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Numbers

WHOLE NUMBERS

Contents

Introduction:

Whole numbers are subset of integers that include all non-negative integers, i.e., positive integers including zero. These numbers are used in various fields of study, such as mathematics, science, economics, and many others. This article aims to provide comprehensive guide to whole numbers, including their properties, operations, and real-world applications.

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Whole Number Definition and Symbol:

Whole numbers are defined as non-negative integers, which means they are positive integers including zero. These numbers are denoted by symbol “W” or “Z.” The whole numbers set can be represented as

    \[\left\{ {0,{\rm{ }}1,{\rm{ }}2,{\rm{ }}3,{\rm{ }}4,{\rm{ }}5,{\rm{ }}6,{\rm{ }} \ldots } \right\}.\]

the whole number which is smallest is zero.

Numbers that are Natural and Numbers that are whole:

Numbers that are Natural are also subset of integers that include all positive integers, i.e., numbers greater than zero. Whole numbers, on other hand, include zero as well as all positive integers. Therefore, every number that are natural is also numbers that are whole, but not every numbers that are whole is number that are natural.

Whole Numbers on Number Line:

Numbers that are whole can be represented on number line, which is visual representation of numbers. The number line starts at zero and then positive integers extend over.

Properties of Whole Numbers:

Property Closure: The property closure of numbers that are whole states that if we add, subtract, or multiply any two numbers that are whole, result will always be numbers that are whole. For example,

    \[4 + 5 = 9\]

, which is whole number.

Associative Property: The associative property of whole numbers states that way we group numbers does not affect final result of operation. (i.e),

    \[\left( {3 + 4} \right) + 5 = 3 + \left( {4 + 5} \right) = 12.\]

Property of Commutative: The commutative property of whole numbers states that order in which we add or multiply numbers does not affect final result of operation. For example,

    \[2 + 3 = 3 + 2 = 5\]

.Distributive Property: The distributive property of whole numbers states that when we multiply number by sum or difference of other numbers, we can distribute multiplication over each number in sum or difference. For example,

    \[2 \times \left( {3 + 4} \right) = 2 \times 3 + 2 \times 4 = 14\]

.

Multiplication by Zero: When we multiply any whole number by zero, result is always zero. For example,

    \[5 \times 0 = 0.\]

Division by Zero: Division by zero is undefined in arithmetic, including for whole numbers. Therefore, it is impossible to divide any whole number by zero.

Can Whole Numbers be Negative?

Whole numbers cannot be negative because they include only non-negative integers. Negative integers are not part of set of whole numbers.

Is 0 Number whole?

Yes, 0 is a number that is whole. It is the smallest number that is whole and is called identity element for addition, which means that adding 0 to any number doesn’t change its value.

Whole Number Operations:

Addition: Addition is combining two or more number that is whole to find their sum. For example,

    \[3 + 5 = 8\]

Subtraction: Subtraction of finding difference between two numbers that is whole. For example,

    \[8--3 = 5\]

Multiplication: Multiplication process of finding product of two or more number that is whole. For example,

    \[3 \times 5 = 15\]

Division (with remainder): Division is process of finding quotient and remainder of dividing 1 number that is whole by another. (i.e),

    \[17 \div 5 = 3\]

with remainder 2.

Examples of each operation:

Addition: If you have 4 apples and you buy 3 more apples, you will have total of 7 apples.

Subtraction: If you have 10 dollars and you spend 4 dollars, you will have 6 dollars left.

Multiplication: If you have 2 boxes of chocolates, and each box contains 12 chocolates, you have total of 24 chocolates.

Division (with remainder): If you have 20 candies and you want to distribute them equally among 5 friends, each friend will get 4 candies, with 0 candies remaining.

Whole Number Concepts in Real-World Scenarios:

Counting Objects: Whole numbers are used to count number of objects in set. For example, counting number the students in class, apples in basket, or cars on street.

Measuring Time and Distance: Whole numbers are used to measure time in seconds, minutes, and hours, and distance in feet and meters. For example, measuring time it takes to complete task or distance between two locations.

Calculating Costs and Prices: Whole numbers are used to calculate cost of goods and services in dollars and cents. For example, calculating total cost of groceries or price of ticket for concert.

Examples of Whole Number Usage in Everyday Life:

Counting money: Whole numbers are used to count money in coins and bills.

Measuring ingredients for cooking: Whole numbers are used to measure amount of ingredients needed in recipes.

Calculating discounts on purchases: Whole numbers are used to calculate discounts on purchases, such as buy one get one free or

    \[\;20\% \]

off.

Conclusion:

Whole numbers are fundamental concept in mathematics that are used in wide range of applications in everyday life. Understanding their properties and rules is essential for solving mathematical problems and making informed decisions. Whole numbers are non-negative integers that include zero, and they have various properties, such as property closure, associative property of associative, property of commutative, property of distributive, and multiplication of zero. Whole numbers are used in various real-world scenarios, such as counting objects, measuring time and distance, and calculating costs and prices

Solved Examples:

1. If you have 10 pencils and you give 3 pencils to a friend, how many pencils do you have left?

Ans:

    \[10--3 = 7\]

You have 7 pencils left.

2. If you have 2 packs of gum, and each pack contains 6 pieces of gum, how many pieces of gum do you have in total?

Ans:

    \[2 \times 6 = 12\]

You have 12 pieces of gum in total.

3.If you have 5 friends and you want to distribute 20 stickers equally among them, how many stickers will each friend get?

Ans:

    \[20 \div 5 = 4\]

Each friend will get 4 stickers.

4.If you have ₹50 and you buy a shirt for ₹20 and a pair of shoes for ₹30, how much money do you have left?

Ans:

    \[50--20--30 = 0\]

You have no money left.

5.If you have 3 bags of apples, and each bag contains 8 apples, how many apples do you have in total?

Ans:.

    \[3 \times 8 = 24\]

You have 24 apples in total

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FAQs:

1. What is the difference between natural numbers and whole numbers?

Ans: Natural numbers are set of positive integers that start from 1 and go up to infinity, while whole numbers are set of non-negative integers that include zero.

2. Can whole numbers be negative?

Ans: No, whole numbers cannot be negative. They only include non-negative integers.

3. Is zero a whole number?

Ans: Yes, zero is a whole number. It is the smallest whole number and is included in the set of whole numbers.

4. What is the distributive property of whole numbers?

Ans: The distributive property of whole numbers states that when you multiply a number by the sum of two other numbers, you can distribute multiplication to each number separately and then add results. For example

    \[3{\rm{ }} \times \left( {4 + 5} \right) = \left( {3 \times 4} \right) + \left( {3 \times 5} \right) = 12 + 15 = 27\]

5. What is the result of dividing any whole number by 1?

Ans: The result of dividing any whole number by 1 is the same number. For example,

    \[8 \div 1 = 8\]

References:

S.Chand maths book

R.S.Agarwal maths book

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