Q: 1
Is zero a rational number? Can you write it in the form
, where p and q are integers and q
0?
Answer
Yes zero is a rational number as it can be represented in the
q
0 as
form, where p and q are integers and
etc.
Concept Insight: Key idea to answer this question is "every integer is a rational number and zero is a
non negative integer". Also 0 can be expressed in
0. simplest is
.
Class 9th - Chapter 1 - Number Systems
form in various ways as 0 divided by any number is
Q: 2
Find six rational numbers between 3 and 4.
Answer
There are infinite rational numbers in between 3 and 4.
3 and 4 can be represented as
respectively.
Now rational numbers between 3 and 4 are
Concept Insight: Since there are infinite number of rational numbers between any two numbers so the
answer is not unique here. The trick is to convert the number to equivalent
form by multiplying and
dividing by the number atleast 1 more than the rational numbers to be inserted.
Class 9th - Chapter 1 - Number Systems
Q: 3
Find five rational numbers between
.
Answer
There are infinite rational numbers between
Now rational numbers between are
Concept Insight: Since there are infinite number of rational numbers between any two numbers so the
answer is not unique here. The trick is to convert the number to equivalent
dividing by the number at least 1 more than the rational numbers required.
Alternatively for any two rational numbers a and b,
a and b.
Class 9th - Chapter 1 - Number Systems
form by multiplying and
is also a rational number which lies between
Q: 4
State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number
(ii) Every integer is a whole number
(iii) Every rational number is a whole number
Answer
(i)
True, since collection of whole numbers contains all natural numbers.
(ii) False, integers include negative of natural numbers as well, which are clearly not whole numbers. For
example -1 is an integer but not a whole number.
(iii) False, rational numbers includes fractions and integers as well. For example
but not whole number.
is a rational number
Concept Insight: Key concept involved in this question is the hierarchy of number systems
Remember the bigger set consists of the smaller one.
Since Mathematics is an exact science every fact has a proof but in order to negate a statement even one
counter example is sufficient.
Class 9th - Chapter 1 - Number Systems
Q: 1
State whether the following statements are true or false. Justify your answers.
(i)
Every irrational number is a real number.
(ii) Every point on the number line is of the form
(iii) Every real number is an irrational number.
, where m is a natural number.
Answer
(i) True, since real numbers consists of rational and irrational numbers.
(ii) False, Since negative integers cannot be expressed as the square root of any natural number.
(iii) False, real number includes both rational and irrational numbers. So every real number can not be an
irrational number.
Concept Insight: Mentioning the reasons is important in this problem. Real Numbers consists of rational
and irrational numbers and not vice versa. Every real number corresponds to a point on number line and
vice versa.
Recall real number includes negative numbers also. Square root of negative numbers is not defined.
Class 9th - Chapter 1 - Number Systems
Q: 2
Are the square roots of all positive integers irrational? If not, give an example of the square root of a
number that is a rational number.
Answer
Square roots of all square numbers are rational.
For example
Thus the square roots of all positive integers are not irrational
Concept Insight: In general only the square root of a prime number is irrational.
Therefore square root of perfect square numbers are rational.
Class 9th - Chapter 1 - Number Systems
Q: 3
Show how
can be represented on the number line.
Answer
Using Pythagoras Theorem: 5=22+12
Taking positive square root we get
1. Mark a point 'A' representing 2 units on number line.
2. Now construct AB of unit length perpendicular to OA. Join OB
3. Now taking O as centre and OB as radius draw an arc, intersecting number line at point C.
4. Point C represents
on number line. [length (OB) = length (OC)]
Concept Insight: For a positive integer n,
can be located on number line , if
using Pythagoras Theorem . If is a perfect square then this method is useful.
To represent the irrational number
of
is located
key idea is to use Pythagoras theorem and create a length
units by constructing a right triangle of base and perpendicular of length 2 and 1 units.
Class 9th - Chapter 1 - Number Systems
Q: 1
Write the following in decimal form and say what kind of decimal expansion each has:
(i)
(ii)
(iv)
(iii)
(v)
(vi)
Answer
(i)
terminating
(ii)
non terminating repeating
(iii)
Terminating
(iv)
non terminating repeating
(v)
non terminating repeating decimal
(vi)
Terminating decimal
Concept Insight: The decimal expansion of a rational number is either terminating or non terminating
recurring.
Decimal expansion terminates in case the prime factors of denominator includes 2 or 5 only.
Class 9th - Chapter 1 - Number Systems
Q: 2
You know that
. Can you predict what the decimal expansion of
are, without actually doing the long division? If so, how?
Answer
Yes it can be done as follows:
Concept Insight: Multiples of the given decimal expansion can be obtained by simple multiplication with
the given constant. Cross check the answer by performing long division.
Class 9th - Chapter 1 - Number Systems
Q: 3
Express the following in the form
, where p and q are integers and
.
Answer
(i)
Let x = 0.666 ... (i)
Multiplying by 10 we
get 10x = 6.666 ... (ii)
(ii) - (i) gives
9x = 6
Or x =
(ii)
Let x = 0.4777 ... (i)
10x = 4.777 ...
100x = 47.777 ...
(ii) (ii) - (i) gives
99 x = 43
x=
(iii)
Let x = 0.001001 ...(i)
1000x = 1.001001 ...(ii)
(ii) - (i) gives
999x = 1
x=
Concept Insight: The key idea to express a recurring decimal in the p/q form is to multiply the number
by the 10n where n = number of digits repeating.
This is done to make the repeating block a whole number part of the decimal. By subtracting the two
expressions x can be expressed in the P/q form
Class 9th - Chapter 1 - Number Systems
Class 9th - Chapter 1 - Number Systems
Q: 4
Express 0.99999 ..... in the form
. Are you surprised by your answer? With your teacher and
classmates discuss why the answer makes sense.
Answer
Let x = 0.9999 .. .. ..(i)
10x = 9.9999 ... ...(ii)
(ii) - (i) gives
9x = 9
x=1
Concept Insight: .9999999 ..... is nothing but 1 when expressed in p/q form.
Class 9th - Chapter 1 - Number Systems
Q: 5
What can be the maximum number of digits be in the repeating block of digits in the decimal expansion
of
? Perform the division to check your answer.
Answer
Expressing
in the decimal form we
There are 16 digits in repeating block of decimal expansion of
.
Concept Insight: Maximum number of digits that can repeat will be 1 less than the prime number in
denominator.
Class 9th - Chapter 1 - Number Systems
Q: 6
Look at several examples of rational numbers in the form
where p and q are
integers with no common factors other than 1 and having terminating decimal representations
(expansions). Can you guess what property q must satisfy?
Answer
Terminating decimal expansion will come when denominator q of rational number
8, 10, and so on ... ...
, is either of 2, 4, 5,
Terminating decimal may be obtained in the situation where prime factorisation of the denominator of
the given fractions are having power of 2 only or 5 only or both.
Concept Insight: A rational number in its simplest form will terminate only when prime factors of its
denominator consists of 2 or 5 only.
Class 9th - Chapter 1 - Number Systems
Q: 7
Write three numbers whose decimal expansions are non-terminating non-recurring.
Answer
3 numbers whose decimal expansion is non terminating non recurring are ... ... ,
0.505005000051509 ... ... ...
0.72012009200011500007200000 ... ... ...
7.03124509761202 ... ... ... ... ... ...
Concept Insight: Recall that a non terminating non recurring decimal is an irrational number. Answer to
such questions is not unique.
Class 9th - Chapter 1 - Number Systems
Q: 8
Find three different irrational numbers between the rational numbers
Answer
3 irrational numbers are 0.73073007300073000073 ... ... ...
0.75075007500075000075 ... ... ...
0.79079007900079000079 ... ... ...
Concept Insight: There is infinite number of rational and irrational numbers between any two
rational numbers. Convert the number into its decimal form to find irrationals between them.
Alternatively following result can be used to answer
Irrational number between two numbers x and y
Class 9th - Chapter 1 - Number Systems
Q: 9
Classify the following numbers as rational or irrational:
Answer
(i)
As decimal expansion of this number is non-terminating non recurring. So it is an irrational number.
(ii)
Rational number as it can be represented in
form.
(iii) 0.3796
As decimal expansion of this number is terminating, so it is a rational number.
(iv)
As decimal expansion of this number is non terminating recurring so it is a rational number.
(v)
As decimal expansion of this number is non terminating non repeating so it is an irrational number.
Concept Insight: A number is rational if its decimal expansion is either terminating or non terminating
but recurring. A number which cannot be expressed in p/q form is irrational. Square root of prime
numbers is always irrational.
Class 9th - Chapter 1 - Number Systems
Q: 1
Visualise 3.765 on the number line using successive magnification.
Answer
3.765 can be represented
Concept Insight: Divide the number line between the number to be represented in 10 parts starting the
whole number part.
Class 9th - Chapter 1 - Number Systems
Q: 2
Visualise
on the number line, up to 4 decimal places.
Answer
= 4.2626
We can visualise 4.2626 as in following steps
Concept Insight: Divide the number line between the number to be represented in 10 parts starting the
whole number part.
Class 9th - Chapter 1 - Number Systems
Q: 1
Classify the following numbers as rational or irrational:
Answer
(i)
As decimal expansion of this expression is non terminating non recurring, so it is an irrational number.
(ii)
It can be represented in
form so it is a rational number.
(iii)
As it can be represented in
form, so it is a rational number.
(iv)
As decimal expansion of this expression is non terminating non recurring, so it is an irrational number.
(v)
As decimal expansion is non terminating non recurring, so it is an irrational number.
Concept Insight: Do the simplifications as indicated and see whether the number is terminating,
non terminating recurring or neither terminating nor repeating. Remember Sum/difference/Product of
a rational and irrational number may or may not be irrational.
Class 9th - Chapter 1 - Number Systems
Q: 2
Simplify each of the following expressions:
Answer
Concept Insight: Apply the algebraic identities (a+b)2, (a-b)2,(a+b)(a-b) etc to simplify the
given expressions.
Equivalent Identities used here are
Class 9th - Chapter 1 - Number Systems
Q: 3
Recall,
is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That
is,
. This seems to contradict the fact that
contradiction?
is irrational. How will you resolve this
Answer
There is no contradiction. Since
here circumference or diameter are not given to be integers . When
we measure a length with scale or any other instrument, we only get an approximate rational value.
We never get an exact value. c or d may be irrational. So, the fraction
irrational.
Concept Insight: A rational number is the number of the form
integers. In
is irrational. Therefore,
where p and q are
c and d are not integers. Also remember that no measurement is exact.
Class 9th - Chapter 1 - Number Systems
is
Q: 4
Represent
on the number line.
Answer
(i)
(ii)
(iii)
(iv)
Mark a line segment OB = 9.3 on number line.
Take BC of 1 unit.
Find mid point D of OC and draw a semicircle on OC while taking D as its centre.
Draw a perpendicular to line OC passing through point B. Let it intersect semicircle at E. Length of
perpendicular BE =
.
(v) Taking B as centre and BE as radius draw an arc intersecting number line at F. BF is
represents
Verification: In
ED2=EB2+DB2
i.e point F
on number line
EDB
Using Pythagoras theorem
Concept Insight: This method based on the application of Pythagoras theorem can be used to represent
root of any rational number on the number line.
Class 9th - Chapter 1 - Number Systems
The key idea to represent
In
is to create a length of
ODB
DB =
Class 9th - Chapter 1 - Number Systems
units.
Q: 5
Rationalise the denominators of the following:
Answer
Concept Insight: Rationalisation of denominator means converting the irrational denominator to
rational i.e . removing the radical sign from denominator.A number of the form
can be
converted to rational form by multiplying with its conjugate. Remember the algebraic identities
Class 9th - Chapter 1 - Number Systems
Class 9th - Chapter 1 - Number Systems
Q: 1
Answer
Concept Insight: Express the number in exponent notation and use the
rule Exponent m must be such that it is divisible by n.
Class 9th - Chapter 1 - Number Systems
Q: 2
Answer
Concept Insight: Express the number in exponent notation and use the rule of exponents.
Class 9th - Chapter 1 - Number Systems
Q: 3
Answer
Concept Insight: Use the rule of exponents
Class 9th - Chapter 1 - Number Systems