imagine 4 rows of tokens. Each row contains 2 green positive tokens and 3 red negative tokens:
Pick the Pattern: Function Machines & Puzzles
Machine 1, Machine 2, Figure It Out Q7, Modified Collatz & Alien Pibs Currency
Machine 1 (Blue Box in Textbook)
| a | b | c | Output |
|---|---|---|---|
| 5 | 8 | 3 | 10 |
| 10 | 11 | 12 | 9 |
| 5 | 8 | -3 | 16 |
| -3 | 10 | 2 | 5 |
| -4 | -1 | -6 | 1 |
| -10 | -12 | -9 | ★ −13 |
Machine 2 (Orange Box in Textbook)
| a | b | c | Output |
|---|---|---|---|
| 4 | 8 | -3 | -29 |
| 6 | -11 | 12 | 54 |
| 5 | 3 | 7 | -22 |
| -3 | 9 | -8 | 35 |
| -7 | 4 | 6 | 22 |
| -10 | -12 | -9 | ★ −111 |
Modified Integer Collatz Explorer
The textbook introduces a signed version of the famous Collatz conjecture: If even, take half; if odd, multiply by −3 and add 1.
Alien Pibs Currency: (+13) and (−9) Coins
An alien society uses coins of +13 pibs and −9 pibs. Can we reach various targets?
Because 13 and 9 are coprime (gcd(13, 9) = 1), every single integer value (even large amounts like +1568 pibs: 122 coins of +13 and 2 coins of −9) can be purchased exactly!
Terhüchü — An Integer & Strategy Game
Traditional Game from Assam & Nagaland Played on a 16-Square Board with Diagonals
Interactive Terhüchü Board
Players take turns moving 1 coin along any line to a neighbouring vacant intersection. Jumping over an opponent coin into a vacant intersection captures and removes it. Multiple jumps in one turn are allowed!
Inside the outer triangular corners, a coin may skip an empty intersection and move directly to the one beyond it. This unique feature makes corners powerful tactical hubs!
Integer Rules — Quick Reference
Common Exam Pitfalls & Misconceptions
(-4) × (-3) = +12 (NOT -12).(-20) ÷ (-4) = +5 (NOT -5).5 - (-3) = 5 + 3 = 8 (which is larger than 5).3 × (-4) = (-4) × 3 (Commutative); 2 × (3 × 4) = (2 × 3) × 4 (Associative).(-1) × 7 = -7 (magnitude is still 7).Practice Zone (40 Graded Questions)
Progress through Foundation, Application, Challenge, and Master tiers
What is the additive inverse of -24?
Evaluate: (+7) - (+18) using the additive inverse rule.
Evaluate: 4 × (-6)
Evaluate: (-8) × (-5)
What is the multiplicative identity for integers?
What is the result of multiplying an integer a by -1?
Evaluate: (-36) ÷ 9
Evaluate: (-56) ÷ (-7)
What is the mathematical result of 15 ÷ 0?
Which property is stated by: a × b = b × a?
Rakesh thinks of two numbers whose sum is 0 and difference is 14. What is the first number?
A carrom coin moves by strikes in order: 1, -2, 3, -4, 5, -6, 7, -8, 9, -10. What is its final position from 0?
In an exam of 50 questions, +5 is awarded for a correct answer and -2 for a wrong answer. Mala gets 30 correct and 20 wrong. What is her score?
An elevator descends into a mine shaft at 3 m/min from ground level (0 m). Where is it after 1 hour?
If the elevator begins at 15 m above ground (+15) and descends at 3 m/min for 45 minutes, what is its final position?
A cement company earns +₹8 profit per white bag and suffers -₹5 loss per grey bag. If it sells 3,000 white and 5,000 grey bags, what is the net financial result?
If the company sells 6,400 grey bags, how many white bags must it sell to have neither profit nor loss?
Using the distributive property, evaluate: (-5) × (18 + (-3))
Evaluate: (-7) × 4 × (-1)
Determine the sign of the product: (-2) × 3 × (-4) × (-5) × (-1)
In the textbook's 4×4 Magic Grid, what is the constant product obtained by circling any 4 numbers with no two in the same row or column?
Machine 1 computes a + b - c. What is its output for inputs [-10, -12, -9]?
Machine 2 computes -((a × b) + c). What is its output for inputs [-10, -12, -9]?
In Figure it Out Q7, the machine rule is a - (b × c). What is the output for [-16, -6, -9]?
If 47 - 56 + 14 - 8 + 2 - 8 + 5 = -4, find the value of -47 + 56 - 14 + 8 - 2 + 8 - 5 without evaluating every term.
In the modified integer Collatz rule (Even: n/2, Odd: -3n + 1), what number immediately follows -7?
Find three consecutive integers whose product is -6.
Using +13 pibs and -9 pibs coins, can you make a total of +85 pibs?
Given that (-548) × 972 = -532656, find (-547) × 972 without multiplying from scratch.
Given that 207 × (-33 + 7) = -5382, what is the value of -207 × (33 - 7)?
Using the numbers 3, -2, 5, -6 and operations '+', '-', '×' exactly once each with brackets, what is the MAXIMUM possible result?
Using the same numbers 3, -2, 5, -6 and '+', '-', '×' once each, what is the MINIMUM possible result?
Arrange in INCREASING order: P = 348 × (-1064), Q = (-348) + (-1064), R = (-348) - (-1064), S = (-348) × (-1064).
Evaluate: (280 × (-7)) ÷ ((-8) × (-35))
Given that (-548) × 972 = -532656, calculate (-547) × 971.
Anita answered all questions in a test with +4 for correct and -2 for incorrect. She got 15 correct and scored 40. How many questions were in the test?
In the same 25-question test, Anil scored -10 marks with 5 correct answers. Did he leave any questions unanswered?
In the traditional board game Terhüchü, what unique movement rule applies inside the triangular corners outside the main square?
Which of the following demonstrates that multiplication is distributive over subtraction?
Using only +13 pibs and -9 pibs coins, is it mathematically possible to purchase an item costing 1568 pibs?
Final Chapter Assessment (20 MCQs)
Evaluate your mastery across integer movement, token logic, sign patterns, applications, and properties
Key Terms & Definitions (Class 7 Integers)
The set of whole numbers and their negatives: {... , -3, -2, -1, 0, 1, 2, 3, ...}.
ℤThe size or absolute distance of a number from zero on the number line, disregarding direction.
|a|The number that, when added to a given integer, yields zero. The additive inverse of a is -a.
a + (-a) = 0A pair consisting of one positive token (+1) and one negative token (-1) whose combined net value is 0.
(+1) + (-1) = 0In the product a × b, a is the multiplier (number of groups) and b is the multiplicand (size of each group).
Multiplier × Multiplicand = ProductThe number 1, which leaves any integer unchanged when multiplied.
1 × a = a × 1 = aChanging the order of the two factors does not alter their product.
a × b = b × aChanging the grouping of three or more factors does not alter the product.
a × (b × c) = (a × b) × cMultiplication distributes over addition (and subtraction) across parentheses.
a × (b + c) = (a × b) + (a × c)In division a ÷ b = c, a is the dividend, b is the divisor (b ≠ 0), and c is the quotient.
Dividend ÷ Divisor = QuotientAncient Indian rules for signed arithmetic using dhana (fortune/positive) and ṛṇa (debt/negative).
debt × debt = fortuneA traditional strategy and movement game from Assam and Nagaland played on a 16-square board with diagonals.
2 players × 9 coinsA product of non-zero integers is positive if the number of negative factors is even.
(-1)^{2k} = 1A product of non-zero integers is negative if the number of negative factors is odd.
(-1)^{2k+1} = -1Dividing any number by 0 has no mathematical meaning because no quotient satisfies q × 0 = a (when a ≠ 0).
a ÷ 0 is undefinedFrequently Asked Questions (FAQ)
What are integers and how are they represented?
Integers are the set of positive whole numbers, negative whole numbers, and zero (... -3, -2, -1, 0, 1, 2, 3 ...). On a horizontal number line, positive integers lie to the right of zero and negative integers lie to the left.
What is an additive inverse?
The additive inverse of an integer a is -a. When an integer and its additive inverse are added, their sum is always zero (e.g., 18 + (-18) = 0). Subtracting an integer is mathematically identical to adding its additive inverse: a - b = a + (-b).
What is a zero pair in the token model?
A zero pair consists of one positive token (+1, green) and one negative token (-1, red). Because +1 and -1 cancel each other out, adding or removing zero pairs does not change the net value of a collection.
Why is the product of two positive integers positive?
Multiplying a positive integer by another positive integer represents repeated addition of positive groups. For example, 4 × 2 means placing 2 positive tokens 4 times, yielding 8 positive tokens (+8).
Why is negative × negative equal to positive?
In the token model, a negative multiplier means removing tokens from an empty bag. To evaluate (-4) × (-2), we remove 2 negative tokens 4 times. Since the bag is initially empty, we first insert zero pairs. Removing the 8 negative tokens leaves behind 8 positive tokens (+8). Additionally, continuing arithmetic patterns like 2 × (-3) = -6, 1 × (-3) = -3, 0 × (-3) = 0 requires that (-1) × (-3) = +3 to maintain the constant step size.
What happens when one integer is positive and the other is negative?
The product of a positive integer and a negative integer is always negative. For example, 4 × (-2) = -8 (placing 2 negative tokens 4 times), and (-4) × 2 = -8 (removing 2 positive tokens 4 times).
What is the rule for the sign of an integer quotient?
Division follows the exact same sign rules as multiplication because division is the inverse of multiplication: Positive ÷ Positive = Positive, Negative ÷ Negative = Positive, Positive ÷ Negative = Negative, and Negative ÷ Positive = Negative.
Why is division by zero not defined?
Division a ÷ b asks: 'What number multiplied by b gives a?' If b = 0 and a = 12, we would need 0 × ? = 12, which is impossible because zero multiplied by any number is 0. If a = 0 and b = 0, every number satisfies 0 × ? = 0, so there is no unique answer. Hence, division by zero is undefined.
What is the effect of multiplying an integer by -1?
Multiplying any integer a by -1 yields its additive inverse: (-1) × a = -a. The magnitude remains unchanged while the sign flips: (-1) × 5 = -5 and (-1) × (-5) = +5.
What is the commutative property of multiplication?
The commutative property states that the order of factors does not affect the product: a × b = b × a for all integers a and b. For example, 3 × (-4) = (-4) × 3 = -12.
What is the associative property of multiplication?
The associative property states that grouping factors differently does not change the product: a × (b × c) = (a × b) × c. For example, 5 × ((-3) × 4) = (5 × (-3)) × 4 = -60.
What is the distributive property of multiplication over addition?
The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend separately and then adding: a × (b + c) = (a × b) + (a × c). For example, 4 × (2 + (-3)) = 4 × 2 + 4 × (-3) = 8 + (-12) = -4.
How do you quickly determine the sign of a product of many integers?
Count the number of negative factors. If there are an EVEN number of negative factors, the product is POSITIVE. If there are an ODD number of negative factors, the product is NEGATIVE. If any factor is zero, the product is immediately 0.
Who was Brahmagupta and what was his contribution to integer arithmetic?
Brahmagupta was an Indian mathematician and astronomer who in 628 CE wrote the Brāhmasphuṭasiddhānta. He gave the world's first formal rules for operations with signed numbers, using the terms dhana (fortune) for positive numbers and ṛṇa (debt) for negative numbers.
What is Terhüchü and how does it connect to mathematics?
Terhüchü is a traditional game from Assam and Nagaland played by 2 players with 9 coins each on a 16-square diagonal board. It involves coordinate reasoning, direction, and tactical spatial problem-solving, concluding the chapter with cultural game-based learning.
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