Operations with Integers | CBSE Class 7 Maths Notes

This note covers integers, signed movements, addition and subtraction with tokens, additive inverses, multiplication and division sign rules, multiplication patterns, commutative, associative and distributive properties, and calculations involving marks, temperatures, profit, loss and positions above or below ground.

How do integers describe numbers and movements?

Integers are positive whole numbers, negative whole numbers and zero. A positive number is greater than zero; a negative number is less than zero. On a number line, positive integers lie to the right of zero and negative integers lie to its left.

The symbol + means addition, − means subtraction or a negative sign, and = means that two quantities are equal. The sum is the result of addition. The difference is the first number minus the second, so the order matters.

How does the number puzzle work?

Rakesh asks for two numbers with sum 25 and difference 11. The numbers 18 and 7 satisfy both conditions: 18 + 7 = 25 and 18 − 7 = 11. Checking just the sum would leave other pairs possible.

For the second puzzle, the sum stays 25 but the difference becomes −11. Swapping the numbers gives first number 7 and second number 18. Their sum remains 25, while 7 − 18 = −11. The order is therefore part of the answer.

How are direction and magnitude recorded?

Magnitude means the size of a movement without its direction. For the carrom coin, rightward movement is positive and leftward movement is negative. A movement of −4 has magnitude 4 and direction leftwards. Its negative sign does not mean a negative distance travelled.

Let a denote the first signed movement, b the second signed movement and P the final position of a coin starting at zero. Then P = a + b. Letters here stand for numbers; the sign of each movement records its direction.

Worked example 1. A carrom coin starts at zero, moves 5 units rightwards, then 7 units leftwards. Find its final position.

Answer: The signed movements are 5 and −7. Thus P = 5 + (−7) = −2. The coin ends 2 units to the left of zero.

For two rightward strikes of 4 and 3 units, the same addition gives 4 + 3 = 7. Signed numbers let one addition rule handle movements in the same direction or in opposite directions.

How do tokens explain subtraction and additive inverses?

A token is a counter used to represent a number. A green token represents +1 and a red token represents −1. One green token together with one red token is a zero pair: their values cancel and their total is zero.

Adding zero pairs changes the tokens available without changing the number represented. This is useful when subtraction asks us to remove more positive or negative tokens than are currently present. We first add enough zero pairs, then remove the required tokens.

How can 18 positives be removed from 7 positives?

Worked example 2. Find (+7) − (+18) using tokens.

Answer: Begin with 7 positive tokens. Add 11 zero pairs, giving 18 positive tokens and 11 negative tokens. Remove the 18 positives. The 11 negatives remain, so 7 − 18 = −11.

The first row shows seven green positive tokens. The next arrangement has eighteen green positives and eleven red negatives. The eighteen positives are crossed out in the last arrangement, leaving eleven negatives.

Reference: NCERT Class 7, page 28, unnumbered diagram

What is an additive inverse?

Definition: The additive inverse of an integer is the number that adds to it to make zero. For an integer a, its additive inverse is written −a.

The additive inverse of 18 is −18. The additive inverse of −18 is 18, written −(−18) = 18. Here the outer minus means taking the additive inverse; it does not force the final value to be negative.

Subtracting a number gives the same result as adding its additive inverse. Therefore, 7 − 18 = 7 + (−18). Similarly, 4 − (−12) = 4 + 12. Brackets group a signed number so its sign is not confused with the subtraction operation.

In the second calculation, removing negative tokens leaves an excess of positive tokens. The token method explains why subtraction of a negative becomes addition of the corresponding positive number. It is a rule about subtraction, distinct from multiplication sign rules.

How does multiplication work when the multiplier is positive?

The symbol × means multiplication. In 4 × 2 = 8, the first number, 4, is the multiplier; the second number, 2, is the multiplicand; and the result, 8, is the product. The two numbers multiplied are also called factors.

For each new token multiplication, begin with an empty bag. A positive multiplier tells us to put tokens into the bag. The multiplicand tells us whether each group contains positive tokens or negative tokens, and how many it contains.

How is repeated addition represented?

For 4 × 2, put two positive tokens into the bag four times. This gives eight positive tokens, so 4 × 2 = 8. Multiplication combines equal groups: the four groups each represent the same number, 2.

Worked example 3. Use tokens to calculate 4 × (−2).

Answer: Put two negative tokens into an empty bag four times. The bag now holds eight negatives. Thus 4 × (−2) = −8, matching the addition (−2) + (−2) + (−2) + (−2).

One arrangement shows four groups of two green positive tokens. Another shows four groups of two red negative tokens. Their accompanying equations give products 8 and −8 respectively.

Reference: NCERT Class 7, page 29, unnumbered diagrams

Does the appearance of a token set change its value?

The same integer can be represented by different collections of tokens because adding a zero pair does not change the total. When different token sets each represent −2, placing any one of those sets into the bag four times still represents four copies of −2.

The number represented is what matters. Count the remaining positives or negatives after cancelling zero pairs, rather than treating the total number of physical tokens as the integer. A mixed collection can represent a negative number even though it contains some positive tokens.

This interpretation connects multiplication to addition. It also separates two jobs: the positive multiplier determines how many groups are added, while the sign of the multiplicand determines the value represented by each group.

Why does a negative multiplier mean removing tokens?

A negative multiplier tells us to remove tokens from the bag. We still begin with an empty bag. When the required tokens are absent, add zero pairs first. Because each pair represents zero, this preparation preserves the starting value.

What happens when positive tokens are removed?

Worked example 4. Model (−4) × 2 with tokens.

Answer: Remove two positive tokens four times. For each removal, place two zero pairs in the empty or existing bag and take out the two positives. After four removals, eight negative tokens remain. Therefore, (−4) × 2 = −8.

The multiplicand 2 determines that positive tokens must be removed. The multiplier −4 determines the removal action and its four repetitions. Confusing these roles can lead to choosing the wrong colour of token or assigning the wrong sign to the product.

What happens when negative tokens are removed?

Worked example 5. Model (−4) × (−2) with tokens.

Answer: Remove two negative tokens four times. Before each removal, add two zero pairs. Each removal leaves two positive tokens, so eight positives remain after four repetitions. Hence (−4) × (−2) = 8.

The upper arrangement crosses out green positive tokens in four groups, leaving red negatives. The lower arrangement crosses out red negative tokens in four groups, leaving green positives. The products shown are −8 and 8.

Reference: NCERT Class 7, page 30, unnumbered diagrams

Removing a number is equivalent to adding its additive inverse. Thus removing two positives four times can also be represented by adding two negatives four times. This explains why (−4) × 2 and 4 × (−2) both give −8.

The signs in multiplication must be interpreted together. Two negative factors produce a positive product because removing negative groups leaves positive tokens. This explanation does not say that adding two negatives gives a positive sum; addition and multiplication are different operations.

What sign rules and patterns govern integer multiplication?

For two non-zero integers, meaning integers other than zero, matching signs give a positive product and different signs give a negative product. Multiply the magnitudes to find the magnitude of the answer, then use the two signs to determine the product’s sign.

Changing the signs of the factors does not change the magnitude of their product when their individual magnitudes remain the same. The four token calculations involving magnitudes 4 and 2 all have product magnitude 8, although their signs differ.

How do patterns extend across zero?

Keep the multiplicand at 3 while reducing the multiplier by one at each step. The products decrease by 3. Keeping the multiplicand at −3 produces the corresponding sequence shown alongside it, whose products rise by 3 at each step.

The zero row joins the positive and negative multipliers. Continuing the same pattern below zero gives negative products in the multiplication-by-3 column and positive products in the multiplication-by-−3 column. This agrees with the removal-of-tokens interpretation.

Worked example 6. Given 123 × 456 = 56088, find (−123) × 456, (−123) × (−456) and 123 × (−456).

Answer: All three products have magnitude 56088. Their signs give −56088, 56088 and −56088 respectively. The given multiplication supplies the magnitude, so no fresh long multiplication is needed.

A sign check and a magnitude calculation answer separate questions. Keeping them separate is especially useful for large numbers, where repeating a multiplication can introduce unnecessary arithmetic errors.

What do multiplication by one and changing factor order show?

What happens when the multiplier is 1 or −1?

Multiplication by 1 preserves an integer: 1 × a = a, where a is any integer. In the token model, the group representing the integer is placed in the bag once. For a = −5, the resulting five negatives still represent −5.

Multiplication by −1 gives the additive inverse: (−1) × a = −a. If a is positive, the result is negative; if a is negative, the result is positive. Its magnitude is unchanged. This connects the multiplication rule to the earlier idea of opposites.

The expression −a means the additive inverse of the number represented by a. It should not be read as a promise that the result is negative. For example, when a itself is negative, taking its additive inverse gives a positive value.

Property: Commutativity of multiplication

Commutativity means that exchanging the order of two factors leaves the product unchanged. For any integers a and b, a × b = b × a. This applies when the signs match and when they differ.

The magnitude stays the same because multiplying the corresponding positive magnitudes is commutative. The sign also stays the same: swapping two matching signs still leaves matching signs, and swapping different signs still leaves different signs.

This property concerns exchanging factors in a multiplication. It does not justify exchanging the numbers in a subtraction. The opening puzzle already shows the distinction: 18 − 7 and 7 − 18 have opposite results, although the sums of those pairs agree.

When explaining commutativity, mention both magnitude and sign. Showing that the unsigned multiplication is unchanged establishes the magnitude, while the sign rules explain why the signed answer is also unchanged.

How is integer division linked to multiplication?

The symbol ÷ means division. The number being divided is the dividend, the number it is divided by is the divisor, and the answer is the quotient. We can find a quotient by identifying a missing factor.

For example, asking for (−100) ÷ 25 means asking which number, multiplied by 25, gives −100. This uses the multiplication sign rules already established, rather than requiring a separate token interpretation for every division.

How do missing factors determine the quotient?

Worked example 7. Calculate (−100) ÷ 25 and (−100) ÷ (−4).

Answer: Since 25 × (−4) = −100, the first quotient is −4. Since (−4) × 25 = −100, the second quotient is 25. Each answer is checked by multiplying it by its divisor.

Similarly, (−25) × (−2) = 50 gives 50 ÷ (−25) = −2. Two negative factors multiply to a positive number, so dividing that positive number by one negative factor recovers the other negative factor.

What condition belongs with the division rules?

Let a and b be positive integers, with b ≠ 0. The symbol ≠ means “is not equal to”. The sign rules can be written a ÷ (−b) = −(a ÷ b), (−a) ÷ b = −(a ÷ b), and (−a) ÷ (−b) = a ÷ b.

Note: These division rules require a non-zero divisor. For non-zero dividend and divisor, matching signs give a positive quotient and different signs give a negative quotient.

Use the multiplication check after deciding the quotient: divisor multiplied by quotient must recover the dividend. This checks the sign as well as the numerical value and makes the connection between the two operations explicit.

How can several integer factors be grouped?

An expression is a combination of numbers, operations and, sometimes, letters. Brackets show a group to be evaluated together. In an expression containing several multiplied integers, it is useful to know whether changing that grouping affects the result.

Property: Associativity of multiplication

Associativity means that regrouping three factors leaves their product unchanged. Let a, b and c denote any three integers. Then a × (b × c) = (a × b) × c. The same factors occur in the same order; only their grouping changes.

Worked example 8. Evaluate 5 × (−3) × 4 using different groupings.

Answer: Multiplying the first pair gives (5 × (−3)) × 4 = (−15) × 4 = −60. Multiplying the second pair gives 5 × ((−3) × 4) = 5 × (−12) = −60.

Commutativity also allows the factors 5 and 4 to be multiplied first. Their product is 20, and 20 × (−3) = −60. Thus changing the order and grouping can make a calculation easier while preserving the product.

How can the sign of several factors be checked?

Pairing two negative factors gives a positive product. An even number of negative factors can be divided into such pairs; an odd number leaves one negative factor unpaired. For non-zero factors, this gives a positive or negative product respectively.

The repeated multiplication (−1) × (−1) gives 1. Multiplying that result by another −1 gives −1; multiplying once more by −1 gives 1. The sign alternates because each new multiplication by −1 takes the additive inverse of the previous result.

This sign-counting method is a consequence of the two-factor sign rules. It helps check a calculation but does not replace finding the magnitude. The worked expression has one negative factor, so its negative result agrees with the sign count.

Regrouping and changing order are related but different actions. Associativity concerns brackets, while commutativity concerns exchanging factors. Stating the correct property makes the explanation of a simplified multiplication clear.

How does multiplication distribute over addition?

Property: Distributivity over addition

Distributivity means multiplying a sum by multiplying each part and then adding the products. For any integers a, b and c, a × (b + c) = (a × b) + (a × c). Each term inside the brackets receives the outside factor.

For 5 × (4 + (−2)), we can first add inside the brackets, or multiply 5 by 4 and by −2 separately. Both approaches describe the same total. Keeping the negative sign attached to −2 is essential in the second approach.

Worked example 9. Evaluate 5 × (4 + (−2)) in two ways.

Answer: First add inside the brackets: 4 + (−2) = 2, so 5 × 2 = 10. Alternatively, distribute: 5 × 4 + 5 × (−2) = 20 + (−10) = 10.

How do rectangular token arrangements explain this?

Four rows each contain two green positive tokens and three red negative tokens. The whole arrangement is labelled 4 × (2 + (−3)); the two parts are labelled 4 × 2 and 4 × (−3).

Reference: NCERT Class 7, page 41, unnumbered diagram

Count the arrangement as four copies of the combined group, or count the green and red parts separately and add their values. These are two ways of describing the same tokens, so they give the same integer.

The outside factor must multiply every part of the sum. Multiplying only the positive term changes the expression. Distributivity remains applicable when a term is negative, provided its sign is carried through its multiplication.

This property complements the other two multiplication properties. Commutativity changes order, associativity changes grouping among factors, and distributivity connects a product involving a sum to a sum of products. Identify which structure is present before selecting a property.

After using distributivity, carry out both multiplications before combining their signed results. In the worked example, the second product is −10; adding it to 20 gives 10. This final addition is as important as the initial distribution.

How are integer operations used in practical problems?

A signed model assigns positive and negative numbers to opposite effects. Correct answers can contribute positive marks and wrong answers negative marks. Positions above ground can be positive and positions below ground negative. A profit is a gain; a loss is a negative contribution.

How are positive and negative marks combined?

Worked example 10. An exam contains 50 questions. A correct answer earns 5 marks and a wrong answer earns −2 marks. Mala answers 30 correctly and 20 wrongly. Find her total.

Answer: Correct answers contribute 30 × 5 = 150 marks. Wrong answers contribute 20 × (−2) = −40 marks. The total is 150 + (−40) = 110 marks.

Multiplication first combines repeated contributions of the same kind. Addition then combines the two signed totals. The negative contribution must remain negative when it is added to the positive contribution.

How is a final position calculated?

Worked example 11. An elevator starts 15 metres above ground and descends at 3 metres per minute for 45 minutes. Find its final position, taking ground level as zero.

Answer: Its signed movement is 45 × (−3) = −135 metres. Add this to the starting position: 15 + (−135) = −120 metres. The elevator finishes 120 metres below ground.

The abbreviation m means metres, a unit of length. Distance travelled describes how far the elevator moves; signed position describes where it ends relative to ground. A negative final position therefore means below ground, not a negative distance travelled.

Starting from ground instead, the elevator descends 180 metres in one hour, which is 60 minutes. The signed calculation is 60 × (−3) = −180, giving a position 180 metres below ground.

How do profit and loss combine?

The symbol ₹ means rupees. A cement company gains ₹8 for each bag of white cement sold and loses ₹5 for each bag of grey cement sold. Represent these contributions as +8 and −5 rupees per bag.

Worked example 12. The company sells 3,000 bags of white cement and 5,000 bags of grey cement. Using the stated gain and loss per bag, find the overall result.

Answer: White cement contributes 3,000 × 8 = ₹24,000. Grey cement contributes 5,000 × (−5) = −₹25,000. Combining them gives −₹1,000, so the company makes an overall loss of ₹1,000.

Each contribution must include both the number of bags and its signed amount per bag. The negative combined total identifies a loss, just as a negative position identifies a location below the chosen zero level.

How should a word problem be organised?

1. Identify the starting value and decide what positive and negative values represent.
2. Write each repeated contribution as a multiplication using its signed value.
3. Combine contributions with the starting value or other signed totals.
4. Interpret the answer in words, including its units and the meaning of its sign.

A calculation is incomplete as an explanation until the signed result is interpreted. For marks, it is the total score; for the elevator, it is a position relative to ground. The same arithmetic structure can represent these different situations.

Glossary

• Integer — A positive whole number, a negative whole number, or zero.
• Magnitude — The size of a number or movement without its positive or negative direction.
• Sum — The result obtained when two or more numbers are added together.
• Difference — The result obtained by subtracting the second number from the first.
• Zero pair — One positive token and one negative token whose combined value is zero.
• Additive inverse — The number that combines with a given integer by addition to make zero.
• Multiplier — The first factor, specifying repetitions and addition or removal in the token model.
• Multiplicand — The number multiplied, represented by the value of each token group.
• Product — The result obtained when two or more numbers are multiplied together.
• Dividend — The number being divided by another number in a division calculation.
• Divisor — The non-zero number by which the dividend is divided in the calculation.
• Quotient — The result obtained when a dividend is divided by a non-zero divisor.
• Commutativity — The property that exchanging the order of two factors leaves their product unchanged.
• Associativity — The property that changing the grouping of three factors leaves their product unchanged.
• Distributivity — The property allowing multiplication of a sum by multiplying each part and adding.

Common errors and misconceptions

• Misconception: Reversing two numbers leaves their difference unchanged. Correct: 18 − 7 = 11, whereas 7 − 18 = −11. The order matters in subtraction.
• Misconception: Adding zero pairs changes the number represented. Correct: Each pair contains +1 and −1, so its net contribution is zero.
• Misconception: A negative multiplier means putting negative tokens into the bag. Correct: It means removing groups; the multiplicand determines which tokens to remove.
• Misconception: Two negative factors have a negative product. Correct: Their product is positive, as (−4) × (−2) = 8 shows.
• Misconception: The expression −a must be negative. Correct: It is the additive inverse of a; for example, −(−18) = 18.
• Misconception: The outside factor multiplies only the first term in brackets. Correct: Distributivity requires it to multiply every term inside the sum.
• Misconception: The elevator’s movement alone gives its final position. Correct: Add the signed movement to its starting position, which may already be above or below ground.

Exam-style questions with model answers

Q1. Find two numbers whose sum is 25 and whose difference, first minus second, is −11. Check both conditions. [2 marks]

1. The first number is 7 and the second is 18. Their sum is 7 + 18 = 25.
2. The difference in the required order is 7 − 18 = −11, so this ordered pair satisfies both conditions.

Q2. Explain with positive and negative tokens why 7 − 18 = −11. [3 marks]

1. Start with seven positive tokens representing 7. Eighteen positive tokens must be removed, but the original collection does not contain enough.
2. Add eleven zero pairs. The collection now has eighteen positive tokens and eleven negative tokens, while its value remains 7.
3. Remove all eighteen positive tokens. Eleven negative tokens remain, representing −11, so the subtraction gives 7 − 18 = −11.

Q3. Given 123 × 456 = 56088, find (−123) × 456, (−123) × (−456) and 123 × (−456), explaining each sign. [3 marks]

1. (−123) × 456 = −56088. The factors have different signs, so the product is negative; its magnitude is the given product 56088.
2. (−123) × (−456) = 56088. Both factors are negative, so their product is positive, with the same magnitude.
3. 123 × (−456) = −56088. One factor is positive and the other negative, giving a negative product without changing the magnitude.

Q4. Find (−100) ÷ 25 and verify the answer by multiplication. [2 marks]

1. The quotient is −4 because a negative dividend divided by a positive divisor gives a negative quotient.
2. The multiplication check is 25 × (−4) = −100, which recovers the given dividend and verifies the result.

Q5. An elevator starts 15 metres above ground and descends 3 metres per minute for 45 minutes. Taking ground as zero and upward as positive, find and interpret its final position. [5 marks]

1. The starting position is +15 metres because the elevator begins above the ground. The ground is the zero position used throughout this calculation.
2. Descending is movement in the negative direction, so the signed movement each minute is −3 metres. The time interval is 45 minutes.
3. The total signed movement is 45 × (−3) = −135 metres. The elevator therefore travels 135 metres downwards during the stated time.
4. Add the signed movement to the starting position: final position = 15 + (−135) = −120 metres. The starting height must be included.
5. The negative final position means that the elevator ends below ground. Its final location is 120 metres below the ground level.

Q6. An exam has 50 questions, with 5 marks per correct answer and −2 per wrong answer. Mala gives 30 correct and 20 wrong answers. Find her total, then the maximum and minimum scores if all 50 are answered. [5 marks]

1. The thirty correct answers contribute 30 × 5 = 150 marks. This part of the score is positive because correct answers earn marks.
2. The twenty wrong answers contribute 20 × (−2) = −40 marks. This negative contribution must be combined with the positive contribution.
3. Mala’s total score is 150 + (−40) = 110 marks. Her thirty correct and twenty wrong answers account for all fifty questions.
4. The maximum score occurs when all fifty answers are correct. It is 50 × 5 = 250 marks, with no negative contribution.
5. The minimum score occurs when all fifty answers are wrong. It is 50 × (−2) = −100 marks, with no positive contribution.

Q7. Evaluate 5 × (4 + (−2)) both directly and by using distributivity. State the property used. [3 marks]

1. Directly, calculate the bracket first: 4 + (−2) = 2. Multiplying this result by 5 gives 5 × 2 = 10.
2. Distributing gives 5 × 4 + 5 × (−2) = 20 + (−10) = 10, the same result as the direct calculation.
3. The distributive property states that multiplying a sum equals multiplying each part by the outside factor and adding the resulting products.

Q8. A room starts at 32°C and its temperature falls by 5°C every hour for 10 hours. Here °C means degrees Celsius. Find the final temperature and explain the signs. [3 marks]

1. The starting temperature is +32°C. Since the temperature falls, the signed change each hour is −5°C rather than +5°C.
2. Over ten hours, the total change is 10 × (−5) = −50°C. The negative sign represents the total decrease in temperature.
3. Add this change to the starting temperature: 32 + (−50) = −18°C. The final room temperature is therefore 18 degrees Celsius below zero.

Key takeaways

• Signed integers record magnitude and direction together, allowing rightward and leftward movements to be combined using addition.
• Zero pairs let us add positive and negative tokens without changing the integer represented by the collection.
• Subtracting an integer is equivalent to adding its additive inverse, including when the integer subtracted is negative.
• Two non-zero factors with matching signs give a positive product; factors with different signs give a negative product.
• Division can be checked through multiplication: the divisor multiplied by the quotient must recover the dividend.
• Commutativity changes factor order, associativity changes factor grouping, and distributivity multiplies each part of a sum.
• Multiplication by one preserves an integer, while multiplication by negative one gives that integer’s additive inverse.
• In practical problems, combine the starting value with the signed change and explain the resulting sign in context.

Test yourself

What is the additive inverse of −18?

It is 18, because adding 18 to −18 gives zero.

What does a carrom coin movement of −4 represent?

It represents a movement of magnitude 4 units in the leftward direction.

Why can zero pairs be added before removing tokens?

Each pair has total value zero, so adding it does not change the represented integer.

What is (−4) × (−2), and why is its sign positive?

The product is 8. Both factors are negative, so the product is positive.

What property allows 3 × (−4) to be written as (−4) × 3?

Commutativity allows the two factors to exchange places without changing their product.

Which multiplication checks 50 ÷ (−25) = −2?

The check is (−25) × (−2) = 50, recovering the dividend.

How are commutativity and associativity different?

Commutativity changes the order of factors. Associativity changes their grouping without changing their product.

In the elevator example, what does a final position of −120 metres mean?

It means the elevator is 120 metres below the ground, which is the zero position.