Investigation Ordering of numbers think! Mathematics 1A, p33–34
Fig. 2.1 shows a thermometer.
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- What are the temperatures indicated by each of the points A, B and C?
A: °C B: °C C: °C
- Which point shows the highest temperature?
- Which point shows the lowest temperature?
- What are the temperatures indicated by each of the points A, B and C?
The number line is used to represent numbers and allows us to visualise how they are related to each other.
The markings on a thermometer enable us to read and compare temperatures. Similarly, we can represent numbers on a diagram called a number line (see Fig. 2.2). The markings are equally spaced and the number 0 is the reference. Numbers on the right of 0 are positive and numbers on the left of 0 are negative. The arrow pointing right indicates the direction in which numbers increase.
In some countries, the number line is drawn with an arrow at each end to indicate that the line goes on indefinitely in both directions.
You can also draw a number line vertically.
- Copy and complete the following:
- The numbers on the right of 0 are positive numbers.
- The numbers on the left of 0 are numbers.
- A number that is on the right of another number is more than (or greater than) that number.
For example, 5 is on the right of 2, so 5 is than (or than) 2. - A number which is on the left of another number is less than (or smaller than) that number.
For example, 2 is on the left of 5, so 2 is than (or than) 5.
Notations, such as >, <, ≥ and ≤, help to represent the relationships between numbers concisely and precisely.
The symbol > represents ‘more than’ (or ‘greater than’), and the symbol < represents ‘less than’ (or ‘smaller than’).
5 is more than 2, so we write: 5 > 2.
2 is less than 5, so we write: 2 < 5.
The symbol ≥ represents ‘more than or equal to’, and the symbol ≤ represents ‘less than or equal to’.
- Look at the number line in Fig. 2.2 and answer each of the following questions. Use ‘>’ or ‘<’ to represent the relationship between the two numbers in each question and explain your answer.
- Is −3 more or less than 2? −3 2
- Is −3 more or less than −5? −3 −5
Practise Now 1B, Question 2
Use a dot to represent each of the following numbers on the number line below. An example is given.
−5, 4, 0, −112, 1.666, −3.8, 34
Each dot below is labelled with its number, but it starts in the wrong place: drag it to where its number belongs (or click it and use the arrow keys; Page Up and Page Down take bigger steps) — while you drag, a magnified view of the line appears above it. The dot for 1.666 is the example, already in place.
Hence, arrange the numbers in ascending order, i.e. from the smallest to the greatest.
Investigation Addition of two negative integers think! Mathematics 1A, p36–37
In this section, we will learn how to add and subtract numbers that involve negative integers using number discs.
Fig. 2.3 shows number discs representing 1 (i.e. +1) and −1 respectively:
To represent the number 2, we use two discs as shown:
To represent the number −2, we use two discs as shown:
A. Addition of two negative integers
(−2) + (−3) can be written as −2 + (−3). The first pair of brackets is not necessary, but the second pair is necessary. Why?
What is (−2) + (−3) equal to?
To represent the number −2, we put two discs as shown in Fig. 2.4(a).
To add the number −3, we add three more discs as shown in Fig. 2.4(b).
∴ (−2) + (−3) = −5
Investigation
Work out the answers using number discs as shown previously. Write down the final answer for each part.
Tap a disc above the pad to add it (or drag it on). Drag discs to line them up. To take a disc away, tap it and press Take away. The Value line under the pad adds up your discs.
For parts (a) and (b), what do you observe about
Explore Zero pair think! Mathematics 1A, p37
B. Zero pair
What is the overall value if we put one disc and one disc together as shown in Fig. 2.5?
The pad starts like Fig. 2.5. Press Zero pairs to see the pair, and press it again to remove it. Add a zero pair puts another and on the pad.
The overall value is
Investigation Addition of a positive and a negative integer think! Mathematics 1A, p37–38
C. Addition of a positive and a negative integer
What is 5 + (−2) equal to?
To represent 5 + (−2), we put five discs and two discs as shown in Fig. 2.6(a).
How many zero pairs can we form? This is shown in Fig. 2.6(b).
Since the zero pairs are equal to 0, we can remove them, as shown in Fig. 2.6(b) and (c).
Removing the two zero pairs leads to removing two discs from the five discs.
In other words, 5 + (−2) = 5 − 2.
∴ 5 + (−2)= 5 − 2= 3
Try Fig. 2.6 yourself: the pad below starts with 5 + (−2). Press Zero pairs once to see the zero pairs, and again to remove them.
Investigation
−3 + 4 is the same as (−3) + 4 which is equal to 4 + (−3).
Work out the answers using number discs as shown above. Write down the final answer for each part.
Press Clear to start each part. Tap a disc and press Take away to take it away.
Investigation Subtraction involving negative integers think! Mathematics 1A, p39–42
Three investigations, one for each kind of subtraction. When there are not enough discs to take away, add zero pairs first: a zero pair is worth 0, so it does not change the number.
D. Subtraction between two positive integers
We can use number discs to find out the value of 5 − 2.
To represent the number 5, we put five discs as shown in Fig. 2.7(a), then subtract or take away 2 as shown in Fig. 2.7(b).
The end result is 3, as shown in Fig. 2.7(c).
∴ 5 − 2 = 3
Now, what is 2 − 5 equal to?
To represent the number 2, we put two discs as shown in Fig. 2.8(a). There are not enough discs to subtract or take away 5, so we add three zero pairs as shown in Fig. 2.8(b). But why can we just add three zero pairs?
Because 2 + 0 + 0 + 0 is still equal to 2.
There are now enough discs to subtract or take away as shown in Fig. 2.8(c) and (d).
∴ 2 − 5 = −3
The answer is negative because we are subtracting a larger value from a smaller value.
Investigation: Subtraction between two positive integers
Work out the answers using number discs as shown above. Write down the final answer for each part.
Try Fig. 2.8 first: the pad starts with 2. Press Add a zero pair three times, then take away the five discs: tap one and press Take away, five times. Press Clear to start each part.
For parts (a) and (b), what do you observe about
E. Subtraction of a positive integer from a negative integer
What is −5 − 2 equal to?
To represent the number −5, we put five discs as shown in Fig. 2.9(a).
Since there are no positive discs to subtract or take away, we add two zero pairs as shown in Fig. 2.9(b). But why can we just add two zero pairs?
Because −5 + 0 + 0 is still equal to −5.
There are now enough discs to subtract or take away as shown in Fig. 2.9(c) and (d).
∴ −5 − 2 = −7
Investigation: Subtraction of a positive integer from a negative integer
Work out the answers to parts (a) and (b) below, using number discs as shown previously. Write down the final answer for each part.
Try Fig. 2.9 first: the pad starts with −5. Press Clear to start each part.
F. Subtraction of a negative integer
What is 5 − (−2) equal to?
To represent the number 5, we put five discs as shown in Fig. 2.10(a), but there are no discs to subtract or take away −2.
So we add two zero pairs as shown in Fig. 2.10(b). We add two zero pairs because 5 + 0 + 0 is still equal to 5.
Now we have two discs to subtract or take away as shown in Fig. 2.10(c) and (d).
Manipulating the number discs from Fig 2.10(b) to 2.10(c), we see that 5 − (−2) = 5 + 2.
∴ 5 − (−2)= 5 + 2= 7
Investigation: Subtraction of a negative integer
Work out the answers using number discs as shown previously. Write down the final answer for each part.
Try Fig. 2.10 first: the pad starts with 5. Press Clear to start each part.
Investigation Negative of an integer think! Mathematics 1A, p46
In primary school, we have learnt how to multiply positive numbers, e.g. 2 × 3 means 2 groups of 3, which is equal to 6.
How do we multiply a positive number by a negative number, or two negative numbers together, e.g. 2 × (−3), (−3) × 2 or (−3) × (−2)?
In this section, we will learn how to multiply numbers that involve negative integers using number discs.
A. Negative of an integer
In Section 2.2, we have learnt how to represent 1 (i.e. +1) and −1 using and respectively.
If we flip the disc, you will realise that the back of the disc is actually .
In other words, the negative of 1 can be obtained by flipping the disc to obtain , and we represent this as follows:
How do we obtain the negative of (−1), i.e. what is −(−1)?
By flipping the disc as shown below, we obtain −(−1) = 1.
Investigation
Work out the answers to parts (a) and (b) below, using number discs as shown above. Write down the final answer for each part.
The pad starts like Fig. 2.11: tap the disc (its group lights up) and press Flip. For each part: press Clear, put the discs on the pad, drag a box round them and press Group, press Negative for the red “−” wedge, then press Flip.
Investigation Multiplication involving negative integers think! Mathematics 1A, p47–48
B. Multiplication involving negative integers
Case 1: 2 × 3
The product 2 × 3 means 2 groups of 3, and it can be represented by number discs as shown in Fig. 2.13, i.e. 2 × 3 = 6.
Case 2: 2 × (−3)
The product 2 × (−3) means 2 groups of −3, and it can be represented as shown in Fig. 2.14, i.e. 2 × (−3) = −6.
Case 3: (−3) × 2
What does the product (−3) × 2 mean?
Does it mean ‘negative 3 groups’ of 2? But what does ‘negative 3 groups’ mean?
Since 2 × 3 = 3 × 2, i.e. we can change the order of the numbers in a product, then (−3) × 2 = 2 × (−3) = −6 from Case 2.
How do we get −6 from (−3) × 2 directly?
Since −6 = −(3 × 2), we can interpret (−3) × 2 as −(3 × 2), i.e. (−3) × 2 means the negative of ‘3 groups of 2’.
So we flip 3 groups of 2 as shown in Fig. 2.15 to get (−3) × 2 = −6.
(−3) × 2 can be written as −3 × 2, i.e. the brackets are not necessary. In other words, we can also write: −3 × 2 = −6.
The middle diagram in Fig. 2.15 shows that (−3) × 2 = 3 × (−2) as well.
Case 4: (−3) × (−2)
What does the product (−3) × (−2) mean?
From Case 3, we have interpreted (−3) × 2 as −(3 × 2).
Similarly, we can interpret (−3) × (−2) as −[3 × (−2)], i.e. (−3) × (−2) means the negative of ‘3 groups of −2’.
So we flip 3 groups of −2 as shown in Fig. 2.16 to get (−3) × (−2) = 6.
We can also write it as: −3 × (−2) = 6.
Investigation
Work out the answers to parts (a) to (d) below, using number discs as shown above. Write down the final answer for each part.
The pad starts like Fig. 2.15: tap the group and press Flip. For each part, make the groups below. For (c) and (d), read the product the way Cases 3 and 4 do — (−4) × 3 is the negative of 4 groups of 3 — so make the 4 groups, then tap the group, press Negative for the red “−” wedges, and Flip.
Using the results from the above Investigation, complete the following. In general,
positive number × positive number = positive number,
positive number × negative number = negative number,
negative number × positive number = number,
negative number × negative number = number.
Practice Adding and subtracting on a number line Sums (a), (b): think! Mathematics 1A, p37
Math Lab’s practice: sums (a) and (b) are the book’s, from p37, now as jumps on a number line; sum (c) is Math Lab’s own. A jump to the right adds a positive number; a jump to the left adds a negative number. Subtracting 4 is the same as adding −4 (the book shows −5 − 2 = −5 + (−2) on p40), so it is a jump of 4 to the left.
- Drag P to the first number (or click it and use the arrow keys).
- Add the jump: type its size in Jump by (for example −3) and press Add jump — or press just above P and slide to where the jump lands.
- Type the answer, then press Check. The line under the number line writes your jumps as a sum.
(a) (−4) + (−3)
(−4) + (−3) =
(b) 5 + (−2)
5 + (−2) =
(c) −3 − 4
−3 − 4 =