The Other Side of Zero

Why This Matters

Think about the numbers you already know. You count 1, 2, 3, 4, and so on. You learnt about 0, which means “nothing”. You even learnt about fractions, the numbers that sit between the counting numbers.

But here is a question. Can a number be less than zero?

At first this sounds silly. How can you have less than nothing? But you meet such numbers all the time in real life.

On a cold winter day in Shimla, the news says the temperature is “minus 4 degrees”. That is colder than zero.

You owe your friend 20 rupees. You don’t have 20 rupees. You have less than zero rupees — you have a debt.

A mall has a basement parking. The ground floor is “0”. The parking is below the ground. So we call it Floor “minus 1”.

All of these need a new kind of number. A number that lives on the other side of zero. This chapter is about those numbers. By the end, you will read them, compare them, put them in order, and add and subtract them — using simple pictures.

The Big Idea

Zero is not the end of the number line. It is the middle. To the right of 0 are the positive numbers (1, 2, 3, …) — the ones you already know. To the left of 0 are the negative numbers (−1, −2, −3, …) — the new ones. Put them together with 0, and you get the integers. They stretch forever in both directions. As you move right, numbers get bigger. As you move left, they get smaller. That one simple rule — left is smaller, right is bigger — explains everything else in this chapter.

Let’s Break It Down

Numbers below zero (where we see them)

A negative number is a number less than zero. We write it with a small minus sign in front, like −3 (we say “minus three” or “negative three”).

A positive number is a number greater than zero, like 3. (We usually don’t bother writing the plus sign. So 3 means the same as +3.)

And 0 is special. It is neither positive nor negative. It sits right in the middle.

Where do we actually meet negative numbers? Everywhere. Here are three everyday places.

1. Temperature. When it is very cold, the temperature drops below 0 degrees. The thermometer in Figure 10.1 shows a reading of −4 degrees. That means 4 degrees below zero — colder than the freezing point of water.

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2. Money you owe. Imagine you have 50 rupees. That is +50. Now you spend it all and borrow 20 rupees more from a friend. You don’t have any money — in fact you are 20 rupees in debt. We can write your money as −20. The minus sign says “this much is owed”.

3. Floors below the ground. Many buildings have floors below the ground, like a basement or parking. The ground floor is Floor 0. The floors above are +1, +2, +3. The floors below the ground are −1, −2, −3. Figure 10.2 shows such a building.

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So a negative number is not a strange, useless idea. It is just a clear way to say “below zero”, “owed”, or “lower down”.

The number line goes both ways — integers

Before we go on, let us quickly remember the number line from earlier classes.

Now we complete the picture. We let the line keep going past 0, to the left. Each step to the left gives us the next negative number: −1, −2, −3, and so on. This full line is shown in Figure 10.3.

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All these numbers together — the positive numbers, the negative numbers, and zero — have a single name: integers.

So the integers are:

… −4, −3, −2, −1, 0, 1, 2, 3, 4 …

The three dots on each side mean “they keep going forever”. Just as the positive numbers never run out, the negative numbers never run out either.

Notice two simple things from the picture. Every positive number is to the right of 0. Every negative number is to the left of 0. So every negative number is less than 0, and every positive number is greater than 0.

Comparing & ordering integers (which is bigger?)

Now the big question. Between two integers, which one is bigger?

There is one golden rule, and it comes straight from the number line:

On the number line, the number on the right is always the bigger one. The number on the left is always the smaller one.

This is easy to believe for positive numbers. 5 is to the right of 2, and indeed 5 > 2. (The sign > means “is greater than”, and < means “is less than”.)

But here is the part that surprises many students. Look at −5 and −2 on the number line in Figure 10.3.

Which one is on the left? −5 is on the left (it is further from 0, deeper into the negatives). −2 is on its right.

So by the rule, −5 is smaller than −2. We write −5 < −2.

Wait — but 5 is bigger than 2! So shouldn’t −5 be bigger than −2? No. And it really helps to see why.

Think about the building in Figure 10.2 again. Floor −5 would be five floors below the ground. Floor −2 is only two floors below. Which floor is lower down? Floor −5, of course — it is deeper. A lower floor is a smaller number. So Floor −5 is smaller than Floor −2.

Or think about cold. −5 degrees is colder than −2 degrees. Colder means a smaller temperature. So −5 < −2.

Or think about money. Owing 5 rupees (−5) is worse than owing 2 rupees (−2). You have less. So −5 < −2.

Here is the key idea to remember: for negative numbers, the bigger the digit, the smaller the number. The minus sign flips things around. The further left you go, the smaller you get.

Once you can compare two integers, you can put a whole list in order. To order them from smallest to biggest, just read them off the number line from left to right.

For example, order these: 3, −4, 0, −1, 2.

Find each on the number line and read left to right: −4, −1, 0, 2, 3. The smallest is −4 (furthest left), the biggest is 3 (furthest right).

Opposites and zero

Here is a beautiful idea. Every number has an opposite.

The opposite of a number is the number that is the same distance from 0, but on the other side.

The opposite of 3 is −3. (Both are 3 steps from 0 — one to the right, one to the left.)

The opposite of −5 is 5. The opposite of 7 is −7. And the opposite of 0 is just 0 itself.

Why do opposites matter so much? Because a number and its opposite always add up to 0.

Picture the building lift. You press +3 to go up 3 floors. Then you press −3 to come back down 3 floors. Where are you? Right back where you started — Floor 0. So:

3 + (−3) = 0

This is why −3 is called the opposite (or inverse) of 3. They cancel each other out perfectly.

We can show this cancelling with tokens. Imagine a green token worth +1 and a red token worth −1. Put one green and one red together. The +1 and the −1 cancel. Together they are worth nothing — they make 0. We call this a zero pair. Figure 10.4 shows it.

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This zero-pair idea is the secret behind adding integers, which we look at next.

Adding integers on the number line

To add integers, we just walk on the number line. Here is the one rule:

Start at the first number. Then move by the second number. Adding a positive number means move right. Adding a negative number means move left. Where you land is the answer.

Why does adding a negative mean moving left? Because a negative number means “go backward” or “go down”. On our line, the negative direction is the left direction. So adding −5 means take 5 steps in the negative (left) direction.

Let us do the famous example: 3 + (−5).

Start at 3. Now add −5, so move 5 steps to the left. Count along: from 3 you reach 2, then 1, then 0, then −1, then −2. You land on −2. Figure 10.5 shows the whole journey.

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See how the line makes it obvious? You started at +3, but you moved a bigger amount (5) in the negative direction. So you crossed over 0 and ended up negative. That is exactly why 3 + (−5) = −2.

Let us try a few more on the line.

Subtracting integers (add the opposite)

Subtracting is just adding’s twin. The simplest way to subtract any integer is this neat trick:

To subtract a number, change it to its opposite and add instead. Subtracting +2 becomes adding −2. Subtracting −2 becomes adding +2.

Why is this true? Think about the building lift again. Subtracting means “take away” or “undo a movement”. If someone went down 2 floors and you want to undo that, you must go up 2 floors. Taking away a “−2” move is the same as adding a “+2” move. They reach the same place.

Look at Figure 10.6. It shows that 5 − (−2) lands in exactly the same spot as 5 + 2.

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Let us use the rule on a normal-looking subtraction too, so you see it always works.

Example: 4 − 7. Here we subtract +7, so we add −7 instead: 4 + (−7). Start at 4, move 7 steps left: 3, 2, 1, 0, −1, −2, −3. We land on −3. So 4 − 7 = −3. (This is why you can subtract a bigger number from a smaller one now — the answer is just negative.)

Common Mistakes

These are the slips almost every student makes at first. Spotting them now will save you a lot of marks later.

Quick Check

Try these to test yourself. Pick an answer, then read the explanation.

Practice Problems

Try each one yourself first. Use the number line if it helps. Then reveal the full solution.

Easy

Medium

Challenge

Summary

You can now do all of these. Try saying each one out loud in your own words.

• A negative number is less than 0. We write it with a minus sign, like −3. We meet them as temperatures below zero, money owed, and floors below the ground.
• 0 is neither positive nor negative — it sits in the middle.
• The integers are the negatives, zero, and the positives all together: … −3, −2, −1, 0, 1, 2, 3 … They go on forever in both directions.
• On the number line, right means bigger and left means smaller. So every negative number is less than 0, and less than every positive number.
• For negative numbers, the bigger the digit, the smaller the number — so −5 < −2.
• The opposite of a number is the same distance from 0 on the other side. A number and its opposite always add to 0 (like 3 + (−3) = 0).
• To add an integer, walk the number line: add a positive → move right; add a negative → move left. Where you land is the answer.
• To subtract an integer, change it to its opposite and add instead — so 5 − (−2) = 5 + 2 = 7.

What’s Next

That’s it — you have reached the end of the Class 6 Maths journey, and what a journey it has been. You started by spotting patterns in numbers and shapes, played with how numbers are built, measured angles, read data, hunted for prime numbers, and now you have crossed over to the other side of zero. Negative numbers used to sound impossible. Now you can read them, compare them, and add and subtract them with a simple picture in your head.

Be proud of how far you have come. Every idea here — the number line, opposites, moving left and right — comes back again in Class 7 and beyond, when you multiply and divide integers and meet even more kinds of numbers. You are ready for all of it.

Want to revisit anything? Head back to the chapter list and pick any chapter — a quick reread is one of the smartest things a learner can do. Well done, and keep going!