Download Secondary 1 Mathematics Chapter 1 Real Numbers Practice 1

Secondary 1 Mathematics
Chapter 1 Real Numbers
Practice 1
History of Real Numbers
In legends, the first use of numbers traces all the way back to 35 000 B.C.
Numerical representation was needed for various purposes (e.g. to keep track
of the goods bought and sold, determine the amount of property ownership etc).
People in the past kept track of quantities using tally marks.
Tally marks used in Europe, North
America and some parts of Asia Pacific.
Tally marks used in China, Japan and
Korea
The Egyptian number system favours the use of picture symbols to represent numerals.
This number system uses bases of 10.
1
10
100
1000
10 000
100 000
1 000 000
Staff
Arch
Coiled rope
Flower
Bent finger
Tadpole
Astonished man
The Roman number system used letter symbols for numerals. These symbols are still
commonly used today in clocks faces, successive political leaders or children with identical names
and the numbering of annual events.
Symbol
I
V
X
L
C
D
M
Number
1
5
10
50
100
500
1000
The Addition Rule:
The Roman numeration system is based on an additive system in which symbols do not
have place values. This means that a numeral is the total sum of the numbers represented by each
symbol. Hence, II = 2, VIII = 8, XVII = 17, XXXVIII = 38.
The Subtraction Rule:
Learning
Objectives:
The subtraction rule states that when a smaller numeral
is placed
on the left of a larger
To
provide
an aim for
numeral, the smaller numeral has to be subtracted from the larger numeral.
For example, 45 is
students to achieve at
written as XLV (−10 + 50 + 5 =
45) instead of XXXXV.
the end of each lesson.
•
•
•
Identifying integers, rational numbers, natural numbers, whole numbers and real numbers;
Representing numbers on a number line;
Perform the four operations on positive and negative integers correctly.
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S1 | Chapter 1 Real Numbers | Practice 1
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Introduction to the world of real numbers
 Natural numbers are numbers we use for counting. {1, 2, 3, 4, 5, 6}
 Whole numbers include the number zero. {0, 1, 2, 3, 4, 5, 6}
 Negative integers are numbers smaller than zero. {−1, − 2, − 3, − 4, − 5, − 6}
 Positive integers are numbers greater than zero. {1, 2, 3, 4, 5, 6}
 Zero, 0, is an integer that is neither positive nor negative.
 Rational numbers are numbers which can be expressed in the form of
 22
are integers, and b ≠ 0 . Eg:  ,
7
7
2
o All integers, Eg:  , − ,
1
1
a
, where a and b
b
2
4
, − 
3
5
5

1
o All terminating or repeating decimals, Eg: 0.375
=
•
3
1
, 0.3
=
8
3
a
, where a
Summary:
b
An overview of key concepts
and b are integers, and b ≠ 0 . They are non-terminating and non-repeating
decimals.
 Irrational numbers are numbers which cannot be expressed in the form of
Eg: π, e,
2,
is provided to strengthen
student’s understanding.
5 etc.
Real Numbers, ℝ
• The set of rational and irrational numbers
Rational Numbers, ℚ
Irrational Numbers
• Numbers which can be expres
sed
a
in the form of , where a and b are
b
integers, and b ≠ 0 .
• Numbers which cannot be expressed in
a
the form of
, where a and b are
b
integers, and b ≠ 0
• Examples are: π,e, 2, 5, etc.
Integers, 
• Consists of positive and negative
integers, and zero
•  = {..., -3, -2, -1, 0, 1, 2, 3, ...}
Negative Integers,  −
•  − = {-1, -2, -3, ...}
Zero
Fractions
• Examples are:
22 1 3 5
, , , etc.
7 2 4 12
Positive Integers,  +
• Also known as natural numbers, ℕ
•  + = {1, 2, 3, ...}
Whole numbers
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S1 | Chapter 1 Real Numbers | Practice 1
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Number line
-5
-4
-3
-2
-1
0
1
Negative Integers
4
3
2
5
Positive Integers / Natural Numbers
Numbers which are smaller in values will be to the left and numbers with greater values will be
to the right of the number line.
Representing the numbers −2, − 10, 0, 1, 3 on a number line will be as follows:
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
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S1 | Chapter 1 Real Numbers | Practice 1
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Example 1
Evaluate the following without using a calculator.
a) 13 + 22
b) 17 − 8
c) 4 − 9
Solution:
a) 13 + 22 =
35
d) −13 − 6
b) 17 − 8 =
9
+ 22
13
-8
35
9
c) 4 − 9 =−5
17
d) − 13 − 6 =−19
-6
-9
-5
-19
4
Example 2
Evaluate the following without using a calculator.
a) −12 + ( −10 )
b) −6 + ( −14 )
Solution:
a) − 12 + ( −10 ) =−12 − 10 =−22
c) −8 − ( −9 )
d) −11 − ( −10 )
b) − 6 + ( −14 ) =−6 − 14 =−20
-10
-22
-14
-12
-20
-6
d) − 11 − ( −10 ) =−11 + 10 =−1
c) − 8 − ( −9 ) =−8 + 9 =1
+ 10
+9
-8
-13
Teachers will go through
examples in details to bring
across key concepts.
-11
1
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-1
S1 | Chapter 1 Real Numbers | Practice 1
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Multiplication of Integers
Rules for signs:
(+)× (+) = (+)
Examples
5× 3 =
15
( −) × (−) = ( +)
( +) × ( −) = ( −)
( −) × (+) = (−)
( −5) × ( −3) =15
5 × ( −3) =−15
( −5) × 3 =−15
Rules for signs:
(+) ÷ (+) = (+)
Examples
15 ÷ 3 =
5
( −) ÷ ( −) = ( +)
( +) ÷ ( −) = ( −)
( −) ÷ ( +) = ( −)
( −15) ÷ ( −3) =5
15 ÷ ( −3) =−5
( −15) ÷ 3 =−5
Division of Integers
Example 3
Evaluate the following.
a) 6 × ( −5)
b)
( −2 ) × ( −7 )
c) 112
d)
( −6 )
e) 83
f)
h)
( −1)
( −11 + 10 ) × ( −3)
a) 6 × ( −5) =−30
b)
( −2 ) × ( −7 ) =14
2
c) 11=
11 × 11
d)
( −6 ) = ( −6 ) × ( −6 )
g)
( −7 ) × 3 − ( −2 ) × 4
2
3
Solution:
= 121
= 36
e) 83 = 8 × 8 × 8
f)
= 512
g)
2
( −1) = ( −1) × ( −1) × ( −1)
3
= −1
( −7 ) × 3 − ( −2 ) × 4 = ( −21) − ( −8 )
=
−21 + 8
= −13
h)
( −11 + 10 ) × ( −3) = ( −1) × ( −3)
=3
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S1 | Chapter 1 Real Numbers | Practice 1
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Example 4
Evaluate the following.
a)
c)
( −21) ÷ ( −3)
0 ÷ ( −267 )
b) 125 ÷ ( −5)
d) 987 ÷ 0
Solution:
a) ( −21) ÷ ( −3) =7
b) 125 ÷ ( −5) =−25
c) 0 ÷ ( −267 ) =0
d) 987 ÷ 0 =∞
Exercise
1)
Arrange the following in ascending order.
3
2
7 3  2
a)
, 0.008, − , ,  − 
5
8 7  3
1
3
b) − 3 216 , − , ( −2 ) , 0, − 49
3
c) −
2)
• •
•
157 π
, , − 0.785, − 0.78,
200 4
11
4
Write down all the rational numbers in descending order.
π
, −
( 7 ) , 0.7, 25
112
3
3)
7
,
81
(
)
4
81 ,
7 + 1, − 7,
7
2
, −1
7
9
Use a calculator to evaluate each of the following.
a)
 3 
7 
c)  −7 −  −7  
 5  10  
e)
2
 3
15 625 −  −1 
 8
7  4  7 4 5
d)
× −  ÷ − × −
8  3  8 6 7
742 − 702
b)
3
3
3
3
4 7 1
×
− ×
5
9 9 3
2 1
8
f)  +  − ÷ 3
125
4 9 5
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S1 | Chapter 1 Real Numbers | Practice 1
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No calculators allowed for Questions 6 and 7. Proper workings must be shown.
4) Evaluate each of the following.
 3  1 
 
 1   5 
a)   +  −1   × 4 + ( −49 )  ÷ 15 ×  −  ÷  −  
 3   8 
 8  4 
 
 1  2    23  1  1   
b)  −  −   ÷ − ÷ 3 −  −2   
 2  5    30  2  4   
 3  1  
1
1 
1 
− 3 −−
c)  −3  × 2  ×  − 3

8  144  
 5  2   −27
 1  3   1    169 3 125 
d)  +  −  ×  −1   ÷  −
×

512 
 4  4   4    100
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S1 | Chapter 1 Real Numbers | Practice 1
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