Operations with Integers

Why This Matters

Numbers are not always about “how many”. Sometimes they are about “which way”.

Think about a cold morning in the hills. The news says it is “−6 °C”. That little minus sign means the temperature went below zero. Or think about money. If you have ₹50, that is +50. But if you owe your friend ₹50, that is a debt — we write it as −50.

A lift in a building can go up to floor +3 or down to the basement at floor −2. A submarine sits below the sea at −40 metres. A cricket run rate can drop. A bank balance can go negative.

In all these cases, the minus sign carries real meaning. It tells us the direction — down, below, owed, lost. The numbers with a minus sign are called negative numbers.

In Class 6 you met these negative numbers and learned to add and subtract them. This chapter takes the next big step. You will learn to multiply and divide integers too. And, most importantly, you will learn why the famous sign rules work — especially the strange-looking one: why does a negative times a negative give a positive? By the end, you will not just remember the rules. You will understand them.

The Big Idea

An integer is a whole number that can be positive, negative, or zero: …, −3, −2, −1, 0, 1, 2, 3, … Every integer has a home on the number line. Once we picture them on that line, every operation has a clear meaning: adding is moving right, subtracting is moving left, and subtracting a negative is the same as adding its positive twin. Multiplying is just repeated adding — and following that idea carefully forces one clear conclusion: same signs make a positive answer, different signs make a negative answer. Nothing here is a trick to memorise. Every rule comes from one simple picture.

Let’s Break It Down

Integers on the number line

Let us first refresh what integers are and where they live.

The picture below sets up the number line we will use again and again, so keep it in mind.

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Adding integers

Adding an integer means moving along the number line.

Here is the simple rule for what each sign tells you to do:

• A positive number means move right (the “warm”, “up”, “gain” direction).
• A negative number means move left (the “cold”, “down”, “loss” direction).

You always start at the first number, then make the move the second number tells you to. Where you land is the answer.

Let us try −3 + 5. We start at −3. The “+5” says move 5 steps to the right. Counting 5 steps right from −3: −2, −1, 0, 1, 2. We land on 2. So −3 + 5 = 2. Figure 2.2 shows this jump.

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What if both moves are in the same direction? Then we just go further that way.

• −4 + (−3): start at −4, then “−3” means move 3 more steps left. We land on −7. So −4 + (−3) = −7. Two leftward moves pile up.
• 6 + 2: start at 6, move 2 right, land on 8. The same as ordinary addition you have always done.

Here is a handy way to think about it without always drawing the line, using money:

• Same signs (both positive, or both negative): add the two amounts and keep the common sign. (−4) + (−3): you owe ₹4, then owe ₹3 more — you owe ₹7 in total, so −7.
• Different signs (one positive, one negative): they fight each other. Find the difference of the two amounts, and keep the sign of the bigger one. −3 + 5: a gain of 5 fights a loss of 3; the gain is bigger by 2, so the answer is +2.

Subtracting integers (= add the opposite)

Subtraction can feel scary with negatives. But there is one golden rule that makes every subtraction easy:

Subtracting a number is the same as adding its opposite.

a − b = a + (−b)

That is it. To subtract, flip the sign of the second number and then just add (which you already know how to do). “Take away” becomes “add the opposite”.

Why is this true? Subtracting means “take away that much”. Taking away a gain is the same as adding a loss. And taking away a loss is the same as adding a gain — if I remove a ₹6 debt from your account, you are ₹6 richer, exactly as if I gave you ₹6.

Let us see the tricky case: 4 − (−6). We are subtracting a negative. By the golden rule, flip the −6 to +6 and add:

4 − (−6) = 4 + 6 = 10.

Figure 2.3 shows both — taking away −6, and adding +6 — landing on the same spot.

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A few more, using the golden rule each time:

• 7 − 18 = 7 + (−18) = −11. (A gain of 7 fights a loss of 18; loss wins by 11.)
• −5 − 3 = −5 + (−3) = −8. (Both moves go left, so they pile up.)
• −5 − (−9) = −5 + 9 = 4. (Flip −9 to +9, then add.)

Multiplying integers & the sign rules

Multiplying is just repeated adding. 4 × 2 means “add 2, four times”: 2 + 2 + 2 + 2 = 8. Easy when everything is positive. Now let us bring in negatives, one case at a time. We will use the token model from Class 6: a green token is +1, a red token is −1, and one green and one red together make zero (they cancel).

Case 1: positive × positive. Just ordinary multiplication. 4 × 3 = 12. Putting 3 positives into a bag, 4 times, gives 12 positives.

Case 2: positive × negative. Here we put in negative tokens. 3 × (−2) means “put 2 red tokens into the bag, 3 times”. That gives 6 red tokens, which is −6. So 3 × (−2) = −6. A positive times a negative is negative. Figure 2.4(a) shows this.

Case 3: negative × negative. This is the famous one. A negative multiplier means “remove tokens instead of putting them in”. So (−3) × (−2) means “remove 2 red tokens from the bag, 3 times”. But the bag is empty! To remove red tokens that are not there, we first drop in some zero-pairs (one green + one red each, which is still zero), then take the red ones out. When the dust settles, the green tokens are the ones left behind. Removing 6 red tokens leaves 6 green tokens, which is +6. So (−3) × (−2) = +6. A negative times a negative is positive. Figure 2.4(b) shows this.

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The token model gives the right answer, but a doubting student might still wonder why removing reds should leave greens. So here is a second, even clearer proof — a pattern. We do not need any new rule; we just trust that multiplication stays steady and consistent.

Look at the times-table for −2, going down one step at a time:

(−2) × 3 = −6 (−2) × 2 = −4 (−2) × 1 = −2 (−2) × 0 = 0 (−2) × (−1) = ? (−2) × (−2) = ?

Each time the second number drops by 1, the answer goes up by 2: from −6 to −4 to −2 to 0… The steps are perfectly even. For this steady pattern to keep going, the next answers must be +2 and +4. There is no other choice. So (−2) × (−1) = +2 and (−2) × (−2) = +4. The pattern itself forces negative × negative to be positive. Figure 2.5 shows this ladder.

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Now let us see why this matters in real life, so it does not feel like an empty rule.

Imagine a tap leaking water from a tank at 2 litres each hour — a loss, so the rate is −2 litres/hour. Where was the water level 3 hours ago — in the past, so −3 hours? Three hours ago the tank had more water, because it has been leaking since. The amount it has lost since then is (−2) × (−3) = +6 litres more than now. A negative rate times a negative time gives a positive (more in the past). The maths matches the real world.

Let us pull all four cases together with a worked example.

There is one neat shortcut hiding here. Multiplying by −1 just flips the sign of a number: (−1) × 7 = −7 and (−1) × (−7) = +7. So −1 acts like a “sign-flipper”.

Dividing integers

Good news: division follows the exact same sign rules as multiplication. You do not have to learn anything new for the signs.

Why are they the same? Because division undoes multiplication. Dividing −100 by 25 is really asking: “What times 25 gives −100?”

25 × (?) = −100.

We know 25 × (−4) = −100. So −100 ÷ 25 = −4. Notice the answer is negative, because we needed a negative number to reach a negative product. Division just reads the multiplication backwards, so the signs behave identically.

Let us check the trickier case: (−100) ÷ (−4) asks “what times −4 gives −100?”. Since (−4) × 25 = −100, the answer is +25. Negative ÷ negative = positive — same as in multiplication.

So the full set of rules:

• (+) ÷ (+) = (+) — same signs → positive. Example: 36 ÷ 9 = 4.
• (−) ÷ (−) = (+) — same signs → positive. Example: (−46) ÷ (−23) = 2.
• (+) ÷ (−) = (−) — different signs → negative. Example: 36 ÷ (−18) = −2.
• (−) ÷ (+) = (−) — different signs → negative. Example: (−16) ÷ 4 = −4.

Now a single table to lock in both multiplication and division, since they share the same sign rules.

Signs of the two numbers	Sign of the answer	× example	÷ example
Both positive (+, +)	Positive (+)	4 × 3 = 12	12 ÷ 3 = 4
Both negative (−, −)	Positive (+)	(−4) × (−3) = 12	(−12) ÷ (−3) = 4
One +, one − (different)	Negative (−)	4 × (−3) = −12	12 ÷ (−3) = −4
One −, one + (different)	Negative (−)	(−4) × 3 = −12	(−12) ÷ 3 = −4

The one-line memory hook: same signs → positive, different signs → negative. Figure 2.6 shows all four cases in one grid.

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Useful properties

Integers behave very politely when you add and multiply them. These nice habits are called properties, and they let you rearrange a calculation to make it easier.

For addition and multiplication (not for subtraction or division):

• Commutative — order does not matter. You can swap the two numbers.
  • 3 + (−7) = (−7) + 3 (both −4).
  • (−5) × 6 = 6 × (−5) (both −30).
• Associative — when there are three numbers, the grouping does not matter. You can decide which pair to combine first.
  • (2 + (−5)) + 4 = 2 + ((−5) + 4) (both 1).
  • (5 × (−3)) × 4 = 5 × ((−3) × 4) (both −60).

These let you pick the easy path. For example, to find (−25) × (−6) × 4, multiply the −25 and 4 first to get −100, then −100 × (−6) = 600. Friendlier than going left to right.

Two more small but handy facts:

• Multiplying by 0 always gives 0: (−9) × 0 = 0. Putting nothing into the bag leaves nothing.
• Multiplying by 1 leaves a number unchanged: 1 × (−5) = −5. And, as we saw, −1 flips the sign: (−1) × (−5) = +5.

Let us use commutativity and associativity to simplify a longer product.

Common Mistakes

These slips are extremely common. Spotting them now will save you marks later.

Quick Check

Test yourself before the practice. Read each question, pick an answer, then check the explanation.

Practice Problems

Try each one yourself first. Then tap to reveal the full solution.

Easy

Medium

Challenge

Summary

• An integer is a positive whole number, a negative whole number, or zero. Each one has a spot on the number line: positives right of zero, negatives left of zero, zero in the middle.
• Adding means moving along the line: a positive number moves you right, a negative number moves you left. Same signs pile up; different signs fight, and you keep the sign of the bigger amount.
• Subtracting is adding the opposite: a − b = a + (−b). Subtracting a negative makes the answer bigger (for example, 4 − (−6) = 4 + 6 = 10).
• Multiplying is repeated adding. The sign rules are: same signs → positive, different signs → negative.
• A negative times a negative is positive — proved both by the token model (removing reds leaves greens) and by the steady +2 pattern that forces it.
• Dividing uses the exact same sign rules as multiplying, because division just undoes multiplication.
• Addition and multiplication are commutative (order does not matter) and associative (grouping does not matter), so you can rearrange a calculation to make it easier. Multiplying by −1 flips the sign; multiplying by 0 gives 0.

What’s Next

You can now move smoothly up, down, and around the number line, and multiply and divide with confidence. Next, in Chapter 11: Finding Common Ground, we look at the factors and multiples that numbers share — ideas like the HCF and LCM — and see how finding “common ground” between numbers helps us compare, simplify, and solve problems neatly.