Hello, and welcome to today's mathematics lesson! Today, we are going to explore a fascinating world of numbers that extends beyond what you have learned so far. We will be studying negative numbers and integers. By the end of this lesson, you will understand why we need negative numbers, what integers are, how to represent them on a number line, how to compare them, and how to add and subtract them.
Let us begin with a simple question. What happens when you try to subtract a larger number from a smaller number? Suppose you have 6 rupees and you need to give away 15 rupees. When you calculate 6 minus 15, you get a result that is less than zero; it is a negative number. You get –9. This is where negative numbers come in. We need negative numbers to represent situations where we owe something, or when we are below a reference point, or when we move in the opposite direction.
Now, let us build the complete picture of integers. Take the natural numbers: 1, 2, 3, 4, 5, and so on. For each of these, we create a matching negative number. So we get –1, –2, –3, –4, –5, and so on. Here, –1 is called the negative of 1, –2 is the negative of 2, and so on. When we combine these negative numbers with all the whole numbers, which include zero, we get a complete new collection called integers.
Integers stretch endlessly in both directions. They look like this: dot dot dot, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, dot dot dot. The positive integers are simply the natural numbers: 1, 2, 3, 4, 5, and so on. The negative integers are –1, –2, –3, –4, –5, and so on. And here is something important: zero is special. Zero is neither positive nor negative. It sits right in the middle, a neutral integer.
Now, here is a beautiful property. When you add a number and its negative, you always get zero. One plus –1 equals zero. Two plus –2 equals zero. Five plus –5 equals zero. This is why we call them opposites of each other. –1 and 1 are opposites. –2 and 2 are opposites. –5 and 5 are opposites.
Let us see how to picture all this on a number line. First, draw a straight horizontal line. Mark equal points along this line. Choose one point and label it zero. Now, starting from zero, move to the right and mark the positive integers: +1, +2, +3, and so on. Starting from zero again, move to the left and mark the negative integers: –1, –2, –3, continuing leftward. The arrows at both ends remind us that the numbers go on forever in both directions.
Negative numbers are not just mathematical ideas. They appear everywhere in daily life. If profit is represented by a positive integer, then loss is represented by a negative integer. If height above sea level is positive, then depth below sea level is negative. If moving 5 metres north is plus 5, then moving 5 metres south is –5. If a rise in temperature of 30 degrees is plus 30, then a fall of 50 degrees is –50. If moving 80 metres west is plus 80, then moving 80 metres east is –80. See how the opposite ideas use opposite signs?
Now comes an essential skill: comparing integers. Here is the key rule. On a number line, any integer on the right side is greater than any integer on the left side. Conversely, any integer on the left side is smaller than any integer on the right side.
Let us verify this. 5 is to the right of 2, so 5 is greater than 2. Zero is to the right of –3, so zero is greater than –3. 2 is to the right of –1, so 2 is greater than –1. –4 is to the left of –3, so –4 is less than –3.
From this, we can draw four important conclusions. First, every positive integer is greater than zero. Second, every positive integer is greater than every negative integer. Third, zero is greater than every negative integer but less than every positive integer. Fourth, every negative integer is smaller than every positive integer.
Here is another useful pattern. If a is greater than b, then the negative of a is less than the negative of b. For example, since 9 is greater than 7, –9 is less than –7. Similarly, if a is less than b, then the negative of a is greater than the negative of b. Since 13 is less than 20, –13 is greater than –20. Remember this: the greater the integer, the lesser its opposite.
Let us work through an example together. Suppose we need to arrange these integers in ascending order: 5, –8, 0, 3, –5, –10, 9, and –2. Imagine placing these on a number line. The leftmost number is –10, then comes –8, then –5, then –2, then 0, then 3, then 5, and finally 9 on the right. So the ascending order is: –10, –8, –5, –2, 0, 3, 5, 9.
For descending order, let us try: 6, –4, 5, –7, 0, –6, –2, and 3. Starting from the rightmost on the number line: 6, then 5, then 3, then 0, then –2, then –4, then –6, and finally –7. So the descending order is: 6, 5, 3, 0, –2, –4, –6, –7.
Now we move to addition of integers. There are different cases to understand.
First, adding two positive integers. This is just like normal addition. +59 plus +32 equals +91. Simply add them and keep the plus sign.
Second, adding a positive and a negative integer. Consider +3 plus –4. On the number line, start at zero, move 3 units right to reach +3. Then from there, move 4 units left for the –4. You land at –1. So +3 plus –4 equals –1.
Third, adding a negative and a positive integer. Consider –4 plus +3. Start at zero, move 4 units left to reach –4. Then move 3 units right. You again land at –1. Notice that –4 plus +3 also equals –1. This shows that addition is commutative: the order does not matter.
Here is the rule for opposite signs. Subtract the smaller numerical value from the larger one, then give the answer the sign of the number with larger numerical value. For –38 plus +72, subtract 38 from 72 to get 34. Since 72 is larger and positive, the answer is plus 34. But for –72 plus +38, subtract 38 from 72 to get 34. Since 72 is larger and negative, the answer is –34.
Fourth, adding two negative integers. Consider –2 plus –4. Start at zero, move 2 units left, then move 4 more units left. You reach –6. When both integers are negative, add their values and keep the minus sign. –43 plus –55 equals –98.
Subtraction has a clever trick. To subtract one integer from another, you can use the number line or follow this simple rule. If the number being subtracted is positive, subtract normally. 8 minus 5 equals 3. –8 minus 5 equals –13.
But if the number being subtracted is negative, change its sign and add instead. 8 minus –5 becomes 8 plus 5, which equals 13. –8 minus –5 becomes –8 plus 5, which equals –3.
Let us visualize with the number line. For +6 minus +2, mark both positions. Count the steps from +2 to +6. Moving right, we take 4 steps. So the answer is +4.
For +5 minus –3, mark +5 and –3. Count from –3 to +5. You move 8 steps to the right. The answer is +8.
For –7 minus +2, mark –7 and +2. Count from +2 to –7. You move 9 steps to the left. The answer is –9.
Let us try some practical examples. What is 5 more than 2? Start at 2, move 5 units to the right, you reach 7. What is 3 less than 4? Start at 4, move 3 units to the left, you reach 1. What is 6 more than –8? Start at –8, move 6 units to the right, you reach –2. What is 3 less than –4? Start at –4, move 3 units to the left, you reach –7.
Let us recap the key takeaways from today's lesson.
First, negative numbers are needed when we subtract a larger whole number from a smaller one, or when we represent opposite quantities like loss, depth, or temperatures below zero.
Second, integers include all positive whole numbers, all negative numbers, and zero. Zero is neutral, neither positive nor negative.
Third, every number has an opposite, and a number plus its opposite always equals zero.
Fourth, on a number line, numbers increase as you move right and decrease as you move left. Every positive integer is greater than every negative integer.
Fifth, when adding integers with opposite signs, subtract the values and keep the sign of the larger number. When adding two negatives, add the values and keep the negative sign.
Sixth, to subtract a negative integer, change it to positive and add.
Excellent work today! You have now entered the world of integers, a fundamental concept that will support your mathematics journey for years to come. Practice visualizing numbers on the number line, and the rules will become second nature. Keep exploring, keep questioning, and I will see you in the next lesson.