Algebra Class 8 (Zambak)

240 Pages·2012·4.48 MB·English

A. RATIONAL NUMBERS: THE SET Q
1. Understanding Rational Numbers
Definition
The ratio of an integer to a non-zero integer is called a rational nnumber. The set of rational
a
numbers is denoted by Q. Q = { | a, b  Z, b  0}
b
2. The Set of Positive Rational Numbers
If a rational number represents a point on the number line on the right side of zero, then it
is called a positive rational number.
a
In short, is a positive rational number if a and b are both positive integers or both nega-
b
tive integers.
2 –2 2
For example, and are positive rational numbers, and denoted by .
7 –7 7
Definition
The set of positive rational nnumbers is denoted by Q+.
a a
Q+ = { | 0 and a, b  , b  0}
b b
3. The Set of Negative Rational Numbers
If a rational number represents a point on the number line on the left side of zero, then it is
called a negative rational number.
a
In short, is a negative rational number if a is a positive integer and bis anegative integer,
b
or if a is a negative integer and b is a positive integer.
–5 5
For example, and are negative rational numbers. We can write negative rational
4 –4
5 –5 5
numbers in three ways: –   .
4 4 –4
Definition
The set of negative rational numbers is denoted by Q–.
a a
Q– = { | 0 and a, b  , b  0}
b b
10 Algebra 8
A. THE SET OF REAL NUMBERS
1. Understanding Real Numbers
In algebra we use many different sets of numbers. For example, we use the natural numbers
to express quantities of whole objects that we can count, such as the number of students in
a class, or the number of books on a shelf.
The set of natural numbers is denoted by N.
N = {1, 2, 3, 4, 5, ...}
(cid:2)(cid:3) (cid:2)(cid:4) (cid:2)(cid:5) (cid:2)(cid:6) (cid:2)(cid:7) (cid:2)(cid:8) (cid:9) (cid:8) (cid:7) (cid:6) (cid:5) (cid:4) (cid:3)
(cid:29)(cid:16)(cid:17)(cid:21)(cid:15)(cid:16)(cid:20)(cid:14)(cid:19)(cid:21)(cid:12)(cid:22)(cid:13)(cid:15)(cid:23)
The set of whole numbers is the set of natural numbers together with zero. It is denoted by
W.
W = {0, 1, 2, 3, 4, 5, ...}
(cid:2)(cid:3) (cid:2)(cid:4) (cid:2)(cid:5) (cid:2)(cid:6) (cid:2)(cid:7) (cid:2)(cid:8) (cid:9) (cid:8) (cid:7) (cid:6) (cid:5) (cid:4) (cid:3)
(cid:27)(cid:28)(cid:11)(cid:20)(cid:13)(cid:14)(cid:19)(cid:21)(cid:12)(cid:22)(cid:13)(cid:15)(cid:23)
The set of integers is the set of natural numbers, together with zero and the negatives of the
natural numbers. It is denoted by Z.
Z = {..., –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, ...}
(cid:2)(cid:3) (cid:2)(cid:4) (cid:2)(cid:5) (cid:2)(cid:6) (cid:2)(cid:7) (cid:2)(cid:8) (cid:9) (cid:8) (cid:7) (cid:6) (cid:5) (cid:4) (cid:3)
(cid:25)(cid:19)(cid:17)(cid:13)(cid:26)(cid:13)(cid:15)(cid:23)
We use integers to express temperatures below zero, distances above and below sea level, and
increases and decreases in stock prices, etc. For example, we can write ten degrees Celsius
below zero as –10°C.
To express ratios between numbers, and parts of wholes, we use rational numbers.
8 2 3 0 17
For example, , , – , , and are rational numbers.
3 5 7 7 1
The set of rational numbers is the set of numbers that can be written as the quotient of two
integers. It is denoted by Q.
a
Q = { | a, b   and b  0}
b
(cid:24) (cid:8) (cid:7) (cid:4) (cid:8)(cid:6)
(cid:7) (cid:7) (cid:6) (cid:5) (cid:7)
(cid:2)(cid:24) (cid:2)(cid:3) (cid:2)(cid:4) (cid:2)(cid:5) (cid:2)(cid:6) (cid:2)(cid:7) (cid:2)(cid:8) (cid:9) (cid:8) (cid:7) (cid:6) (cid:5) (cid:4) (cid:3) (cid:24)
(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:11)(cid:19)(cid:16)(cid:20)(cid:14)(cid:19)(cid:21)(cid:12)(cid:22)(cid:13)(cid:15)(cid:23)
Radicals 11
We can write every rational number as a repeating or terminating decimal. Conversely, we
can write any repeating or terminating decimal as a rational number.
3 321
For example, 0.6, and 0.324 = 0.324242424...
5 990
––
0.6 is a terminating decimal, and 0.324 is a repeating decimal.
There are some decimals which do not repeat or terminate.
For example, the decimals 0.1012001230001234000 ...
3.141592653 ...= 
2.71828 ... = e
R= R+{0} R–
R+ is the set of positive 1.4142135 ... = ñ2
real numbers
do not terminate and do not repeat. Therefore, we cannot write these decimals as rational
R– is the set of negative
real numbers numbers. We say that they are irrational.
Definition
A number whose decimal form does not repeat or terminate is called an irrational number.
The set of irrational numbers is denoted by Q or I.
Definition
The union of the set of rational numbers and the set of irrational numbers forms the set of
all decimals. This union is called the set of real nnumbers.
The set of real numbers is denoted by R.
R = Q  Q
Real NNumbers For every real number there is a point on the number line. In
other words, there is a one-to-one correspondence between
the real numbers and the points on the number line.
(cid:2)(cid:6)(cid:9)(cid:5)(cid:3) (cid:2)(cid:8)(cid:9)(cid:3) (cid:8)(cid:9)(cid:10) (cid:11) (cid:2)
(cid:2)(cid:3) (cid:2)(cid:4) (cid:2)(cid:5) (cid:2)(cid:6) (cid:2)(cid:7) (cid:8) (cid:7) (cid:6) (cid:5) (cid:4) (cid:3)
(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:3)(cid:11)(cid:8)(cid:12)(cid:6)(cid:11)(cid:13)(cid:4)(cid:14)(cid:5)(cid:7)(cid:15)
The real numbers fill up the number line.
We can summarize the relationship between the different sets
of numbers that we have described in a diagram. As we know,
the set of natural numbers is a subset of the set of whole
numbers, the set of whole numbers is a subset of the set of
integers, the set of integers is a subset of the set of rational
numbers, and the set of rational numbers is a subset of the
N W Z Q R
set of real numbers. This relationship is shown by the dia-
QR
gram on the left.
12 Algebra 8
1. Understanding Square Roots
Remember that we can write aaas a2. We call a2the square of a, and multiplying a number
by itself is called squaring the number. The inverse operation of squaring a number is called
finding the square root of the number.
Objectives
After studying this section you will be able to:
1. Understand the concepts of square root and radical number.
2. Use the properties of square roots to simplify expressions.
3. Find the product of square roots.
4. Rationalize the denominator of a fraction containing square roots.
Definition
If a2 = b then a is the square rroot of b (a  0, b  0).
We use the symbol ñ to denote the square root of a number. ñbis read as ‘the square root of
b’. So if a22 = b then a = ñb (a  b, b  0).
Here are the square roots of all the perfect squares from 1 to 100.
12 = 1  ñ1 =1 62 = 36  ò36 = 6
22 = 4  ñ4 =2 72 = 49  ò49 = 7
32 = 9  ñ9 =3 82 = 64  ò64 = 8
42 = 16  ò16 =4 92 = 81  ò81 = 9
52 = 25  ò25 =5 102 = 100  ó100 = 10
The equation x2 = 9 can be stated as the question, ‘What number multiplied itself is 9?’
There are two such numbers, 3 and –3.
Rule
If x  R then
x if x0.
x2 |x|
–x if x0.
In other words, if x is a non-negative real number, then x2 x, and
if x is a negative real number, then x2  –x.
Radicals 13
For example, 32 3, ( 32  9 3), and
(–3)2 –(–3) 3 ( (–3)2  9 3).
We can conclude that the square root of any real number will always be greater than or equal
to zero. ò–9 is undefined. Negative numbers have no square root because the square of any
real number cannot be negative.
ò–9  3, since 32 is9, not (–9).
ò–9  –3, since (–3)2 is 9, not (–9).
Note
x = ñ9 and x2 = 9 have different meanings in the set of all real numbers.
 ñ9 = 32 = |3| = 3
 If x2 = 9 then x = 3 or x = –3.
1
EXAMPLE Evaluate each square root.
a. ò81 b. ñ1 c. ñ0 d. ò64 e. ñ9 f. ó0.64
g. –ó100 h. –ó0.09 i. ò–4 j. (–4)2 k. –42
Solution a. ò81 = 9 b. ñ1 =1 c. ñ0 =0
4 2
d. ò64 =8 e.  f. ó0.64 = 0.8
9 3
g. –ó100 = –10 h. –ó0.09 = –0.3 i. ò–4 is undefined
j. (–4)2  16 4 k. –42  –16 is undefined
2
EXAMPLE Evaluate each square root.
a. ó100 b. ó121 c. ó144 d. ó169 e. ó225 f. ó361
g. ó400 h. ó625 i. 1225 j. 10000
Solution a. ó100 =10 b. ó121 =11 c. ó144 = 12
d. ó169 =13 e. ó225 =15 f. ó361 =19
g. ó400 =20 h. ó625 =25 i. 1225 35
j. 10000 100
14 Algebra 8
2. Properties of Square Roots
Property
For any real number aand b, where a0, and b0,
ñañb= óab.
For example, 2516  25  16 5 4 20,
3  27  327  81 9,
36a2  36 a2 6a (a0), and
5  5  55  25 5.
Note
b 0 óa.6 =
If a  0 then a 0
 ña .ñ6
ña  ña =óa  a = a2 a.
Mathematics is a universal language.
3
EXAMPLE Simplify each of the following.
a. ñ2ñ8 b. ñ7ñ7 c. ò50ñ2 d. ò25ñ1 e. ó576 f. ò10ò90
Solution a. ñ2ñ8 = ó28= ò16 = 4 b. 7  7  77  49 7
c. ò50ñ2 = ó502 = ó100 = 10 d. 25  1 251 25 5
e. 576  36 16  36  16 6 4 24
f. 10  90  10 90  900 30
Property
For any real numbers aand b, where a0, and b> 0,
a a
 .
b b
24 24
For example,   4 2, and
If a > 0 then 6 6
a a
= = 1 =1. 1 1 1
a a   .
49 49 7
Radicals 15
4
EXAMPLE Simplify the expressions.
25 50 16 1 1 625
a. b. c. d. e. – f.
9 2 49 64 100 144
24a3 a5b6 x y
g. h. i.
6a ab2 x3y3
25 25 5 50 50
Solution a. = = b.   25 5
9 9 3 2 2
16 16 4 1 1 1
c.   d.  
49 49 7 64 64 8
1 1 1 625 625 25
e. –  – – f.  
100 100 10 144 144 12
24a3 24a3
g.   4a2  4  a2 2a
6a 6a
a5b6 a5b6
h.   a4b4  (a2b2)2 a2b2
ab2 ab2
x y xy 1 1 1
i.    
x3 y3 x3y3 x2y2 x2  y2 xy
Property
For any real number a and n  Z, ( a)n  an (a  0).
Proof
( a)n  a  a  a... a a a a ... a an
nfactors of ña nfactors of a
For example, ( a)2  a2 a,
( 5)3  53  125, and
( 2)8  28  256 16.
16 Algebra 8
5
EXAMPLE Evaluate
(ñ2)4 + (ñ5)4 – (ñ5)2 – (ñ2)6.
Solution ( 2)2 ( 5)4 –( 5)2–( 2)6  24  54 – 52 – 26
 (22)2  (52)2 – 52 – (23)2
22 52 –5–23 425–5–8 16
3. Working with Pure and Mixed Radicals
Definition
A radical eexpression is an expression of the form na.
index ñna radical sign
radicand
Square roots have index 2. However, we usually write square roots in their shorter form, ña:
2 a  a
Definition
A mixed rradical is a radical of the form
x na (x  Q, x  {–1, 0, 1})
For example, 3ñ2, 6ñ7, and 9ó115 are mixed radicals.
ò55, ò99, and ò27 are not mixed radicals. We say that they are pure radicals.
We can convert between mixed and pure radical numbers to simplify radical expressions.
Property
For any real numbers a and b, where a  0 andb  0,
a2b  a b and a b a2 b.
For example, 8  42  222  22  2 2 2,
27  93  323  32  3 3 3,
32  16 2  422  42  2 4 2, and
50  252  522  52  2 5 2.
Radicals 17
6
EXAMPLE Simplify the expressions.
a. ñ8+2ò32–ò18+ò72–ò98
b. 2ò48+3ò27–ó108+ó243
Solution a. 8= 222=2 2 


2 32=2 422=8 2

82 32  18  72  98

18= 322=3 2  
 2 28 2 3 2 6 2 7 2
72= 622=6 2 
  2(28367)6 2

98= 722=7 2

b. 2 483 27– 108  243 2 423 3 323– 62 3  92 3
8 39 3–6 3 9 3
(89–69) 3 20 3
7
EXAMPLE Write the numbers as pure radicals.
a. 2ñ2 b. 3ñ5 c. 5ñ3 d. 10ò10 e. xñy
Solution a. 2 2  22  2  22 2 4 2 8
b. 3 5  325  9 5 45
c. 5 3  52 3 25 3 75
d. 10 10  10210  10010  1000
e. x y  x2y
Property
For any non-zero real numbers a, b, c, and x,
añx + bñx – cñx =(a + b – c)ñx.
Note
ña + ñb óa+b
For example,
ñ9 + ò16 = 3 + 4 = 7, but ó9 + 16 = ò25 = 5.
18 Algebra 8