Grade 9 Number Systems Natural numbers The counting numbers 1, 2, 3 … are called natural numbers. The set of natural numbers is denoted by N. N = {1, 2, 3, …} Whole numbers
If we include zero to the set of natural numbers, then we get the set of whole numbers. The set of whole numbers is denoted by W. W = {0, 1, 2, …} Integers The collection of numbers … – 3, 3, – 2, 2, – 1, 1, 0, 1, 2, 3 … is called integers. This collection is denoted by Z, or I. 3, – 2, 2, –1, 0, 1, 2, 3, …} Z = {…, – 3,
Rational numbers
Rational numbers are those which can be expressed in the form integers and q Example:
p q
, where p, q are
0.
1 3 6 , , , etc. 2 4 9
Note:
1.
12
12 3
4
, where the HCF of 4 and 5 is 1
15 15 3 5 12 4 and are equivalent rational numbers (or fractions) 15 5 a , where a, b are integers Thus, every rational number ‘x ’can be expressed as x b such that the HCF of a and b = 1 and b 0.
2. Every natural number is a rational number. 3. Every whole number is a rational number. [Since every whole number W can be expressed as
W
1
].
4. Every integer is a rational number. There are infinitely many rational numbers between any two given rational numbers.
Example:
Find 5 rational numbers
3 8
and
5 12
.
Solution: 3
3 3
9
9 6
54
8 8 3 24 24 6 144 5 5 2 10 10 6 60 12
12 2
24
24 6
144
It can be observed that: 54
55
56
144 144 3 55
57
58
59
60
144 144 144 144 144 7 19 29 59 5
8
144 18 48 72 144 12 3 5 55 7 19 29 59 . Thus, , , , and are 5 rational numbers between and 8 12 144 18 48 72 144
Irrational Numbers
Irrational numbers are those which cannot be expressed in the form are integers and q Example:
p q
, where p, q
0.
2, 7, 14, 0.0202202220.......
There are infinitely many irrational numbers. Real Numbers
The collection of all rational numbers and irrational numbers is called real numbers. So, a real number is either rational or irrational. Note: Every real number is represented by a unique point on the number line (and vice versa). So, the number line is also called the real number line. Example:
Locate 6 on the number line. Solution:
It is seen that: 6
5
2
12
To locate 6 on the number line, we first need to construct a length of
5
.
22 1
5
By Pythagoras Theorem: OB2
OA2
OB
AB2
22
12
4 1 5
5
Steps:
(a) Mark O at 0 and A at 2 on the number line, and then draw AB of unit length 5 perpendicular to OA. Then, by Pythagoras Theorem, OB (b) Construct BD of unit length perpendicular to OB. Thus, by Pythagoras Theorem, OD
5
2
12
6
(c) Using a compass with centre O and radius OD, draw an arc intersecting the number line at point P. Thus, P corresponds to the number 6 .
Real numbers and their decimal expansions:
The decimal expansion of a rational number is either terminating or non-terminating recurring (repeating). Moreover, a number whose decimal expansion is terminating or non-terminating repeating is rational. Example: 3 2 15 8 4 3 24 13
1.5 1.875
Terminating Terminating
1.333....... 1.3
Non – terminating recurring
1.846153846153 1.846153
Example:
Non-terminatingrecurring
Show that 1.23434 …. can be written in the form and q
p q
, where p and q are integers
0.
Solution: Let x
1.23434..... 1.234
1
Here, two digits are repeating. Multiplying (1) by 100, we get: 100x = 123.43434……… =122.2 + 1.23434 …….. Subtracting (1) from (2), we get: 99 x
x
(2)
122.2 122.2
1222
99
990 661 495
Thus,1.234
611 495
The decimal expansion of an irrational number is non-terminating non-recurring. Moreover, a number whose decimal expansion is non-terminating non- recurring is irrational. Example:
2.645751311064……. is an irrational number Representation of real numbers on the number line Example: Visualize 3.32 on the number line, upto 4 decimal places. Solution: 3.32
3.3232...... 3.3232
approximate upto 4 decimal place
Now, it is seen that 3 < 3.3232 < 4. Divide the gap between 3 and 4 on the number line into 10 equal parts and locate 3.3232 between 3.3 and 3.4 [as 3.3 < 3.3232 < 3.4]. To locate the given number between 3.3 and 3.4 more accurately, we divide this gap into 10 equal parts. It is seen that 3.32 < 3.3232 < 3.33. We continue the same procedure by dividing the gap between 3.32 and 3.33 into 10 equal parts. It is seen that 3.323 < 3.3232 < 3.324. Now, by dividing the gap between 3.323 and 3.324 into 10 equal parts, we can locate 3.3232.
Operation on real numbers
Some facts (a) The sum or difference of a rational number and an irrational number is always irrational. (b) The product or quotient of a non-zero rational number with an irrational number is always irrational. (c) If we add, subtract, multiply or divide two irrational numbers, then the result may be rational or irrational. Illustrations 2
3 is irrational
2
2
3
5
2
2
6
2
15 is irrational
2 is rational
3 is irrational
2 2
0 is rational
1 is rational
Identities
If a and b are positive real numbers, then ab a b a. b.
a
a
b
b
c.
a
d.
b
a
a
b
a
e.
a
b
f.
a
b
b
a b
b
a
c 2
a
The denominator of
2
2
b
d
ac
2 ab
ad
bc
bd
b
a
b
x
y
can be rationalised by multiplying both the
numerator and the denominator by x
y , where a, b, x, y are integers.
Laws of exponents
1) a p .a q 2) 3) 4)
a a
q
p
a
p q
a
pq
p
a
a
q
ab
p
p q
p p a b , where a > 0 is a real number and p, q are rational numbers.
Note: 1 x
a
ax
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