Cheatsheet: Number System | Mathematics (Maths) Class 9 PDF Download

 Page 1


Number S ystems - Class 9 Cheatsheet
1. Number Line and Number Types
Subtopic Brief
Number Line
A line representing numbers as points, starting at
zero, extending infinitely in positive and negative
directions, used to visualize number types.
Natur al Numbers
Positive integers (1, 2, 3, ...), denoted b y N ,
infinitely man y .
Whole Numbers
Natur al numbers plus zero (0, 1, 2, ...), denoted b y
W . Zero is a whole number but not a natur al
number .
Integers
Whole numbers plus negative numbers (..., -2, -1,
0, 1, 2, ...), denoted b y Z . Every integer m is
r ational as
m
1
(e.g.,-25 =
-25
1
).
Rational Numbers
Numbers of the form
p
q
, where p and q are
integers, q?= 0 , denoted b y Q (e.g.,
1
2
,
-2005
2006
).
Includes natur al numbers, whole numbers, and
integers.
3
5
is r ational but n ot an integer .
Equivalent
Rationals
Rational numbers have multiple forms (e.g.,
1
2
=
2
4
=
10
20
). On the number line, use the form
where p and q are co-prime (no common factors
other than 1).
Finding Rationals
T o find r ational numbers between two numbers
(e.g., 1 and 2), use: (1) A ver age method:
1+2
2
=
3
2
,
repeat to get
5
4
,
11
8
,
13
8
,
7
4
. (2) Denominator method:
Write 1 =
6
6
, 2 =
12
6
, yielding
7
6
,
8
6
,
9
6
,
10
6
,
11
6
. There
are infinitely man y r ationals between an y two
r ationals.
1
Page 2


Number S ystems - Class 9 Cheatsheet
1. Number Line and Number Types
Subtopic Brief
Number Line
A line representing numbers as points, starting at
zero, extending infinitely in positive and negative
directions, used to visualize number types.
Natur al Numbers
Positive integers (1, 2, 3, ...), denoted b y N ,
infinitely man y .
Whole Numbers
Natur al numbers plus zero (0, 1, 2, ...), denoted b y
W . Zero is a whole number but not a natur al
number .
Integers
Whole numbers plus negative numbers (..., -2, -1,
0, 1, 2, ...), denoted b y Z . Every integer m is
r ational as
m
1
(e.g.,-25 =
-25
1
).
Rational Numbers
Numbers of the form
p
q
, where p and q are
integers, q?= 0 , denoted b y Q (e.g.,
1
2
,
-2005
2006
).
Includes natur al numbers, whole numbers, and
integers.
3
5
is r ational but n ot an integer .
Equivalent
Rationals
Rational numbers have multiple forms (e.g.,
1
2
=
2
4
=
10
20
). On the number line, use the form
where p and q are co-prime (no common factors
other than 1).
Finding Rationals
T o find r ational numbers between two numbers
(e.g., 1 and 2), use: (1) A ver age method:
1+2
2
=
3
2
,
repeat to get
5
4
,
11
8
,
13
8
,
7
4
. (2) Denominator method:
Write 1 =
6
6
, 2 =
12
6
, yielding
7
6
,
8
6
,
9
6
,
10
6
,
11
6
. There
are infinitely man y r ationals between an y two
r ationals.
1
2. Irr ational Numbers
Subtopic Brief
Definition
Numbers that cannot be written as
p
q
, wherep and
q are integers, q?= 0 (e.g.,
v
2 ,
v
3 ,
v
15 , p ,
0.10110111011110...). Infinitely man y exist.
Real Numbers
Rational and irr ational numbers together form
real numbers, denoted b y R . Every point on the
number line corresponds to a unique real
number , and vice versa.
Locating Irr ationals
T o locate
v
2 : Use a square with side 1, diagonal
v
2 (Pythagor as theorem), tr ansfer to number line
with compass arc at point P . F or
v
3 : From point P ,
construct a right triangle with sides
v
2 and 1,
yielding
v
3 at point Q . F or an y positive integer n ,
locate
v
n after
v
n-1 .
Square Root Spir al
Construct a spir al starting with point O , segment
OP
1
of unit length, then P
1
P
2
perpendicular to OP
1
(unit length), continuing to form points P
2
, P
3
, ...,
depicting
v
2 ,
v
3 ,
v
4 , ...
3. Decimal Expansions
Subtopic Brief
Rational Decimals
Rational numbers have either terminating
decimals (e.g.,
7
8
= 0.875 ,
1
2
= 0.5 ,
639
250
= 2.556 ) or
non-terminating recurring decimals (e.g.,
10
3
= 3.3 ,
1
7
= 0.142857 ). Remainders become zero or repeat
(repeating block size < divisor).
Irr ational Decimals
Irr ational numbers have non-terminating
non-recurring decimals (e.g.,
v
2 = 1.414213562... ,
p = 3.141592653... ). Numbers lik e
0.10110111011110... are irr ational due to
non-repeating pattern.
Converting
Decimals
T erminating: 3.142678 =
3142678
1000000
. Non-terminating
recurring: 0.3 = x , s o 10x = 3.3 , 10x-x = 3 , x =
1
3
.
F or 1.27 , 100x = 127.27 , 100x-x = 126 , x =
14
11
. F or
0.235 , 100x = 23.535 , 100x-x = 23.3 , x =
233
990
.
Irr ational Between
Rationals
Between
1
7
= 0.142857 and
2
7
= 0.285714 , an
irr ational is 0.150150015000... (non-terminating
non-recurring). Infinitely man y such numbers
exist.
2
Page 3


Number S ystems - Class 9 Cheatsheet
1. Number Line and Number Types
Subtopic Brief
Number Line
A line representing numbers as points, starting at
zero, extending infinitely in positive and negative
directions, used to visualize number types.
Natur al Numbers
Positive integers (1, 2, 3, ...), denoted b y N ,
infinitely man y .
Whole Numbers
Natur al numbers plus zero (0, 1, 2, ...), denoted b y
W . Zero is a whole number but not a natur al
number .
Integers
Whole numbers plus negative numbers (..., -2, -1,
0, 1, 2, ...), denoted b y Z . Every integer m is
r ational as
m
1
(e.g.,-25 =
-25
1
).
Rational Numbers
Numbers of the form
p
q
, where p and q are
integers, q?= 0 , denoted b y Q (e.g.,
1
2
,
-2005
2006
).
Includes natur al numbers, whole numbers, and
integers.
3
5
is r ational but n ot an integer .
Equivalent
Rationals
Rational numbers have multiple forms (e.g.,
1
2
=
2
4
=
10
20
). On the number line, use the form
where p and q are co-prime (no common factors
other than 1).
Finding Rationals
T o find r ational numbers between two numbers
(e.g., 1 and 2), use: (1) A ver age method:
1+2
2
=
3
2
,
repeat to get
5
4
,
11
8
,
13
8
,
7
4
. (2) Denominator method:
Write 1 =
6
6
, 2 =
12
6
, yielding
7
6
,
8
6
,
9
6
,
10
6
,
11
6
. There
are infinitely man y r ationals between an y two
r ationals.
1
2. Irr ational Numbers
Subtopic Brief
Definition
Numbers that cannot be written as
p
q
, wherep and
q are integers, q?= 0 (e.g.,
v
2 ,
v
3 ,
v
15 , p ,
0.10110111011110...). Infinitely man y exist.
Real Numbers
Rational and irr ational numbers together form
real numbers, denoted b y R . Every point on the
number line corresponds to a unique real
number , and vice versa.
Locating Irr ationals
T o locate
v
2 : Use a square with side 1, diagonal
v
2 (Pythagor as theorem), tr ansfer to number line
with compass arc at point P . F or
v
3 : From point P ,
construct a right triangle with sides
v
2 and 1,
yielding
v
3 at point Q . F or an y positive integer n ,
locate
v
n after
v
n-1 .
Square Root Spir al
Construct a spir al starting with point O , segment
OP
1
of unit length, then P
1
P
2
perpendicular to OP
1
(unit length), continuing to form points P
2
, P
3
, ...,
depicting
v
2 ,
v
3 ,
v
4 , ...
3. Decimal Expansions
Subtopic Brief
Rational Decimals
Rational numbers have either terminating
decimals (e.g.,
7
8
= 0.875 ,
1
2
= 0.5 ,
639
250
= 2.556 ) or
non-terminating recurring decimals (e.g.,
10
3
= 3.3 ,
1
7
= 0.142857 ). Remainders become zero or repeat
(repeating block size < divisor).
Irr ational Decimals
Irr ational numbers have non-terminating
non-recurring decimals (e.g.,
v
2 = 1.414213562... ,
p = 3.141592653... ). Numbers lik e
0.10110111011110... are irr ational due to
non-repeating pattern.
Converting
Decimals
T erminating: 3.142678 =
3142678
1000000
. Non-terminating
recurring: 0.3 = x , s o 10x = 3.3 , 10x-x = 3 , x =
1
3
.
F or 1.27 , 100x = 127.27 , 100x-x = 126 , x =
14
11
. F or
0.235 , 100x = 23.535 , 100x-x = 23.3 , x =
233
990
.
Irr ational Between
Rationals
Between
1
7
= 0.142857 and
2
7
= 0.285714 , an
irr ational is 0.150150015000... (non-terminating
non-recurring). Infinitely man y such numbers
exist.
2
4. Oper ations on Real Numbers
Subtopic Brief
Closure Properties
Rational numbers are closed under addition,
subtr action, multiplication, and division (except
b y zero). Irr ationals follow commutative,
associative, distributive laws but are not alwa ys
closed (e.g.,
v
6+(-
v
6) = 0 ,
v
3·
v
3 = 3 ).
Rational and
Irr ational
Sum, difference, product, or quotient (non-zero
r ational) of a r ational and irr ational is irr ational
(e.g., 2+
v
3 , 2
v
3 , 7
v
5 = 15.652... ,
7
v
5
= 3.1304... ,
v
2+21 = 22.4142... , p-2 = 1.1415... ).
Irr ational
Oper ations
Sum, difference, product, or quotient of two
irr ationals ma y be r ational or irr ational (e.g.,
(2
v
2+5
v
3)+(
v
2-3
v
3) = 3
v
2+2
v
3 ,
6
v
5×2
v
5 = 60 , 8
v
15÷2
v
3 = 4
v
5 ).
5. Identities and Rationalising the Denominator
Subtopic Brief
Identities
F or positive real numbers a,b : (i)
v
ab =
v
a
v
b , (ii)
v
a
b
=
v
a
v
b
, (iii) (
v
a+
v
b)(
v
a-
v
b) = a-b , (iv)
(a+
v
b)(a-
v
b) = a
2
-b , (v)
(
v
a+
v
b)(
v
c+
v
d) =
v
ac+
v
ad+
v
bc+
v
bd , (vi)
(
v
a+
v
b)
2
= a+2
v
ab+b .
Rationalising the
Denominator
T o r ationalise
1
v
a+b
, multiply b y
v
a-b
v
a-b
: e.g.,
1
v
2
×
v
2
v
2
=
v
2
2
,
1
2+
v
3
×
2-
v
3
2-
v
3
= 2-
v
3 ,
5
v
3-
v
5
×
v
3+
v
5
v
3+
v
5
=
-5
2
(
v
3+
v
5) ,
1
7+3
v
2
×
7-3
v
2
7-3
v
2
=
7-3
v
2
31
.
6. Laws of Exponents for Real Numbers
Subtopic Brief
Definitions
F or a > 0 , n a positive integer ,
n
v
a = a
1
n
, where
b
n
= a , b > 0 . F or r ational exponents
m
n
(m , n
integers, n > 0 , no common factors),
a
m
n
= (
n
v
a)
m
=
n
v
a
m
(e.g., 4
3
2
= (4
1
2
)
3
= 2
3
= 8 ).
Exponent Laws
F or a > 0 , p,q r ational: (i) a
p
·a
q
= a
p+q
, (ii)
(a
p
)
q
= a
pq
, (iii)
a
p
a
q
= a
p-q
, (iv) a
p
b
p
= (ab)
p
. Also,
a
0
= 1 ,
1
a
n
= a
-n
.
Simplification
Examples
2
2
3
·2
1
3
= 2
2
3
+
1
3
= 2
1
= 2 ,
(
3
1
3
)
4
= 3
4
3
,
7
1
3
7
1
5
= 7
1
3
-
1
5
= 7
2
15
,
13
1
3
·17
1
3
= (13·17)
1
3
= 221
1
3
.
3