Download Grade 9 Number System

ID : kr-9-Number-System [1]
Grade 9
Number System
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Answer t he quest ions
(1)
(2)
Write the rational number that are equal to its negative.
Find 29 rational numbers between
-3
and
11
(3)
(4)
3
.
11
Write the rational number that does not have a reciprocal.
Write a multiple of
-3
rational number ?
-4
(5)
if
-4
X
=
X
, then X is ________ a rational number
4
Choose correct answer(s) f rom given choice
(6) Choose an irrational number among the f ollowing :
a. π
b. -9
c.
d.
(7) T he number
(8)
is
a. both
b. Rational
c. Irrational
d. Can't say
T he average of the middle two rational numbers if
1
,
5
4
5
,
-4
30
,
1
are arranged in
10
ascending order.
a.
3
b.
20
c.
1
2
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-1
60
d.
1
3
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ID : kr-9-Number-System [2]
(9)
Express 0.0275 in the f orm of
p
and reduce it to the lowest terms.
q
275
a.
b.
10000
c.
11
400
400
d.
11
275
100000
(10) T he sum of a rational and an irrational numbers is ________________
a. integer
b. an irrational number
c. a rational or an irrational number
d. a rational number
(11) Is
a rational or irrational number?
a. Rational
b. Irrational
(12) Regarding rational numbers:
A. T he quotient of two integers is always a rational number and
B.
5
is a rational number
0
Which of the f ollowing statements is correct ?
a. A is correct but B is incorrect.
b. A is f alse but B is a correct explanation of A
c. A is correct and B is correct explanation of A d. Both A and B are f alse
(13) T he product of two irrational numbers is ________________
a. an irrational number
b. a rational number
c. neither rational or irrational number
d. a rational or an irrational number
Fill in t he blanks
(14)
T he decimal f orm of an irrational number is neither
nor
.
(15) Find the multiplicative inverse of the f ollowing.
A)
-14
→
13
B)
-6
→
11
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ID : kr-9-Number-System [3]
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ID : kr-9-Number-System [4]
Answers
(1)
0
Step 1
Rational numbers: A rational number is any number that can be expressed as the quotient
or f raction p/q of two integers, p and q, with the denominator q not equal to zero. Since q
may be equal to 1, every integer is a rational number.
Rational number zero(0) is the only rational number that is equal to its negative.
Step 2
T heref ore zero(0) is the only rational number that is equal to its negative.
(2)
-14
-13
,
55
0
-12
,
55
1
,
55
55
2
,
55
-11
,
55
3
,
55
-10
,
,
55
-9
,
55
4
55
55
5
,
-8
,
55
55
6
,
-7
,
55
7
,
55
-6
,
55
55
8
,
-5
,
55
55
9
,
-4
,
,
55
-3
,
55
10
,
55
-2
,
55
11
55
12
,
55
-1
,
,
55
,
55
13
,
55
14
,
55
15
55
Step 1
p
A rational number is a number of the f orm
, where p and q are integers, and q≠0.
q
Step 2
-3
3
and
11
can be represented as
-15
11
15
and
55
-3
T heref ore, 29 rational numbers between
3
and
11
-10
,
55
4
55
-9
,
55
,
5
55
-8
,
55
,
6
55
-7
,
55
,
7
55
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-6
,
55
,
8
55
-5
,
55
,
9
55
-4
,
55
,
10
55
respectively.
55
11
-3
,
55
,
are
11
55
-2
12
55
,
-13
55
,
55
,
-14
-1
,
55
,
13
55
55
0
,
55
,
-12
,
14
55
55
1
,
55
,
-11
,
15
2
55
,
55
,
3
,
55
.
55
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ID : kr-9-Number-System [5]
(3)
0
Step 1
Rational numbers: A rational number is any number that can be expressed as the quotient
or f raction p/q of two integers, p and q, with the denominator q not equal to zero. Since q
may be equal to 1, every integer is a rational number.
2
For example: reciprocal of
is
3
3
.
2
Now,
0
is a rational number.
a
(where a is non-zero integer)
reciprocal of
0
is
a
a
which is not a rational number because denominator is not equal
0
to zero(0) in rational numbers.
Step 2
T heref ore rational number zero(0) does not have a reciprocal.
(4)
9
(Answers may vary)
4
Step 1
Multiples of rational number means the multiplication of the rational number with some
given number or any number.
x1
For example, suppose
is the rational number.
x2
multiple of
x1
x2
are
2x1
x2
,
3x1
, etc.
x2
Step 2
T heref ore, one example of multiple of
-3
-4
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is
9
.
4
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ID : kr-9-Number-System [6]
(5)
not
Step 1
Rational numbers: A rational number is any number that can be expressed as the quotient
or f raction p/q of two integers, p and q, with the denominator q not equal to zero. Since q
may be equal to 1, every integer is a rational number.
For example: reciprocal of
p
q
is
q
.
p
Step 2
If you look at the question caref ully, you will notice that you have
-4
X
Now,
-4
X
=
=
X
.
4
X
4
⇒ -4 × 4 = X × X
-16 = X2
X = √-16
√-16 is not a rational number.
Step 3
T heref ore X is not a rational number, if
-4
X
=
X
.
4
(6) a. π
Step 1
Irrational Number: A real number that can not be written as a simple f raction.
Decimal numbers which are either terminating or repeating can be written as f raction
and are not irrational,
e.g. 1.23 = 123/100, 1.33333333.... = 4/3
On the other hand numbers, which are neither terminating nor repeating cannot be
written as f raction and are irrational numbers.
.e.g. π = 3.141592654..... , √2 = 1.4142135623....
Step 2
Now, if we look at the all options, we notice that π can not be written as a simple f raction
and hence π is an irrational number.
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ID : kr-9-Number-System [7]
(7) c. Irrational
Step 1
We know that √6 is an irrational number as it cannot be represented in p/q f orm.
Step 2
√6 is used in a f raction in the given expression. T hus we cannot be sure if the given
expression is rational or irrational until we simplif y it.
Step 3
Let's simplif y the given expression:
=
⇒=
⇒=
⇒=
Step 4
Now we can see that the f irst part,
42
, is rational as it is in p/q f orm. T he second part,
30
on the other hand, has √6 which is in its simplif ied f orm.
Step 5
T heref ore we are now certain that the given number is an Irrational number
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ID : kr-9-Number-System [8]
(8)
3
a.
20
Step 1
T o compare f ractions, f irst we need to make sure that all denominators are same, so we
can just compare the numerators of f ractions.
Step 2
T he LCM of the denominators 5, 5, 30 and 10 = 30
Step 3
Now, divide the LCM by the denominators and multiply the result with the numerator and
denominator as f ollowing:
1×6
,
5×6
6
or
4 ×6
-4 × 1
,
5×6
,
30
24
,
30
30 × 1
-4
,
30
,
1×3
10 × 3
3
30
Step 4
Let’s arrange the given numbers in ascending order, we get:
-4
,
30
3
,
30
6
30
,
24
30
Step 5
3
Now, the average of the middle two rational numbers =
+
30
6
30
2
=
9
30
=
×
1
2
3
20
Step 6
T hus, the average of the middle two rational numbers is
3
.
20
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ID : kr-9-Number-System [9]
(9)
b.
11
400
Step 1
T he simplest way to solve such problems is to multiple by a f raction
n
such that the
n
numerator becomes an integer.
T he next step is to reduce the f raction to the simplest f orm.
Step 2
Here, we can see that, if we multiply the numerator by 10000, it will become an integer.
So we do the f ollowing, 0.0275 = 0.0275 x
10000
10000
Step 3
T his reduces to
275
,
10000
Which in it's simplest f orm is =
11
400
(10) b. an irrational number
Step 1
Let's assume that the sum of rational and irrational number is a rational number.
rational number + irrational number = rational number
m
a
+x=
n
⇒x=
b
a
-
b
⇒x=
m
n
mb - na
nb
[n,b are integer and product of two integers are must be an integer.]
[mb and na are integer and dif f erence of two integer must be an integer.]
Now x is a rational number but we assume that x is an irrational number.
T heref ore our assumption that sum could be rational number, is not correct
Step 2
T hus, the sum of a rational number and an irrational number is an irrational number.
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(11) b. Irrational
Step 1
We know that 2 is a prime number. Since square root of a prime number is an irrational
number, √2 is an irrational number.
Step 2
Similarly, since 3 is also a prime number, √3 is also an irrational number
Step 3
Also if an irrational number is added to an irrational number, result is an irrational number
(unless irrational parts cancel with each other), theref ore √2 + √3 is an irrational number.
(12) d. Both A and B are f alse
Step 1
Rational Numbers: A rational number is a number that can be expressed as a f raction. A
rational number said to have numerator and denominator.
Step 2
Condition f or rational number: T he quotient of two integers is always a rational number
provided the denominator is non-zero.
Step 3
According to the condition of rational numbers the f irst statement that the quotient of the
two integers is always a rational number is not satisf ied, that is, statement is f alse.
Step 4
5
is the example against the condition of rational numbers. T heref ore
0
5
is not a
0
rational number
Step 5
T heref ore, Both A and B are f alse.
(13) d. a rational or an irrational number
Step 1
T he right answer is that it could be either. T his can be seen with the help of an example
Step 2
A) Consider the two irrational numbers
and
. T heir product =
x
=
is an irrational number
Step 3
B) Consider the two irrational numbers
T heir product =
x
= 12 +
and
.
2 = 1 + 9 = 10.
10 is a rational number
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ID : kr-9-Number-System [11]
(14)
T he decimal f orm of an irrational number is neither repeating
terminating
.
(15) A)
-14
nor
13
→
13
-14
Step 1
Multiplicative Inverse: When we multiply a number by its "Multiplicative Inverse",
we get 1.
Mathematically,
n×
1
=1
n
Step 2
T he multiplicative inverse of
-14
is
13
B)
-6
13
, since
-14
-14
×
13
13
= 1.
-14
11
→
11
-6
Step 1
Multiplicative Inverse: When we multiply a number by its "Multiplicative Inverse",
we get 1.
Mathematically,
n×
1
=1
n
Step 2
T he multiplicative inverse of
-6
11
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is
11
-6
, since
-6
11
×
11
= 1.
-6
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