The method to represent and work with numbers is understood as a numeration system or number system. A number system may be a system of writing to represent numbers. It’s the notation to represent numbers of a given set by using digits or other symbols. It allows us to work on arithmetic operations like division, multiplication, addition, subtraction. Some important number systems are the decimal number system, binary number system, octal number system, and hexadecimal number system.
Whole Numbers
The whole numbers are the numbers without fractions, decimals and are a collection of positive integers from 0 to infinity. All the whole numbers exist in number lines. All the entire numbers are real numbers but we will not say that each one of the important numbers is a whole number. Whole numbers cannot be negative. The whole numbers are represented by the symbol “W”. The examples are: 0, 23, 34, 45, 67, 867, 345, 56754, etc.
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Properties of Whole Numbers
Properties of whole numbers help to determine the numbers better. Moreover, they create calculations under certain operations like addition, subtraction, multiplication, and division very simple. The different sorts of properties of whole numbers are follows,
Closure property for Addition and Multiplication
From the instance, it is concluded that once any two whole numbers are added or subtracted, an entire number is obtained. Whole numbers are closed under addition and multiplication.
15 + 6 = 21, 9 + 88 = 97, 25 + 0 = 25.
Note Division by zero is not defined.
Commutative property for addition and multiplication
Whole numbers can be added in any order. The addition is commutative for whole numbers. This property is understood as commutativity for addition.
6 + 12 = 12 + 6
18 = 18
Two whole numbers can be multiplied in any order. Thus, multiplication is commutative for whole numbers. Multiply 9 and 7 in several orders, an equivalent answer is obtained.
9 × 7 = 63
7 × 9 = 63
∴ 9 × 7 = 7 × 9
Note Subtraction is not commutative (6 – 5 ≠ 5 – 6), Division is not commutative (4 ÷ 2 ≠ 2 ÷ 4).
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Associative property of addition and multiplication
Observe the following examples in order to understand the associative property of addition and multiplication,
- (5 + 7) + 3 = 12 + 3 = 15
- 5 + (7 + 3) = 5 + 10 = 15
In the 1st, add 5 and 7 first and then add 3 to the sum and in the 2nd, add 7 and 3 first and then add 5 to the sum. The result in both cases is the same.
For Addition:
This property usually does the addition in a straightforward and fast way. Observe the example, 234 + 197 + 203. In the example, if 197 and 203 first are first added, then it’ll be easier as the unit (ones) digit has become zero.
234 + (197 + 203)
= 234 + 400
= 634
For Multiplication:
Multiplication is true for associative property. Observe the example, 8 × 125 × 1294. Here, if multiply 125 and 1294 are multiplied, then it’ll be hard and time-consuming. So multiply 8 and 125 then with 1294.
(8 × 125) × 1294
= 1000 × 1294
= 1,294,000 This arrangement of numbers is understood as associative property.
Distributive property of Multiplication over Addition
Lets look at some samples of distributive property of multiplication over addition, these examples have utilized distributive property in one or the other manner,
- 35 × (98 + 2) = 35 × 100 = 3500
- 65 × (48 + 2) = 65 × 50 = 3250
- 297 × 17 + 297 × 3 = 297 × (17 + 3) = 297 × 20 = 5940
Example of distributive property to make the calculation simpler, 854 × 102. To make this multiplication simpler, write 102 as 100 + 2 then use distributive property.
854 × (100 + 2)
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= 854 × 100 + 854 × 2 ⇢ (distributive property)
= 85,400 + 1,708
= 87,108
Identity property for Addition and Multiplication
The collection of whole numbers is different from the collection of natural numbers because of just the presence of zero. This number zero has a special role in addition. When zero is added to any whole number, the same whole number again. Zero is named an Identity for the addition of whole numbers or additive identity for whole numbers. Zero has a special role in multiplication too. Any number when multiplied by zero becomes zero.
- 56 × 0 = 0
- 0 × 346 = 0
An additive identity for whole numbers is found, variety remains unchanged when added zero thereto. Similar case for the multiplicative identity for whole numbers. A number remains unchanged once we multiply by 1. So 1 is named identity for multiplication of whole numbers or multiplicative identity for whole numbers.
What type of number is 25,747?
Answer:
Whole numbers are positive numbers from 0 to infinity. Hence, 25747 is a big number therefore it is a whole number, a natural number, an integer, and a rational number but it is not an irrational number. The number is defined as a whole number since whole numbers start from 0 and go up to infinity and since 25,747 comes between this, it is considered as a whole number. The number is defined as a natural number since natural numbers start from 1 and go up to infinity and since 25,747 comes between this, it is considered as a natural number. Integers are the numbers that go from -∞ to +∞, and no doubt, 25,747 lies in between, hence, it is an integer as well. A rational number is the one that is terminating or repeating, since 25,747 is terminating in nature, it is rational too.
Sample Problems
Question 1: What type of number is 55,345?
Answer:
Whole numbers are positive numbers from 0 to infinity. Hence, 55,345 is a big number therefore it is a whole number, an integer and rational number but it is not an irrational number.
Question 2: What type of number is 3,45,433?
Answer:
Whole numbers are positive numbers from 0 to infinity. Hence, 3,45,433 is a big number therefore it is a whole number, an integer and rational number but it is not an irrational number.
Question 3: What type of number is 1,345?
Answer:
Whole numbers are positive numbers from 0 to infinity. Hence, 1,345 is a big number therefore it is a whole number, an integer and rational number but it is not an irrational number.
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