The Other Side Of Zero

Hello, my dear students! Welcome to today's mathematics lesson. I am so happy to see you all here, ready to learn something new and exciting. Today, we are going to study a very interesting chapter called "The Other Side of Zero" from your NCERT Mathematics book. This chapter is about integers, and I promise you, by the end of this lesson, you will understand numbers in a completely new way!

So students, let's begin our journey into the world of integers.

You remember that when we first started learning mathematics, we learned about counting numbers. These are 1, 2, 3, 4, and so on. We use these numbers every day to count things — like how many apples we have, how many friends came to our birthday party, or how many marks we got in our test.

Then, we learned about another very special number — the number 0, which represents nothing. Zero is a truly remarkable number, and it has a very important history in India. Did you know that the concept of zero was first developed in India? Around the world, we now write numbers using the Indian number system, which uses the digits 0 to 9. With just these ten digits, we can write numbers as large as we want or as small as we want!

After that, we learned about fractions — numbers like 1/2, 3/2, and 13/6. These are numbers that exist between the whole numbers. For example, 1/2 is between 0 and 1.

But here is a very interesting question, students: Are there still more numbers that we haven't discovered yet? We know that 0 comes before 1, and it is less than 1. But are there numbers that come before 0 and are less than 0?

Let me ask you this way. Think about a number line. You have seen a number line before, haven't you? It looks like this: 0——1——2——3——4——5——6——7——8——9——10 and it goes on towards the right. But students, this is actually only half of a number line! This is what we call a number ray — it starts at 0 and goes forever to the right. But what about numbers to the left of 0? Can we complete this to form a true number line that goes in both directions? That is exactly what we are going to investigate in this chapter!

Now, let me introduce you to a wonderful story that will help us understand negative numbers. This is the story of Bela's Building of Fun.

Bela has an ice cream factory, and children love to visit it. To make it even more fun, Bela purchased a multi-storied building and filled it with attractions. She named it Bela's Building of Fun.

But this is no ordinary building, students! Look carefully at this building. Some of the floors are below the ground. What do you find on these floors below the ground? There are shops like the Toy Store and the Video Games shop. And what is on the ground floor? The ground floor is called the Welcome Hall.

Now, there is a lift in this building with two buttons: a plus button (+) to go up and a minus button (–) to go down. To go to the Art Centre from the Welcome Hall, you must press the plus button twice. We say that the button press is plus plus, or simply plus 2. To go down two floors, you must press the minus button twice, which we write as minus 2.

So, if you press plus 1, that means you go up one floor. If you press minus 1, that means you go down one floor.

Let me give you more examples. If you press plus plus plus, that is written as plus 3. If you press minus minus minus minus, that is written as minus 4.

Now, students, here is a question for you. What do you press to go four floors up? That would be plus 4, or plus plus plus plus. And what do you press to go three floors down? That would be minus 3, or minus minus minus.

Now, let's number the floors in the Building of Fun. The entry to the building is at the ground floor level, and we call this the Welcome Hall. Starting from the ground floor, if you press plus 1, you reach the Food Court. So, the Food Court is on Floor plus 1. If you press plus 2 from the ground floor, you reach the Art Centre. So, the Art Centre is on Floor plus 2.

Now, what about the floors below the ground? Starting from the ground floor, if you press minus 1, you reach the Toy Store. So, the Toy Store is on Floor minus 1. Similarly, starting from the ground floor, if you press minus 2, you reach the Video Games shop. So, the Video Games shop is on Floor minus 2.

And what is the ground floor called? It is called Floor 0. Can you see why? Because it is the starting point, the reference point from which we measure everything.

So, students, let's recap what we have learned so far. The floors above the ground are numbered with positive numbers: plus 1, plus 2, plus 3, and so on. The floors below the ground are numbered with negative numbers: minus 1, minus 2, minus 3, and so on. And the ground floor itself is 0, which is neither positive nor negative.

A number with a plus sign in front, like plus 3, is called a positive number. A number with a minus sign in front, like minus 3, is called a negative number. Zero is neither positive nor negative.

Now, students, let's understand addition using this building. Imagine you start from the Food Court, which is on Floor plus 1, and you press plus 2 in the lift. Where will you reach?

We can describe this using an expression: Starting Floor plus Movement equals Target Floor.

The starting floor is plus 1, which is the Food Court. The number of button presses is plus 2. So, we add them: plus 1 plus plus 2 equals plus 3. That means we reach the Book Store, which is on Floor plus 3.

Let me give you another example. If you start from Floor plus 2 and press minus 3, where will you reach? The expression is plus 2 plus minus 3. Plus 2 minus 3 equals minus 1. So, you will reach the Toy Store, which is on Floor minus 1.

Now, students, let's practice some more examples. I want you to think of each expression as Starting Floor plus Movement.

First one: plus 1 plus plus 4. Starting from Floor plus 1, if you press plus 4, you go up 4 floors. So, plus 1 plus plus 4 equals plus 5.

Second one: plus 4 plus plus 1. Starting from Floor plus 4, if you press plus 1, you go up 1 floor. So, plus 4 plus plus 1 equals plus 5. Notice that plus 1 plus plus 4 and plus 4 plus plus 1 both give us plus 5. This is because addition is commutative — the order doesn't matter!

Third one: plus 4 plus minus 3. Starting from Floor plus 4, if you press minus 3, you go down 3 floors. So, plus 4 plus minus 3 equals plus 1.

Fourth one: minus 1 plus plus 2. Starting from Floor minus 1, if you press plus 2, you go up 2 floors. From minus 1, going up 2 floors takes you to plus 1. So, minus 1 plus plus 2 equals plus 1.

Fifth one: minus 1 plus plus 1. Starting from Floor minus 1, if you press plus 1, you go up 1 floor to reach Floor 0, the ground floor. So, minus 1 plus plus 1 equals 0.

Sixth one: 0 plus plus 2. Starting from Floor 0, if you press plus 2, you go up 2 floors to Floor plus 2. So, 0 plus plus 2 equals plus 2.

Seventh one: 0 plus minus 2. Starting from Floor 0, if you press minus 2, you go down 2 floors to Floor minus 2. So, 0 plus minus 2 equals minus 2.

Great job, students! Now, let's learn about combining button presses. Imagine Gurmit is in the Toy Store, which is on Floor minus 1. He wanted to go down two floors, but by mistake he pressed the plus button two times. He realized his mistake and quickly pressed the minus button three times. How many floors below or above the Toy Store will Gurmit reach?

Let's think about this carefully. He first pressed plus 2, which means he went up 2 floors from Floor minus 1. So, minus 1 plus plus 2 equals plus 1. Then, he pressed minus 3, which means he went down 3 floors from Floor plus 1. So, plus 1 plus minus 3 equals minus 2.

Alternatively, we can combine all the button presses at once: plus 2 plus minus 3. Plus 2 plus minus 3 equals minus 1. So, Gurmit will end up on Floor minus 1, which is one floor below the Toy Store. In other words, he goes one floor down from where he started.

Now, students, let's learn about the concept of inverse. Imagine Basant is on the ground floor, Floor 0. By mistake, he presses plus 3. How can he cancel this and get back to the ground floor? He can press minus 3! That is, plus 3 plus minus 3 equals 0.

We call minus 3 the inverse of plus 3. Similarly, the inverse of minus 3 is plus 3. In general, the inverse of a number is what you add to it to get zero.

Let me give you more examples. If Basant presses plus 4 and then presses minus 4, where will he reach? Plus 4 plus minus 4 equals 0. So, he will be back on the ground floor.

If you are at Floor plus 4 and you press its inverse minus 4, you go back to zero, the ground floor! If you are at Floor minus 2 and you press its inverse plus 2, you go to minus 2 plus plus 2, which equals 0 — again the ground floor!

Now, students, I want you to write the inverses of these numbers: plus 4, minus 4, minus 3, 0, plus 2, and minus 1.

The inverse of plus 4 is minus 4. The inverse of minus 4 is plus 4. The inverse of minus 3 is plus 3. The inverse of 0 is 0 itself. The inverse of plus 2 is minus 2. And the inverse of minus 1 is plus 1.

Excellent! Now, let's learn about comparing numbers using the building. Imagine four children are in different places in the building. Jay is in the Art Centre, so he is on Floor plus 2. Asin is in the Sports Centre, so she is on Floor plus 5. Binnu is in the Cinema Centre, so she is on Floor minus 3. And Aman is in the Toys Store, so he is on Floor minus 1.

Now, who is on the lowest floor? Let's compare: plus 2, plus 5, minus 3, and minus 1. We know that negative numbers are below zero, and among negative numbers, the one with the larger negative number is lower. So, minus 3 is less than minus 1, which is less than plus 2, which is less than plus 5. Therefore, Binnu is on the lowest floor, at Floor minus 3.

Now, let's understand how to compare numbers. Floor plus 3 is lower than Floor plus 4. So, we write plus 3 is less than plus 4, or plus 3 < plus 4. We can also write plus 4 > plus 3, which means plus 4 is greater than plus 3.

Now, what about negative numbers? Should we write minus 3 < minus 4 or minus 4 < minus 3? Let's think about the building. Floor minus 4 is lower than Floor minus 3. So, minus 4 < minus 3. It is also correct to write minus 3 > minus 4.

Notice something very important, students. All negative number floors are below Floor 0. So, all negative numbers are less than 0. All positive number floors are above Floor 0. So, all positive numbers are greater than 0.

Now, let's practice comparing some numbers. First, compare minus 2 and plus 5. Minus 2 is less than plus 5 because negative numbers are below zero and positive numbers are above zero. So, minus 2 < plus 5.

Compare minus 5 and plus 4. Minus 5 < plus 4 because minus 5 is below zero and plus 4 is above zero.

Compare minus 5 and minus 3. Both are negative, but minus 5 is further to the left on the number line, so minus 5 < minus 3.

Compare plus 6 and minus 6. Plus 6 is above zero and minus 6 is below zero, so plus 6 > minus 6.

Compare 0 and minus 4. Zero is above all negative numbers, so 0 > minus 4.

Compare 0 and plus 4. Zero is below all positive numbers, so 0 < plus 4.

Great job, students! Now, let's move on to subtraction. In earlier classes, you learned that subtraction means 'take away'. For example, if there are 10 books on a shelf and you take away 4 books, how many are left? We write 10 minus 4 equals 6.

But there is another meaning of subtraction, which is related to comparison or finding the missing number. Let me explain with an example. Suppose I have ₹10 and my sister has ₹6. I can ask: "How much more money should my sister get in order to have the same amount as me?" We can write this as 6 plus question mark equals 10, or 10 minus 6 equals question mark. Here, we see the connection between finding the missing number to be added and subtraction.

For subtraction of positive and negative numbers, we will use this meaning of subtraction as 'finding the missing number to be added' or 'making equal'.

Now, let's apply this to our building. Imagine you are at the Art Centre, which is on Floor plus 2, and you want to go to the Sports Centre, which is on Floor plus 5. What button should you press?

We need to find the movement needed. The starting floor is plus 2, and the target floor is plus 5. We can write this as: Target Floor minus Starting Floor equals Movement needed.

So, plus 5 minus plus 2 equals plus 3. You need to press plus 3 to go from Floor plus 2 to Floor plus 5.

Let me give you more examples.

Example a: If the target floor is minus 1 and the starting floor is minus 2, what button should you press? We need to go from minus 2 to minus 1. That means we need to go one floor up. So, we should press plus 1. The expression is minus 1 minus minus 2 equals plus 1.

Example b: If the target floor is minus 1 and the starting floor is plus 3, what button should you press? We need to go from plus 3 to minus 1. That means we need to go four floors down. So, we should press minus 4. The expression is minus 1 minus plus 3 equals minus 4.

Example c: If the target floor is plus 2 and the starting floor is minus 2, what button should you press? We need to go from minus 2 to plus 2. That means we need to go four floors up. So, we should press plus 4. The expression is plus 2 minus minus 2 equals plus 4.

Now, students, let's practice some subtraction problems. Remember to think of them as finding the movement needed to reach the target floor from the starting floor.

First one: plus 1 minus plus 4. Starting from Floor plus 4, we want to go to Floor plus 1. We need to go down 3 floors. So, plus 1 minus plus 4 equals minus 3.

Second one: 0 minus plus 2. Starting from Floor plus 2, we want to go to Floor 0. We need to go down 2 floors. So, 0 minus plus 2 equals minus 2.

Third one: plus 4 minus plus 1. Starting from Floor plus 1, we want to go to Floor plus 4. We need to go up 3 floors. So, plus 4 minus plus 1 equals plus 3.

Fourth one: 0 minus minus 2. Starting from Floor minus 2, we want to go to Floor 0. We need to go up 2 floors. So, 0 minus minus 2 equals plus 2.

Fifth one: plus 4 minus minus 3. Starting from Floor minus 3, we want to go to Floor plus 4. We need to go up 7 floors. So, plus 4 minus minus 3 equals plus 7.

Sixth one: minus 4 minus minus 3. Starting from Floor minus 3, we want to go to Floor minus 4. We need to go down 1 floor. So, minus 4 minus minus 3 equals minus 1.

Seventh one: minus 1 minus plus 2. Starting from Floor plus 2, we want to go to Floor minus 1. We need to go down 3 floors. So, minus 1 minus plus 2 equals minus 3.

Eighth one: minus 2 minus minus 2. Starting from Floor minus 2, we want to go to Floor minus 2. We don't need to move at all! So, minus 2 minus minus 2 equals 0.

Ninth one: minus 1 minus plus 1. Starting from Floor plus 1, we want to go to Floor minus 1. We need to go down 2 floors. So, minus 1 minus plus 1 equals minus 2.

Tenth one: plus 3 minus minus 3. Starting from Floor minus 3, we want to go to Floor plus 3. We need to go up 6 floors. So, plus 3 minus minus 3 equals plus 6.

Excellent work, students! Now, let's apply what we've learned to a larger context. Imagine a mine where minerals are extracted. The truck is at the ground level, which is marked as 0. Levels above the ground are marked by positive numbers, and levels below the ground are marked by negative numbers. The number indicates how many metres above or below the ground level it is.

Just like in the Building of Fun, we have the formula: Starting Level plus Movement equals Target Level.

For example, plus 40 plus plus 60 equals plus 100. This means if you start at 40 metres above ground and go up 60 metres, you reach 100 metres above ground.

Another example: minus 90 plus minus 55 equals minus 145. This means if you start at 90 metres below ground and go down 55 metres more, you reach 145 metres below ground.

We also have the formula: Target Level minus Starting Level equals Movement needed.

For example, plus 40 minus minus 50 equals plus 90. This means if you start at 50 metres below ground and want to go to 40 metres above ground, you need to go up 90 metres.

Another example: minus 90 minus plus 40 equals minus 130. This means if you start at 40 metres above ground and want to go to 90 metres below ground, you need to go down 130 metres.

Now, students, let's think about how many negative numbers there are. Bela's Building of Fun had numbers from minus 5 to plus 6. The mine had numbers from minus 200 to plus 180. But we can imagine larger buildings or mineshafts. Just as positive numbers plus 1, plus 2, plus 3, and so on keep going up without an end, negative numbers minus 1, minus 2, minus 3, and so on keep going down without an end.

Positive and negative numbers, along with zero, are called integers. They go both ways from 0: minus 4, minus 3, minus 2, minus 1, 0, 1, 2, 3, 4, and so on.

Now, let's learn about converting subtraction to addition. We have learned that Target Floor minus Starting Floor equals Movement needed, or equivalently, Target Floor equals Starting Floor plus Movement needed.

If we start at 2 and wish to go to minus 3, what is the movement needed?

First method: Looking at the number line, we see we need to move minus 5, that is, 5 steps backward. Therefore, minus 3 minus 2 equals minus 5. The movement needed is minus 5.

Second method: We can break the journey from 2 to minus 3 into two parts. From 2 to 0, the movement is 0 minus 2, which is minus 2. From 0 to minus 3, the movement is minus 3 minus 0, which is minus 3. The total movement is the sum of the two movements: minus 3 plus minus 2, which equals minus 5.

Look at the two expressions. In the second method, there is no subtraction! We converted subtraction to addition.

In general, we can always convert subtraction to addition. The number that is being subtracted can be replaced by its inverse and then added instead.

Similarly, a number that is being added can be replaced by its inverse and then subtracted. In this way, we can also convert addition to subtraction.

Let me give you some examples of converting subtraction to addition.

Example a: plus 7 minus plus 5. We can write this as plus 7 plus minus 5. This equals plus 2.

Example b: minus 3 minus plus 8. We can write this as minus 3 plus minus 8. This equals minus 11.

Example c: plus 8 minus minus 2. We can write this as plus 8 plus plus 2. This equals plus 10.

Example d: plus 6 minus minus 9. We can write this as plus 6 plus plus 9. This equals plus 15.

Notice something very interesting, students. In examples c and d, subtracting a negative number is the same as adding the corresponding positive number! This is a very important rule that will help you in solving many problems.

Now, let's learn about the token model. In Bela's Building of Fun, the lift attendant keeps a box containing lots of positive tokens, which are green, and negative tokens, which are red. Each time he presses the plus button, he takes a positive token from the box and puts it in his pocket. Similarly, each time he presses the minus button, he takes a negative token and puts it in his pocket.

He starts on the ground floor, Floor 0, with an empty pocket. After one hour, he checks his pocket and finds 5 positive tokens and 3 negative tokens. On which floor is he now?

He must have pressed plus 5 times and minus 3 times. So, the calculation is plus 5 plus minus 3. We can compute this as plus 2. So, he is at Floor plus 2 now.

Here is another way to do the calculation using tokens. A positive token and a negative token cancel each other because the value of this pair of tokens together is zero. We call a positive and a negative token together a 'zero pair'. When you remove all the zero pairs, you are left with the result.

In this case, we have 5 positive tokens and 3 negative tokens. We can form 3 zero pairs by pairing 3 positive tokens with 3 negative tokens. These cancel each other. We are left with 2 positive tokens. So, plus 5 plus minus 3 equals plus 2.

Let's try another example: Add plus 5 and minus 8.

We have 5 positive tokens and 8 negative tokens. We can form 5 zero pairs by pairing 5 positive tokens with 5 negative tokens. These cancel each other. We are left with 3 negative tokens. So, plus 5 plus minus 8 equals minus 3.

Now, let's learn how to do subtraction using tokens.

Example: Subtract plus 5 minus plus 4.

This is easy to do. From 5 positives, take away 4 positives. We are left with 1 positive. So, plus 5 minus plus 4 equals plus 1.

Example: Subtract minus 7 minus minus 5.

We have 7 negative tokens, and we want to take away 5 negative tokens. From 7 negatives, we take away 5 negatives. We are left with 2 negatives. So, minus 7 minus minus 5 equals minus 2.

Example: Subtract plus 5 minus plus 6.

This is a tricky one! We put down 5 positives. But there are not enough tokens to take out 6 positives. What do we do?

We can put out an extra zero pair, which is one positive and one negative token together. Adding a zero pair does not change the value of the set of tokens because they cancel each other out.

Now, we have 6 positives and 1 negative. We can take out 6 positives. What is left? We are left with 1 negative token. So, plus 5 minus plus 6 equals minus 1.

Example: Subtract plus 4 minus minus 6.

We start with 4 positives. We have to take out 6 negatives. But there are not enough negatives! So, we add some zero pairs. We have to take away 6 negatives, so we put down 6 zero pairs. Now we have 4 positives and 6 negatives. We can take out 6 negatives. What is left? We are left with 4 positives and 6 negatives that we added. But wait, we also had the original 4 positives. So, we have 4 plus 6, which is 10 positives in total. Therefore, plus 4 minus minus 6 equals plus 10.

Great job, students! Now, let's see how integers are used in real life. One important application is in banking, with credits and debits.

Suppose you open a bank account with ₹100 that you had been saving. Your bank balance starts at ₹100.

Then you make ₹60 at your job the next day and deposit it in your account. This is shown in your bank passbook as a credit. Your new bank balance is ₹100 plus ₹60, which is ₹160.

The next day, you pay your electric bill of ₹30 using your bank account. This is shown as a debit. Your bank balance is now ₹160 minus ₹30, which is ₹130.

The next day, you make a major purchase for your business of ₹150. Again, this is shown as a debit. What is your bank balance now? It is ₹130 minus ₹150, which is minus ₹20.

Is this possible? Yes, some banks do allow your account balance to become negative, temporarily! They may charge you an additional amount if your balance becomes negative, in the form of interest or a fee.

The next day, you make ₹200 at your business. What is your balance now? It is minus ₹20 plus ₹200, which is ₹180.

You can think of credits as positive numbers and debits as negative numbers. The total of all your credits and debits is your total bank account balance, which can be positive or negative!

In general, it is better to try to keep a positive balance in your bank account!

Another application of integers is in geography, when we measure the height of geographical features like mountains, plateaus, and deserts from sea level. The height at sea level is 0 metres. Heights above sea level are represented using positive numbers, and heights below sea level are represented using negative numbers.

For example, Mount Everest is at height plus 8848 metres above sea level. The Dead Sea is at height minus 430 metres below sea level.

Integers are also used in measuring temperature. We use Celsius as the unit of measure. The freezing point of water is 0°C. Temperatures above 0°C are positive, and temperatures below 0°C are negative.

For example, during a hot summer day, the temperature might be 40°C, which is a positive number. During a cold winter night in some places in India, the temperature might be minus 4°C, which is a negative number.

Now, students, let's learn about some interesting patterns with integers. There is something special about the numbers in these grids.

Look at the first grid. The top row is 4, minus 1, minus 3. If we add them: 4 plus minus 1 plus minus 3 equals 4 minus 1 minus 3, which equals 0. The bottom row is minus 1, minus 1, 2. If we add them: minus 1 plus minus 1 plus 2 equals 0. The left column is 4, minus 3, minus 1, which also adds up to 0. The right column is minus 3, 1, 2, which also adds up to 0.

In each grid, the numbers in each of the two rows and each of the two columns add up to give the same number. We call this sum the 'border sum'. The border sum of the first grid is 0.

Now, let's learn about an amazing grid of numbers! Below is a grid having some numbers. The game is to circle any number, then strike out the row and column of the chosen number, then circle any unstruck number. When there are no more unstruck numbers, stop. Add the circled numbers.

In the example, the circled numbers are minus 1, 9, minus 7, and minus 2. If we add them, we get minus 1.

Now, students, let's learn about the history of integers. Like fractions, integers including zero and negative numbers were first conceived and used in Asia, thousands of years ago, before they spread across the world.

The first known use of negative numbers was in the context of accounting. In China, around the first or second century CE, in a mathematical work called "The Nine Chapters on Mathematical Art," positive and negative numbers were represented using red and black rods, much like we represent them using green and red tokens!

In India, there was a strong culture of accounting. The concept of credit and debit was written about extensively by Kautilya in his "Arthaśāstra" around 300 BCE. He recognized that an account balance could be negative.

The first general treatment of positive numbers, negative numbers, and zero — all on an equal footing — was given by Brahmagupta in his "Brāhma-sphuṭa-siddhānta" in the year 628 CE. Brahmagupta gave clear and explicit rules for operations on all numbers — positive, negative, and zero — that essentially formed the modern way of understanding these numbers!

Some of Brahmagupta's key rules for addition are:

First, the sum of two positives is positive. For example, 2 plus 3 equals 5.

Second, the sum of two negatives is negative. To add two negatives, add the numbers without the signs, and then place a minus sign. For example, minus 2 plus minus 3 equals minus 5.

Third, to add a positive number and a negative number, subtract the smaller number without the sign from the greater number without the sign, and place the sign of the greater number. For example, minus 5 plus 3 equals minus 2, 2 plus minus 3 equals minus 1, and minus 3 plus 5 equals 2.

Fourth, the sum of a number and its inverse is zero. For example, 2 plus minus 2 equals 0.

Fifth, the sum of any number and zero is the same number. For example, minus 2 plus 0 equals minus 2, and 0 plus 0 equals 0.

Brahmagupta's key rules for subtraction are:

First, if a smaller positive is subtracted from a larger positive, the result is positive. For example, 3 minus 2 equals 1.

Second, if a larger positive is subtracted from a smaller positive, the result is negative. For example, 2 minus 3 equals minus 1.

Third, subtracting a negative number is the same as adding the corresponding positive number. For example, 2 minus minus 3 equals 2 plus 3, which is 5.

Fourth, subtracting a number from itself gives zero. For example, 2 minus 2 equals 0, and minus 2 minus minus 2 equals 0.

Fifth, subtracting zero from a number gives the same number. For example, minus 2 minus 0 equals minus 2, and 0 minus 0 equals 0. Subtracting a number from zero gives the number's inverse. For example, 0 minus minus 2 equals 2.

Once you understand Brahmagupta's rules, you can do addition and subtraction with any numbers whatsoever — positive, negative, and zero!

Now, students, let's summarize everything we have learned in this chapter.

First, we learned that there are numbers less than zero. These are written with a minus sign in front, like minus 2, and are called negative numbers. They lie to the left of zero on the number line.

The numbers minus 4, minus 3, minus 2, minus 1, 0, 1, 2, 3, 4, and so on are called integers. The numbers 1, 2, 3, 4, and so on are called positive integers. The numbers minus 4, minus 3, minus 2, minus 1, and so on are called negative integers. Zero is neither positive nor negative.

Every given number has another number associated with it which, when added to the given number, gives zero. This is called the additive inverse of the number. For example, the additive inverse of 7 is minus 7, and the additive inverse of minus 543 is plus 543.

Addition can be interpreted as Starting Position plus Movement equals Target Position. Addition can also be interpreted as the combination of movements or increases and decreases: Movement 1 plus Movement 2 equals Total Movement.

Subtraction can be interpreted as Target Position minus Starting Position equals Movement.

In general, we can add two numbers by following Brahmagupta's rules. If both numbers are positive, add the numbers and the result is positive. If both numbers are negative, add the numbers without the signs and then place a minus sign. If one number is positive and the other is negative, subtract the smaller number without the sign from the greater number without the sign, and place the sign of the greater number. A number plus its additive inverse is zero. A number plus zero gives back the same number.

We can subtract two integers by converting the problem into an addition problem and then following the rules of addition. Subtraction of an integer is the same as the addition of its additive inverse.

Integers can be compared. Smaller numbers are to the left of larger numbers on the number line. So, minus 3 is less than minus 2, which is less than minus 1, which is less than 0, which is less than plus 1, which is less than plus 2, and so on.

We can give meaning to positive and negative numbers by interpreting them as credits and debits in banking. We can also interpret positive numbers as distances above a reference point like ground level, and negative numbers as distances below ground level. When measuring temperatures, positive temperatures are above the freezing point of water, and negative temperatures are below the freezing point of water.

Students, you have now learned about integers, which are a fundamental concept in mathematics. You have learned how to add, subtract, and compare integers. You have seen how integers are used in real-life situations like banking, geography, and temperature. And you have learned about the rich history of these numbers, which were first developed in India thousands of years ago.

This brings us to the end of our lesson on Chapter 10, "The Other Side of Zero." I hope you have enjoyed learning about integers as much as I have enjoyed teaching you. Remember, mathematics is all around us, and integers help us describe and understand many things in our world.

Thank you for your attention, and keep practicing!