Teachers Eligibility Test - Paper 2
Class – VI Mathematics
NUMBER SYSTEM
WHOLE
NUMBERS
The collection of counting numbers
{1,2,3...} is called Natural numbers, denoted by N. If this collection includes
0 as well, then the collection
{0, 1,2,3...} is called Whole numbers, denoted by W.
Facts on Natural and Whole Numbers
v The smallest
natural number is 1.
v The smallest whole
number is 0.
v Every number has
a successor. The number that comes just after the given number is its
successor.
v Every number has
a predecessor. The number 1 has a predecessor in W namely ‘0’, but it has no
predecessor in N. The number ‘0’ has no predecessor in W.
v There is an
order to numbers. By comparing the two given numbers the larger of the two can
be identified.
v Numbers are
endless. By adding 1 to any chosen large number, the next number can be found.
Properties of Whole Numbers
Commutatively of addition and multiplication
When two numbers are added (or multiplied),
the order of the numbers does not affect the sum (or the product). This is
called commutatively of addition (or multiplication).
Example
43 + 57 = 57 + 43
12 × 15 = 15 × 12
35,784 + 48, 12,
69,841 = 48, 12, 69,841 + 35,784
39,458 × 84,321 = 84,321 × 39,458
Example
7 – 3 = 4 but 3
– 7 will not give the same answer.
Similarly, the
answers of 12 ÷ 6 and 6 ÷ 12 are not equal.
That is, 7 – 3 ≠
3 – 7 and 12 ÷ 6 ≠ 6 ÷ 12
Hence, subtraction and
division are NOT commutative.
Associativity of addition and multiplication
When several numbers are
added, the order in which the numbers are added does not matter. This is called
associativity of addition. Similarly, when several numbers are to be
multiplied, the order in which the numbers are multiplied does not matter. This
is called associativity of multiplication.
Example
(43 + 57) + 25 = 43 + (57 + 25)
12 × (15 × 7) = (12 ×
15) × 7
35,784 + (48, 12, 69,841 + 3) = (35,784 + 48,
12, 69,841) + 3
(39,458 × 84,321) × 17 = 39,458 ×
(84,321 × 17)
It
is to be noted that here too, subtraction and division are NOT
associative.
Distributive of multiplication over
addition or subtraction
An interesting fact relating to
addition and multiplication comes from the following patterns:
(72 × 13) + (28 × 13) = (72 + 28)
× 13
37 × 102 =
(37 × 100) + (37 × 2)
37 × 98
= (37 × 100) – (37 × 2)
In the last two
cases, we are actually noting down the following equations:
37 × (100 + 2) =
(37 × 100) + (37 × 2)
37 × (100 − 2) = (37
× 100) – (37 × 2)
It can be noted that the product of a number
and a sum of numbers can be written as the sum of two products.
Similarly,
the product of a number and a number got by subtraction can be written as the
difference of two products. This property is called as property of distributive
of multiplication over addition or subtraction. It is a very useful property to
group numbers in a convenient way.
Identity for addition and multiplication
When zero is added to any number,
we get the same number.
|
0 |
+ |
9 |
= |
9 |
|
0 |
+ |
10 |
= |
10 |
|
16 |
+ |
0 |
= |
16 |
|
37 |
+ |
0 |
= |
37 |
|
50 |
+ |
0 |
= |
50 |
Similarly,
when we multiply any number by 1, we get the same number.
So,
zero is called the additive identity and one is called the multiplicative
identity for whole numbers.
|
1 |
x |
55 |
= |
55 |
|
25 |
x |
1 |
= |
25 |
|
400 |
x |
1 |
= |
400 |
|
1 |
x |
12 |
= |
12 |
Finally, these
are some simple observations that are important.
When we add any two
natural numbers, we get a natural number. Similarly when we multiply any two
natural numbers, we get a natural number.
When we add any two
whole numbers, we get a whole number. Similarly when we multiply any two whole
numbers, we get a whole number.
When
we add a natural number to a whole number, we get a natural number.
When we
multiply a natural number by a whole number, we get a whole number.
NOTE:
Any number multiplied
by zero gives zero.
Division by
zero is not defined.
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