2.1
Concept of Negative Numbers and the Number Line
ANegative Numbers
In our daily lives, we often come across numbers with the plus '+' or minus '−' signs. We use them to represent quantities with opposite directions or meanings. For example, we use 40°C to represent a temperature that is 40°C above 0°C and −40°C to represent 40°C below 0°C. We read '+40°C' as 'positive forty degrees Celsius' and '−40°C' as 'negative forty degrees Celsius'.
Objective: To identify the use of negative numbers in the real world.
Take a look at your surroundings or search on the Internet for examples of the use of negative numbers. In what contexts are these numbers used? Discuss with your classmates what the '−' sign means in each of your examples.
From Activity 1, we see how negative numbers are used in real life.
Positive numbers include positive integers, fractions and decimals. Negative numbers include negative integers, fractions and decimals. We can omit the '+' sign when we write positive numbers.
Positive integers: 1, 2, 3, 4, ...
Negative integers: −1, −2, −3, −4, ...
Positive numbers: 1, 2, $\tfrac{1}{5}$, $2\tfrac{1}{4}$, 0.3, 1.8, ...
Negative numbers: −1, −3, $-\tfrac{1}{5}$, $-3\tfrac{3}{8}$, −0.8, −4.6, ...
We read '−1' as 'negative 1'. The number 0 is neither positive nor negative.
The collection of positive integers, 0 and negative integers is called the set of integers.
Integers: ..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...
The summit of Mount Everest is about 8850 m above sea level and the surface of the Dead Sea is about 430 m below sea level. Represent their altitudes using positive and negative numbers.
Solution
Let the altitude above sea level be positive.
The altitude of the summit of Mount Everest = 8850 m
The altitude of the surface of the Dead Sea = −430 m (altitude is negative as it is below sea level)
An aeroplane is 1320 m above sea level and a submarine is 56 m below sea level. Represent their altitudes using positive and negative numbers.
Aeroplane: m Submarine: m
BThe Number Line
We can use a number line to show the order of numbers. When drawing a number line, we choose a suitable unit length to mark the positions of numbers at equal unit intervals. We use an arrow at each end of the number line to indicate that the line can be extended further from both ends.
- all the positive numbers are to the right of zero (0),
- all the negative numbers are to the left of zero (0),
- numbers are arranged in ascending (i.e., increasing) order from left to right,
- every number is less than any number on its right,
- every number is greater than any number on its left.
Look at the number line below.
We can see that 3 is to the right of −2. We say that '−2 is less than 3' and denote this by '−2 < 3'.
We can also see that −5 is to the left of −2. We say that '−2 is greater than −5' and denote this by '−2 > −5'.
There are four inequality signs as shown in the table.
| Inequality sign | Meaning |
|---|---|
| < | less than |
| > | greater than |
| ≤ | less than or equal to |
| ≥ | greater than or equal to |
(a) Represent the numbers 0, 2.3 and −3 on a number line.
(b) Arrange the given numbers in ascending order.
Solution
(a) The representation is shown below.
(b) From the number line, we can tell that 2.3 is greater than 0 and −3. So, 2.3 is the greatest number. Similarly, we can tell that −3 is smaller than 0 and 2.3. So, −3 is the smallest number.
The numbers in ascending order are −3, 0 and 2.3.
(a) Represent the numbers −1, −3.5 and 2½ on a number line.
(b) Arrange the given numbers in descending order.
Descending:
In December, the temperature in Alaska can reach −27°C. In March, its temperature can reach −1°C.
(a) Represent −27 and −1 on a number line.
(b) Express the relationship between the two temperatures using an inequality sign.
Solution
(a) It is important to choose a suitable scale to mark the required numbers on a number line. In this example, we can use 1 cm to represent 5 units.
The temperatures of −27°C and −1°C are marked as shown.
(b) Since −1 is to the right of −27, −1 is greater than −27 and we write −1 > −27. We can also write −27 < −1, which means −27 is less than −1.
(a) Represent the numbers 5, −8 and −17 on a number line.
(b) Express the relationship between each pair of numbers using the '>' sign.
(i) 5 and −8:
(ii) −17 and 5:
(iii) −17 and −8:
(c) Can you write a number greater than −8 AND less than −17? Explain your answer.
-
Consider a gain in mass to be positive. Write down the following changes in mass using positive and negative numbers.
(a) John gained 2 kg during the Lunar New Year festive period. kg
(b) Meidi lost 3 kg after joining a fitness club. kg
-
Consider the amount of money deposited into a bank account as a positive number. Write down the amounts for the following transactions using positive and negative numbers.
(a) A withdrawal of $2800 $
(b) A deposit of $1650 $
-
(a) If −4 km/h means 4 km/h below the speed limit, what does +12 km/h mean?
12 km/h the speed limit
(b) If −3 km denotes a distance of 3 km due South, what does +5 km denote?
5 km due
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If −7°C denotes a temperature drop of 7°C, what does +5°C denote?
A temperature of 5°C
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(a) State the numbers represented by the points A, B and C on the number line.
A = B = C =
(b) Arrange the three numbers in descending order.
-
Represent each pair of numbers on a number line and express their relationship using
(a) the '<' sign:
(i) 3, −5 →
(ii) 0, −4.5 →
(iii) 5, −1½ →
(b) the '>' sign:
(i) −4, 1 →
(ii) 0.8, −3.2 →
(iii) 3⅓, −7/2 →
-
Fill in each box with '<' or '>'.
(a) 0 −3 (b) −2.1 1.2
(c) −15 15 (d) 23 −24
(e) −14 12 (f) −11 −33
-
Describe the meaning of each quantity.
(a) The adjustment to the hourly wage of a worker is −$5.
自评：(b) The movement of a lift is +2 levels.
自评： -
A machine packs rice in packets of intended mass of 1000 g each. In a random inspection, a note written '+8' denotes 1008 g. If a packet of rice has a mass of 996 g, how do you denote the shortage in its mass?
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(a) Represent the numbers −2.7, 1⅓, −4¾, 1 and −0.4 on a number line.
(b) Arrange the given numbers in ascending order. -
(a) Represent the numbers 5, −3½, −0.9, −3.2 and 0.3 on a number line.
(b) Arrange the given numbers in descending order. -
Arrange the numbers −50, 210, 0, −160 and −300 in ascending order.
-
Arrange the numbers −22, 4, 7, −15 and −9 in descending order.
-
The table shows the average monthly night time temperatures in the Gobi Desert for some months.
Jan Mar May Jul Nov Dec −20°C −10°C +9°C +18°C −9°C −15°C Arrange the months in descending order of average monthly night time temperature.
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The altitudes of some places are as follows:
- Death Valley, USA: −86 m
- Mount Fuji, Japan: 3376 m
- Mount Faber, Singapore: 105 m
- Turfan Depression, China: −154 m
Arrange the above places in ascending order of their altitudes.
自评：
-
[OPEN] The following shows a page from a savings account passbook.
Date Deposit Withdrawal Balance 03 JAN 2020 $3000.00 $3000.00 05 JAN 2020 $200.00 $2800.00 11 JAN 2020 $150.00 $2950.00 18 JAN 2020 $400.00 $2550.00 Design a page that shows the deposits and withdrawals under the same column called 'Transaction'.
自评： -
Mr Rahim has 3 food stalls A, B and C. In a particular month, the account balances for stalls A, B and C are −$3250, +$760 and −$2180 respectively. Based on these figures, which stall
(a) makes the most money? Stall
(b) loses the most money? Stall
-
The notation −9 ≤ a < 5 is read as '−9 is less than or equal to a and a is less than 5'. Suppose a is an integer.
(a) List all the possible values of a.
(b) If a is a prime number, what are the possible values?
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We can consider the scale on a thermometer as a vertical number line.
(a) What is the temperature shown on the thermometer?
(b) Is −18°C higher or lower than the temperature shown?
(c) Explain how you can tell whether a number is greater or smaller on the vertical number line.
自评： -
Discuss whether the following numbers exist. If they do, write down their values.
(a) The largest positive integer
(b) The smallest positive integer
(c) The largest negative integer
(d) The smallest negative integer
章末概念检查 · Concept Checkpoints
5 道封闭题，自动判分。负数概念是第 2 章其他节的基础——先把这 5 道做对。