Course:
Mathematics Form 1
In our last lesson, we discovered that whole numbers are also called integers, and they include positive numbers, negative numbers, and zero. 🟰 Last time, we focused on positive numbers, but now it’s time to dive into the world of negative numbers! 🌍
Understanding directed numbers is super important, especially when it comes to solving graphs, equations, and even navigating real-life situations. Whether you’re tracking temperatures 🌡️, checking your bank balance 💰, or exploring elevation 🌄, directed numbers are everywhere. So, let’s explore how these numbers work in our day-to-day lives!
By the end of this lesson, you’ll be able to:
We’ll wrap up everything you’ve learned and make sure you’re ready to take on the world of integers! 💪
Take on our Self-Assessment Exercises and see how much you’ve mastered. Let’s see what you’ve got! 🚀🌟
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As already stated, directed numbers are positive and negative numbers including zero, even though zero is neither positive nor negative. Directed numbers can also be called integers. Directed numbers can be best illustrated using a number line as shown in the diagram below.
We can use a number line for the addition and subtraction of directed numbers. Let us do few examples of these.
Example 1 📝
Work out 3 + 6
When you add the numbers above using a number line, you start at three then move 6 steps to the right as shown in diagram below;
The answer is 9. So for addition we always move to the right. ✨
NB: when counting the steps do not start with the first number. In this particular example, the starting number is 3 and our first step is at 4.
Example 2 📝
Let us now look at another example of subtraction.
Workout 4 – 10
In this case the starting point is 4 and since we have a minus sign there we do not move to the right. Instead, we move 10 steps to the left as shown in the diagram below.
Our answer there is -6. Remember that all the numbers on the left of zero are negative.
It is important to note that when adding directed numbers we always move to the right, and when we subtract directed numbers we always move to the left of the number line. Let us now continue with examples of real life situations where directed numbers are used.
Example 3: 📝 Thermometer ✨
A thermometer is an instrument used for measuring temperature. The temperature may be above zero or below zero (when it is really very cold and water is frozen). If it is above zero, let us say 23oC above zero, it will be written as +23oC or 23oC. If it is below zero, let us say 15oC below zero, it will be written as -15oC. The diagram below shows a case where the temperature is 15oC below zero.
Example 4:📝 Money balance at banks
If you deposit or put P700 in a bank account, then later you withdraw or take the money, let us say you take P800, that means you have taken more than you have. This is called an over-draft. That means you owe the bank. In this case, you owe the bank P100 and it is normally written with a negative (-) sign. It will be written –P100.
I hope you have learned something from these examples, now try the following activity.
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Please refer to Assignment Activity 1 Assignment 1.0 Number Lines ✨
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Numbers become greater (bigger) as you move to the right of the number line and they become smaller as you move to the left of the number line. Therefore, the numbers -8, -7, -6, -5, -4, -3 and –2 are all smaller than –1. Negative one (–1) itself is less than 0, 1, 2, 3, 4 and other positive numbers.
Ordering directed numbers means arranging them in order of size, starting with the smallest or starting with the largest.
Example 5 📝
Arrange the numbers 12, -3, 9, 0, -5 in order of size, starting with the largest.
Solution ✅
When you arrange the numbers, starting with the largest, this is called descending order. At this stage I do not think you have problems with ordering 12, 9 and 0. You may have problems with –5 and –3. But –5 is on the left of –3 in a number line so must be smaller. Therefore, your answer is,
12, 9, 0, -3, -5.
Special signs are used to compare any two numbers, to tell which one is smaller or greater than the other.
> means greater than
< means less than
You know that 3 is less than 20. This is written as 3 < 20 or 20 is greater than 3, and this is written as 20 > 3
Example 6 📝
(a) which is greater, -10 and 2
(b) which is smaller, -5 and –1
Solution: ✅
(a) –10 < 2, i.e. 2 is greater than –10
(b) –5 < -1, i.e. –5 is smaller than –1
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In this part of the lesson, you are going to add, subtract, multiply and divide directed numbers.
Remember directed numbers are positive (+) and negative (-) numbers. Most of the time when we write positive numbers, we don’t write the positive (+) sign. For example, +26 is written as 26. Therefore, any number written without a sign attached to it, except zero (0), is a positive number. Zero is not positive and it is not negative either.
Example 7 📝
Workout the following additions and subtractions using a number line:
1
(a) +8 +4 =
(b) +2 +5 =
(c) –7 +6 =
(d) –3 +9 =
2
(a) 10 –3 =
(b) 4 –7 =
(c) –2 –6 =
(d) –1 –2 =
Look at example 7, number (a). It is +8 +4. The sign in front of 8 is positive (+) and that in front of 4 is also positive (+). The result 12 is also positive (+). Now add the numbers and write the total, then you attach the same sign to this total.
When the numbers that are going to be added have a positive (+) sign attached to them, you should add the numbers and attach the same positive (+) sign to the number or result.
+10 +5 = +15 =15
+21 +9 = +30 = 30
Now look at example 2(c). It is –2 –6. Because both numbers have a negative (-) sign, the result, or answer should have the same negative (-) sign. Therefore, -2 –6 = -8
When the numbers have a negative (-)sign attached to them, you should add the numbers and give the answer or result the same negative (-) sign.
(a) –50 –20 = -(50 +20) = -70
(b) –39 –61 = -(39 +61) = -100
(c) –7 –3 –6 = -(7 +3 +6) = -16
What about Numbers that have different signs, such as –7 + 6?
In –7 + 6, you will get the answer as –1 when you use a number line. When you workout without using a number line, you ignore the negative (-) sign on seven, and you subtract the smaller number from the larger number. That is, -7 + 6 now becomes 7 – 6 = 1. Then you attach the sign that was attached to the larger number. 7 is larger than 6, and it is the one that had a negative (-) sign, hence, you attach the negative (-) sign to the answer.
Therefore, -7 +6 = -(7 – 6) = -1
In short, if two numbers have different signs attached to them, you neglect or ignore the signs and subtract the smaller number from the larger number. The answer then carries the sign of the larger number.
(a) –18 +5 = -(18 – 5) = -13
(b) +6 – 4 = +(6 – 4) = +2 = 2
(c) –2 + 17 = +(17 – 2) = +15 = 15
(d) –34 + 4 = -(34 – 4) = -30
In the previous section you learnt that when you add a positive number you move to the right on a number line. When you subtract a positive number you move to the left on a number line. Now when you add or subtract a negative number you move in the opposite direction to that of adding or subtracting a positive number.
For example: 📝
(i) +8 + (-3)
If it was +8 + (3), we could be moving 3 steps to the right, but we are adding -3, so we move to the left.
So, +8 + (-3) is the same as +8 – 3 =5 (ii) +6 – (-5)
Again if it was +6 – (5) we could be moving 5 steps to the left, but we are subtracting -5, so move to the right.
So, +6 – (-5) is the same as +6 + 5 = +11
Note that when a number does not have any sign before it then that number is positive. That is, 3 is the same as +3, 100 is +100 and so on. Therefore,
(iii) 8 +(+4) = 8 + 4 = 12
(iv) -7 – (+6) = -7 – 6 = -13
The rule is; in addition and subtraction of directed numbers, like or same signs next to each other make one single positive (+) sign all the time, while unlike signs make one single negative (-) sign all the time.
| Thats is | -(-) = + |
| +(+) = + | |
| And: | -(+) = – |
| +(-) = – |
Sometimes you will have to workout several numbers that have different signs. To make the work easier, you may add the positive numbers separately, then subtract the total of the negative numbers. This is called re-arranging with addition first.
Example 8 📝
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You should have learnt from primary school that multiplication is repeated addition.
e.g. 5×3=5+5+5=15 or 3+3+3+3+3=15
+5×+3= +5+5+5=+15=15
Hence; +×+ = +
Also, -5×3=(-5)+(-5)+(-5)= -15
Hence; -×+ = –
Therefore, when multiplying directed numbers, the rules are:
1. A positive (+) number multiplied by a positive (+) number gives a positive result or answer.
i.e. (+)×(+)= +
e.g. (a) +6×+5= +30=30
(b) 7×4= +28=28
2. A positive number multiplied by a negative (-) number gives a negative (-) result or answer.
i.e. (+)×(-) = –
e.g. (a) +3×-4 = -12
(b) 9×-2 = -18
3. A negative (-) number multiplied by a negative (-) number gives a positive (+) result or answer.
i.e. (-)×(-) = +
e.g. (a) -4×-9 = +36 = 36
(b) -3×-5 = +15 = 15
4. Since 2×3 is the same as 3×2, that means +6×-3 is the same as -3×+6.
For the same rule that (+)×(-) = -, it is also the same as (-)×(+) = –
Example 9 📝
Use the rules you have just learnt to work out the following:
(a) -7×(+6)
(b) 6×(-7)
(c) +8×(-5)
(d) -5×8
(e) -10×(-3)
Solution ✅
(a) -7×(+6)
Rule: negative number × positive number = negative answer
(-) ×(+) = –
Hence; -7×(+6) = -42
(b) 6×(-7)
Rule: positive number × negative number = negative answer
(+) × (-) = –
Hence; 6×(-7) = -42
(c) +8 × (-5) = – 40
(d) -5 × 8 = – 40
(e) -10 × (-3)
Rule: negative number × negative number = positive answer
Hence; -10×(-3) = +30 = 30
The rules for multiplication may be summarised as follows:
(+) × (+) = +
(-) × (-) = +
(+) × (-) = –
(-) × (+)= –
Further Examples
Example 10 📝
Work out the products of the following, by working with two numbers at a time.
(a) -1×-1
You have learnt that (-)×(-) = +, therefore, -1×-1 = +1 = 1.
There are two negative signs multiplied, and the result is positive.
(b) –1×-1×-1
Work with two numbers at a time.
-1×-1×-1 = -1×-1×-1 = There are three negative signs multiplied, and the
= +1 × -1 result is a negative
= -1 since (+)×(-) = –
(c) -1×-1×-1×-1 = There are four negative signs multiplied and the result is positive
= -1×-1×-1×-1
= +1 × -1×-1 = +1×-1×-1 = -1×-1
= +1
=1
(d) -1×-1×-1×-1×-1 = There are six negative signs multiplied, and the result is positive
= -1×-1×-1×-1×-1
= +1×-1×-1×-1
= +1×-1×-1×-1
= -1×-1×-1
= -1×-1×-1
= +1×-1 = -1
(e) -1×-1×-1×-1×-1×-1
= -1×-1×-1×-1×-1×-1
= +1×-1×-1×-1×-1
= +1×-1×-1×-1×-1
= -1×-1×-1×-1
= -1×-1×-1×-1
= +1×-1×-1
= +1×-1×-1
= -1×-1
= +1
=1
In short,
(a) (-)×(-) = +
(b) (-)×(-)×(-)×(-) = +
(c) (-)×(-)×(-)×(-)×(-)×(-) = +
What do you notice? Yes, when the number of negative signs or numbers to be multiplied is even, the product or answer is positive.
And,
(a) (-)×(-)×(-) = –
(b) (-)×(-)×(-)×(-)×(-) = –
(c) (-)×(-)×(-)×(-)×(-)×(-)×(-) = –
When the number of negative signs or numbers to be multiplied is odd, the product is negative.
Example 11 📝
Work out the following. (a) -3×-5×3
(b) 6×-4×-1×-3
(c) -2×-3×-3×-2
(d) 4×(-5)×(+1)×(-2)×(-3)
Solution ✅
(a) -3×-5×3
There are two negative signs, and two is an even number. Remember, when the number of negative signs to be multiplied is even, the result is positive
Therefore, -3×-5×3 = +45 = 45
(b) 6×-4×-1×-3
There are three negative signs. Hence, 6×-4×-1×-3 = -72
(c) -2×-3×-3×-2 = +36 = 36
(d) 4×(-5)×(+1)×(-2)×(-3) = 4×-5×1×-2×-3 = -120
In dividing directed numbers, the rules are:
1. A positive number divided by a positive number gives a positive quotient.
(Remember, when you divide a number, the result you get is called quotient.) That is; (+)÷(+) = +
Example 12 📝
(a) +6÷+2 = +3
(b) 8÷4 = 2
2. In Lesson 1, you learnt that 14÷7 = 2, and this can be written in terms of multiplication as 7 x 2 = 14.
In the same way, +3×-4 = -12, which can be written as -12÷-4 = +3 = 3. Therefore, the rule is; a negative number divided by a negative number gives a positive quotient or answer.
That is; (-)÷(-) = +
Example 13 📝
(a) -15÷-3 = +5 = 5
(b) -40÷-20 = +2 = 2
If you look at the two rules, you will notice that, dividing numbers with like/same signs gives a positive quotient.
3. Remember again that if 3×4 = 12, then 12 ÷ 4 = 3.
Likewise, +3×-4 = -12,
hence -12÷+3 = -12÷3 = -4.
A negative number divided by a positive number gives a negative quotient.
That is; (-)÷(+) = –
Example 14 📝
(a) -24÷8 = -3
(b) -30÷6 = -5
Again, a positive number divided by a negative number gives a negative quotient.
That is; (+)÷(-) = –
Example 15 📝
(a) +15÷-3 = -5
(b) 28÷-7 = -4
We have done some examples involving multiplication and division, now try the exercise below.
Here, you are now going to be given real life situations or problems involving negative and positive numbers. Go on to Activity 4.
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Negative numbers, positive numbers, and zero are called directed numbers. They are called directed numbers because they have direction.
Negative numbers always have a negative (-) sign attached to them, whereas positive numbers can be written with or without the positive (+) sign.
Addition and subtraction of directed numbers (rules)
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Like signs + (+) = + – (-) = + |
Un-like signs – (+) = – + (-) = – |
Rules for multiplication of directed numbers
+×+ = +
-×- = +
+×- = –
-×+ = –
Rules for division of directed numbers (they are the same as for multiplication)
+ ÷+ = +
– ÷- = +
+ ÷- = –
– ÷+ = –
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