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| DigitsThe symbols used to represent a number are called digits. The name "digit" comes from the fact that the ten digits (Latin digiti meaning fingers) of the hands correspond to the ten symbols of the common base 10 numeral system, i.e. the decimal digits.
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| NumeralA group of digits denoting a number is called a numeral. For example, 456, 73456, 3487 etc.
Single digit number: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Two digits number: 10, 11, 12, .................99 Three digits number: 100, 101, 102, ............999 Four digits number: 1001, 1002, 1003, ..........9999 Five digits number: 10001, 10002, ..............99999 Greatest single digit number + 1 = smallest 2-digit number Greatest 2-digit number + 1 = smallest 3-digit number Greatest 3-digit number + 1 = smallest 4-digit number Greatest 4-digit number + 1 = smallest 5-digit number Greatest 5-digit number + 1 = smallest 6-digit number | |||||||||||||||
| NotationThe method of representing a number in digits or figures is called notation. | |||||||||||||||
| NumerationThe method of expressing a number in words is called numeration. | |||||||||||||||
| SuccessorThe number which comes immediately after a particular number is called its successor.
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| PredecessorThe number which comes just before a particular number is called its predecessor.
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PeriodIn decimal number system, the value of a digit depends on its place value, or positional value, in the number. Each place has a value of 10 times the place to its right.
In Indian System of Numeration numbers are seperated into groups of two digits using commas, therefore in this system period is of group of two digits. | |||||||||||||||
| Commutative Property
An operation is commutative if changing the order of the operands does not change the result.
It is expressed as, a + b = b + a
4 + 5 = 5 + 4
9 = 9
When we change the order of the numbers that we are adding the sum remains the same. This shows addition is commutative.
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a – b ≠ b – a
8 – 3 ≠ 3 – 8
5 ≠ –5
When we change the order of the numbers that we are subtracting the result changed. This shows subtraction is not commutative.
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a b = b a
4 5 = 5 4
20 = 20
When we change the order of the numbers that we are multiplying the product remains the same. This shows multiplication is commutative.
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a ÷ b ≠ b ÷ a
8 ÷ 4 ≠ 4 ÷ 8
2 ≠
When we change the order of the numbers that we are dividing the result changed. This shows division is not commutative.
1/2 | |||||||||||||||
| Associative Property
An operation in which changing the order of the operation does not affect the result is called associative property. Consider the following equations:
a +
(b +
c) =
(a +
b) +
c
3 + (5 + 6) = (3 + 5) + 6 3 + 11 = 8 + 6 14 = 14 Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition. This shows addition is associatative. | |||||||||||||||
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Subtraction:
a –
(b –
c) ≠
(a –
b) –
c
3 – (7 – 6) ≠ (3 – 7) – 6 3 – 1 ≠ – 4 – 6 2 ≠ –10 This shows subtraction doesn’t follow the associative property. Regrouping the numbers resulted in two different answers. This shows subtraction is not associatative. | |||||||||||||||
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Multiplication:
a
(b
c) =
(a
b)
c
3 (5 6) = (3 5) 6 3 30 = 15 6 90 = 90 Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing multiplication. This shows multiplication is associatative. | |||||||||||||||
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Division:
a ÷
(b ÷
c) ≠
(a ÷
b)÷
c
18 ÷ (6 ÷ 3) ≠ (18 ÷ 6) ÷ 3 18 ÷ 2 ≠ 3 ÷ 3 9 ≠ 1 Division doesn’t follow the associative property. Regrouping the numbers resulted in two different answers. This shows division is not associatative. | |||||||||||||||
| Distributive Property
An operation in which multiplying the sum by a number gives the same result as first multiplying each addend by the number and then adding the products is called Distributive Property. Consider the following equations:
a
(b +
c) =
ab +
ac
3 (5 + 6) = (3 5) + (3 6) 3 11 = 15 + 18 33 = 33 So the "3" can be "distributed" across the "5+6" into 3 times 5 and 3 times 6. This shows multiplication is distributive over addition. | |||||||||||||||
| Subtraction:
a
(b –
c) =
ab –
ac
3 (5 – 6) = (3 5) – (3 6) 3 1 = 15 – 18 3 = 3 So the "3" can be "distributed" across the "5 – 6" into 3 times 5 and 3 times 6. This shows multiplication is distributive over subtraction. | |||||||||||||||
| Additive Identity
An operation in which adding zero to any number results same number is called additive identity. Consider the following equations:
0 +
a =
a
0 +
5 =
5
If 0 is added to 5, result is same 5. | |||||||||||||||
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Zero is an additive identity for Natural Numbers, Whole Numbers, Integers, Rational numbers, Irrational numbers, Real Numbers and Complex numbers.
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| Multiplicative Identity
An operation in which a number is multiplying by 1 results same number is called multiplicative identity. Consider the following equations:
1
a =
a
1
5 =
5
When 5 is multiplied by 1 yields the same result 5. | |||||||||||||||
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1 is multiplicative identity for Natural Numbers, Whole Numbers, Integers, Rational numbers, Irrational numbers, Real Numbers and Complex numbers.
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| Additive Inverse
An operation in which a number added to it yields zero is called additive inverse. This number is also known as the opposite number. The opposite to a positive number is negative, and the opposite to a negative number is positive. Zero is the additive inverse of itself. Consider the following equations:
a + (
-a) =
0
5 + (
-5) =
0
Here opposite of 5, i.e. -5 added to 5 yields 0. | |||||||||||||||
| Multiplicative Inverse
An operation in which a number multiplied by it yields 1 is called multiplicative inverse. This number is also known as the reciprocal of same number. For example reciprocal of a is denoted by
1
/
a
or a-1. Consider the following equations:
a(
1
/
a
) = 1, Where a ≠ 0 Here a is multiplied by reciprocal of a yields 1. When we multiply a number by its "Multiplicative Inverse" we get 1. |
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