F.3 Mathematics Supplementary Notes
Chapter 4 More about Rational and Irrational Numbers
Chapter 4 More about Rational and Irrational Numbers
Important Terms
rational number
irrational number
integer
fraction
improper fraction
rationalization of the
denominator
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Name:___________(
P. 1
) Class: F.3 (
surd
decimal
terminating decimal
recurring decimal
real number
infinite non-repeating
decimal
Revision Notes
1. Rational Numbers
(a) A rational number is a number that can be expressed in the form
a
where a and b are integers and b> 0.
b
(b) Integers, fractions, terminating decimals and recurring decimals are rational numbers.
2. Irrational Numbers
(a) If a number is not a rational number, then it is an irrational number.
(b) Any irrational number can be expressed as an infinite non-repeating decimal.
(c) Square roots such as 2 , 7 and 15 are irrational numbers.
These kinds of square roots are called surds.
B
(d) We can construct irrational numbers like 2 and 7 on the number line.
e.g. In the figure, OA = AB = 1 and OAB  90  .
1
1
Therefore, OB = OC = 2 .
O
A
C
1
2
0
Number System:
Real Numbers
Rational Numbers
Irrational Numbers
e.g.
2 ,
Rational Numbers
which are not integers
2 2
Fractions : e.g. ,
3 5
Terminating decimals:
e.g. 3.56, 0.14
Recurring decimals:

 
e.g. 0. 6 , 1. 81
Integers
Negative
Integers
e.g.–1,
–13,–8
Zero
0
Positive
Integers
e.g.113,
18,22
2
)
F.3 Mathematics Supplementary Notes
Chapter 4 More about Rational and Irrational Numbers
11/2005
P. 2
3. Surds
(a) For any positive numbers a and b.
ab  a  b
(i)
(ii)
a

b
a
b
(b) Surds like 3 2 are called surds in their simplest form, which means that the number inside the radical is
the smallest integer possible.
(c) Two surds are called like surds if the integers in the radicals of the two surds in their simplest form are the
same. Otherwise they are unlike surds.
(d) Like surds can be added or subtracted together to become one term.
e.g.
3  12  3  2 2  3
=
32 3
= 3 3
(e) Surds can be multiplied together no matter whether they are like or unlike surds.
e.g.
2  3  5  2  3 5
=
30
(f) The process of changing the irrational number in the denominator of a fraction to a rational number
is called rationalization of the denominators.
2
e.g. The steps to rationalize
are :
3
2
3

2 3
3 3
=
2 3
3
Exercise (Level I)
a
where a and b are integers and b > 0.
b
1.
Express the following numbers in the form
(a)
8
 
 
(c)
0. 93
(d) 1. 81
6
7
(b)

0. 6
F.3 Mathematics Supplementary Notes
Chapter 4 More about Rational and Irrational Numbers


11/2005

(f)  0. 873
(e)
0.13
2.
Find the values of the following surds and express your answers as integers or fractions.
(a)
0.09
(b)
(c)
75
108
(d) 
3.
784
1.5
0.54
Express the following surds in their simplest forms.
(a)
12
(b)
150
(c)
24
216
(d)
24
4
(b)
8  50
4.
Simplify the following expressions.
(a)
3 2  2 5 2
P. 3
F.3 Mathematics Supplementary Notes
(c)
(e)
5.
(a)
Chapter 4 More about Rational and Irrational Numbers
3 18  32
2 ( 18  98 )
(d)
18  24  27
(f)
(3  5 )( 5  2)
Rationalize the following expressions.
5
(b)
2
4 7
2
Level II
28
in terms of a and b.
9
6.
Suppose a  5 , b  7 . Express
7.
Simplify the following expressions.
(a)
4 63  3 28  5 175
(b) (2 5  3 )(3 3  5 )
(c)
(4 3  7)( 4 3  7)
(d) ( 2  3 ) 2
80 
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F.3 Mathematics Supplementary Notes
(e)
4 8
2
1
3
(g)
3
(i)
5
1

3 2
3 2
8.
Chapter 4 More about Rational and Irrational Numbers
(f)
(h)
(j)
(a) Evaluate ( 10  3)( 10  3) .
(b) Hence find the value of ( 10  3) 5 ( 10  3) 3 .
6 2
3 2
2
5
45
5

5 1
5 1

5 1
5 1
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F.3 Mathematics Supplementary Notes
Chapter 4 More about Rational and Irrational Numbers
11/2005
P. 6
**************************************************************************************
Level III (Optional)
1. (Rationalization of denominator) Simplify the following and express with rational denominators.
(a) e.g.
1
32
1
=
=
32

32
(b)
32
2
=
2 3 1
32
 3
2
 2 
2
32
34
=
= 2 3
(c)
2 3 3 5
2 3 3 5

1
(d)
2 1

1
3 2
(Ans: (b)
2.
Solve ( 2  1) x  3 and rationalize the denominator of the answer.
3.
Simplify
4.
Simplify 7  13  7  13 .
1
2 1

1
3 2

1
4 3

1
5 4
.

4 15  19
4 32
(c)
(d) 1  3 )
11
11
(Ans: x  3( 2  1) )
(Ans: 1 
(93AMCSen Ans:
5)
2)
F.3 Mathematics Supplementary Notes
Chapter 4 More about Rational and Irrational Numbers
11/2005
P. 7
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內容簡介
李學數是美國聖荷西大學數學系李信明教授的筆名。多年來他為香港廣角鏡雜誌撰寫數學
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和數學家的故事，希望透過故事的形式破除一般人對數學的恐懼。十多年來，其作品陪伴不少
香港青少年的成長，引發他們對數學的興趣。
本書是繼《數學家傳奇》之後，收錄其多篇精采之文章加以重編及修正，使散篇之作品連
成一氣，形成體系。
「把數學書寫得像童話一樣好看」是李教授的宿願，本書以親切流暢的筆調，娓娓道出數
學的內涵，使讀者如促膝於老祖父之旁，聆聽其和煦的叮嚀。
資料來源：
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F.3 Mathematics Supplementary Notes
Chapter 4 More about Rational and Irrational Numbers
11/2005
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