Download 1. Revision Description Reflect and Review Teasers

1. Revision
Description
Recall of Rational Numbers:





Reflect and Review
Teasers
To find the value of
:
1)
1) Reduce
to
Recall ofstandard
rational numbers
and
operations on
form.
A rational number is of the form ,
where p and q are integers and q≠ 0.
Addition or subtraction of rational
numbers is possible only when they
have the same denominator.
The Product of two rational
numbers is obtained by multiplying
the respective numerators and
denominators with each other.
The Multiplicative Inverse or
Reciprocal of a rational number
Answers
2) Find the product
of and
2)
Since LCM of 17 and 34 is 34,

is , where a and b are non-zero
integers.
Dividing a rational number by
another rational number is same as
multiplying the dividend by the
reciprocal of the divisor.
2. Properties of Rational Numbers
Description
Reflect and Review

Closure Property:
1. Addition: When two
rational numbers are added,
the sum is also a rational
number. Hence, rational
numbers are closed under
addition.
2. Subtraction: When two
rational numbers are
subtracted, the difference is
also a rational number.
=
which is a
rational number.
=
which is a
rational number.
Teasers
Answers
1) a) Multiply
with additive
inverse of
.
1) a)
b) 1
2) a)
b)
b) Add
multiplicative
inverse of
and
(
)
1
Hence, rational numbers
are closed under
subtraction.
3. Multiplication: When two
rational numbers are
multiplied, the product is
also a rational number.
Hence, rational numbers
are closed under
multiplication.
4. Division: When we divide
two rational numbers except
for division by zero (which is
also a rational number), we
get a rational number. So,
rational numbers are not
closed under division. But, if
zero is excluded, we can say
that the collection of other
rational numbers is closed
under division.

= which is a
rational number.
2. Subtraction: For any two
rational numbers ‘a’ and ‘b’,
a - b  b – a. Hence,
subtraction of rational
numbers is not
commutative.
3. Multiplication: For any
two rational numbers ‘a’ and
‘b’, a × b = b × a. Hence,
multiplication of rational
numbers is commutative.
4. Division: For any two
rational numbers ‘a’ and ‘b’,
to get
the sum as
zero?
b) What
must be
multiplied to
= which is a
rational number.
Commutative Property:
1. Addition: For any two
rational numbers ‘a’ and ‘b’,
a + b = b + a. Hence,
addition of rational
numbers is commutative.
2
2) a) What must
be added to
=
and
=
So,
=
=
and
=

So,
=
and
=
So,
=
=
and
to get
the product
1?
a ÷ b  b ÷ a. Hence, division
of rational numbers is not
commutative.

2. Subtraction: For any three
rational numbers a, b and c,
(a – b) – c  a – (b – c).
Hence, subtraction is not
associative for rational
numbers.
3. Multiplication: For any
three rational numbers a, b
and c, (a × b) × c = a × (b ×
c). Hence, multiplication is
associative for rational
numbers.
4. Division: For any three
rational numbers a, b and c,
(a ÷ b) ÷ c  a ÷ (b ÷ c).
Hence, division is not
associative for rational
numbers.
(
)
Hence, 0 is the additive
identity for rational
numbers.
=
(
and
) =
So, (
)
(
=
(
)
)
=
(
and
) =
So, (
)

(
(
)
)
=
(
)
(
=
(
and
) =
So, (
)
)
= and
(
) =
So, (
)

(
Additive Identity:
For any rational number a,
a + 0 = 0 + a = a.


So,
Associative Property:
1. Addition: For any three
rational numbers a, b and c,
(a + b) + c = a + (b + c).
Hence, addition is
associative for rational
numbers.

=
)
+0=
0+
=
So,
=
=
Multiplicative Identity:
For any rational number a,
×1=
3
a × 1 = 1 × a = a.
1×
Hence, 1 is the
multiplicative identity for
rational numbers.

=
So,
=
=
Additive Inverse:
+
+
If is a rational number,
then there exists another
rational number
=
So,
=
+
=
+
=
such
that
(
)
We say
(
)
is the additive
inverse of
and
additive inverse of
is the
.
=
=

Multiplicative Inverse:
So,
=
=
=
=
If is a (non-zero) rational
number, then there exists a
rational number , such that
.
We say is the
+
multiplicative inverse or
reciprocal of .
(
)
=( )

Distributive Property:
So,
For all rational numbers a, b
and c,
a × ( b + c) = (a × b) + (a × c)
4
( )=

+
=(
)
3. Representation of Rational Numbers on the Number Line
Description
Reflect
Teasers
Answers
and
Review

Rational numbers can be
1) Represent
on See below
See
represented on the number
the number line.
below
the table
line the same way as
the table
integers and fractions.
Reflect and Review

To represent
on the number line
Divide the whole number line between 0 and -1 into 7 equal parts and the third point of division
to the left of zero marked as A represents
.
A
𝟑
𝟕
-1
0
Answers
1)
P
-2
𝟏𝟎
𝟗
-1
5
4. Rational Numbers between Two Rational Numbers
Description
Reflect and Review

There are
infinitely
many rational
numbers
between two
given rational
numbers.

To find 5 rational number between
and :
Convert both the numbers into
rational numbers with same
denominator.
LCM of 5 and 3 is 15.
So,
and
The rational numbers between
and
are
We can take any five of them as the
solution.

To find 3 rational numbers between
– 2 and 1:
The mean of two rational numbers ‘a’
and ‘b’ is given by
.
Here a = -2, b = 1
Mean =

Now, we can find the mean of -2 and
.
(
Mean = [
)]
(

Next, we can find the mean of
and
.
Mean = [(
(

6
)
)
(
(
)]
)
)
Teasers
1) Write 5
rational
numbers
between -1
and 2.
Answers
1)