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Whole numbers

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Exploring Whole Numbers: Definition and Examples

Comprehensive Definition, Description, Examples & Rules 

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Maths is synonymous with whole numbers. You can’t think of one without the other. And an understanding of what is whole number is can sort out a number of your maths problems! Edulyte’s maths Maths tutors have created this resource to enable a firmer grasp of the concept. Check it out and benefit from it!

Introduction to Whole Numbers: What Is A Whole Number?

Whole numbers include all the natural numbers (positive integers) and zero, like 0, 1, 2, 3, and so on. They do not have fractions or decimals.

Whole numbers are fundamental building blocks for various mathematical operations and mathematical concepts. They are used in arithmetic for basic calculations like addition, subtraction, multiplication, and division. Whole numbers are also essential in algebraic expressions and equations.

In everyday life, whole numbers are crucial in various contexts, such as counting items, measuring quantities, and representing discrete values.

What Are Whole Numbers? : Definition Of Whole Number

Definition of a whole number states that it is any number starting from 0 and includes all natural numbers. They are denoted by the set W={0,1,2,3,…}.

Whole numbers differ in specific ways from other numbers, such as integers, rational numbers, and real numbers.

Whole numbers example includes numbers like :0, 2, 115, 2357, and so on.

Characteristics that differentiate whole numbers from others

  1. Inclusion of Zero: Whole numbers include zero (0) and all the natural numbers (1, 2, 3, …). It distinguishes them from natural numbers, which begin at 1.
  2. Positivity: Whole numbers are non-negative integers greater than or equal to zero. Unlike integers, which include negative, positive, and zero values, they do not have negative numbers.
  3. No Fractions or Decimals: Whole numbers do not include fractions or decimals. They are discrete values representing counts of items or units.
  4. Counting and Cardinality: Whole numbers are for counting and cardinality. They represent the number of objects in a set or the position of an element in a sequence.
  5. Basic Arithmetic Operations: Whole numbers are handy in basic arithmetic operations such as addition, subtraction, multiplication, and division. They provide a foundation for more complex mathematical concepts.
  6. Application in Algebraic Expressions: Whole numbers are valuable in algebraic expressions and equations. They are often variables or constants in mathematical formulations.

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Understanding Whole Numbers: Concept of “Whole” And Its Applications

In mathematics, “whole” refers to an undivided complete entity. When talking about what is  a whole number, we refer to any number, including zero and natural numbers (1, 2, 3, and so on).

Here’s a breakdown of the concept of “whole” in mathematics:

  1. Indivisibility: The term “whole” underscores the indivisible nature of the numbers in this set. Each number is a complete unit, not a fraction or part of a larger whole.
  2. Zero as a Starting Point: Including zero in the set of whole numbers is crucial. Zero represents the absence of quantity and serves as a reference point, making the set complete in positive and non-negative directions.
  3. Counting and Enumeration: Whole numbers are for counting and enumeration. When you count objects or items, you often start with one and move through the natural numbers and zero, creating a sequence of whole numbers.

Real-world relevance and applications of whole numbers

  1. Counting and Quantification: Whole numbers are fundamental to counting discrete objects. Whether it’s counting the number of people in a room, items in a grocery cart, or days in a week, whole numbers are used to quantify these entities.
  2. Measuring Units: In various measurements, whole numbers play a crucial role. For example, measuring length in metres or determining the quantity of ingredients in a recipe often involves whole numbers.
  3. Financial Transactions: Whole numbers find extensive use in financial transactions. When dealing with money, quantities are often expressed in whole numbers, representing complete currency units.
  4. Timekeeping: Whole numbers express time in hours, minutes, and seconds. They are essential for scheduling, planning, and organising daily activities.
  5. Inventory Management: In business and manufacturing, whole numbers are used to keep track of inventory. The number of products in stock, units sold, and items produced are all expressed using whole numbers.

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Examples of Whole Numbers: Simple Arithmetic Operations

How can you use whole numbers? Well here are a few examples of simple arithmetic operations that you can use whole number operations. 

Counting People:

  • Scenario: Counting the number of people in a room.
  • Whole Numbers: 1, 2, 3, …

Measuring Length:

  • Scenario: Measuring the length of a desk in metres.
  • Whole Numbers: 1, 2, 3, …

Quantifying Items:

  • Scenario: Counting the number of apples in a basket.
  • Whole Numbers: 0, 1, 2, …

Money Transactions:

  • Scenario: Counting the number of dollars in a wallet.
  • Whole Numbers: 0, 1, 2, …

Simple Arithmetic Operations

Addition:

  • 3+4=7

Subtraction:

  • 10−5=5

Multiplication:

  • 2×6=12

Division:

  • 18÷3=6

Combining Operations:

  • (5×3)+2=17

Scenarios Involving Whole Numbers

These examples demonstrate what is whole number and  how whole numbers work in various real-life situations and how simple arithmetic operations can be applied to solve problems involving counting, measuring, and transactions.

Bus Passengers:

  • If there are 30 passengers on a bus and 8 more board at the next stop, how many passengers are there in total?

Library Books:

  • A library has 150 books on its shelves. If 25 books are borrowed, how many books remain on the shelves?

Gardening:

  • A gardener plants 5 flowers, with 7 flowers in each row. How many flowers are there in total?

Savings:

  • If you save $20 each week, how much money will you have after 8 weeks?

Properties of Whole Numbers: Key Properties With Important Details

Whole numbers possess several vital properties that govern their behaviour in mathematical operations. Understanding these properties is fundamental for working with whole numbers and forms the basis for more advanced mathematical concepts. Here are the essential properties:

 Addition:

  • Definition: The sum of two whole numbers is always a whole number.
  • Example: For any whole numbers like 3 and 4, the sum will be a whole number: 3+4= 7

 Subtraction:

  • Definition: The difference between two whole numbers is always a whole number.
  • Example: For any whole numbers 7 and 3(where 7≥3), the difference will be a whole number: 7 – 3 =4

Multiplication:

  • Definition: The product of two whole numbers is always a whole number.
  • Example: For any whole numbers like 3 and 4, the product will be a whole number: 3 x 4 =12

Associative Property of Addition:

  • Definition: How numbers are grouped in an addition operation does not affect the result.
  • Example: For any whole numbers 2,3 and 4, Changing the grouping of numbers in addition does not affect the sum: (2+3)+4=2+(3+4)

Commutative Property of Addition:

  • Definition: The order in which numbers are added does not affect the result.

Example: For any whole numbers, The order in which numbers are added does not affect the sum: 5+7=7+5

Identity Element for Addition:

  • Definition: The number 0 is the identity element for addition.
  • Example: For any whole number 4+0=4+0=4.

Associative Property of Multiplication:

  • Definition: How numbers are grouped in a multiplication operation does not affect the result
  • Example: For any whole numbers, grouping numbers in a multiplication does not affect the product: (2×3)×4 =2×(3×4)

Commutative Property of Multiplication:

  • Definition: The order in which numbers are multiplied does not affect the result.
  • Example: For any whole numbers, The order in which numbers are multiplied does not affect the product:  4×6=6×4

Identity Element for Multiplication:

  • Definition: The number 1 is the identity element for multiplication.
  • Example: For any whole number a, a×1=a×1=a

Importance of Whole Numbers in Mathematics

Whole numbers are foundational and essential in the mathematical landscape, serving as fundamental building blocks for various mathematical concepts. Their significance is evident in several aspects of mathematics:

Basic Arithmetic Operations:

  • Whole numbers are crucial for basic arithmetic operations, including addition, subtraction, multiplication, and division.

Number Theory:

  • Whole numbers are fundamental to number theory, the branch of mathematics that deals with the properties and relationships of numbers.

Algebraic Expressions:

  • Whole numbers serve as variables and constants in algebraic expressions. Algebraic manipulation, solving equations, and understanding algebraic structures rely on a solid understanding of whole numbers.

Integers and Rational Numbers:

  • Whole numbers provide the foundation for integers, including positive and negative whole numbers. Integers, in turn, form the basis for rational numbers, expanding the mathematical landscape to include fractions.

Geometry and Measurement:

  • In geometry, whole numbers are used for counting and specifying the number of sides, angles, or vertices in geometric shapes.

Data and Statistics:

  • Whole numbers are used to represent counts in data and statistics. They are essential for organising and analysing data, calculating frequencies, and understanding the distribution of values.

Computer Science and Algorithms:

  • Whole numbers are fundamental in computer science and algorithms. The discrete nature of whole numbers is well-suited for representing and manipulating information in computer systems.

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Key Takeaways

  1. Inclusion of Zero and Positive Integers: Whole numbers include zero and all positive integers (1, 2, 3, …). They do not have negative numbers, fractions, or decimals.

  2. Fundamental for Arithmetic Operations: Whole numbers are the foundation for basic arithmetic operations, including addition, subtraction, multiplication, and division. They provide a simple and intuitive framework for mathematical calculations.

  3. Building Blocks for Mathematics: Whole numbers are considered the basic building blocks of the number system. They are a starting point for more advanced mathematical concepts, providing a solid foundation for further exploration.

  4. Discreteness and Counting: Whole numbers are discrete values, representing distinct and separate units. They are particularly suitable for counting and measuring quantities in entire units, making them integral to various real-world applications.

  5. Application in Everyday Life: Whole numbers have practical applications in everyday life, from counting items to measuring quantities. They play a crucial role in finance, timekeeping, and various scenarios requiring discrete counting. Understanding whole numbers is essential for navigating daily mathematical challenges.

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Frequently Asked Questions

Whole numbers differ from other numbers, such as natural numbers, integers, rational numbers, and real numbers, regarding their inclusion, characteristics, and properties. Here’s a brief comparison:

Natural Numbers:

  • Whole Numbers: Include natural numbers (1, 2, 3, …) along with zero.
  • Natural Numbers: Begin at 1 and do not include zero.
  • Example: 0 is a whole number but not a natural number.

Integers:

  • Whole Numbers: Include zero and all positive integers.
  • Integers: Include zero, positive integers, and their negative counterparts.
  • Example: -3 is an integer but not a whole number.

Rational Numbers:

  • Whole Numbers: These are a subset of rational numbers.
  • Rational Numbers: Include whole numbers and fractions.
  • Example: 2 is a whole and rational number (2/1).

Real Numbers:

  • Whole Numbers: These are a subset of real numbers.
  • Real Numbers: Include whole, fractions, decimals, and irrational numbers.
  • Example: √2 (an irrational number) is a real number but not a whole number.

Fractions and Decimals:

  • Whole Numbers: Do not include fractions or decimals.
  • Fractions/Decimals: Involve parts of a whole and are not considered whole numbers.
  • Example: 1/2 is not a whole number.

Negatives and Positives:

  • Whole Numbers: Are non-negative integers (greater than or equal to zero).
  • Integers: Include both negative and positive numbers.
  • Example: -5 is an integer but not a whole number.

Counting vs. Measuring:

  • Whole Numbers: Primarily used for counting discrete objects.

Real Numbers: Used for measuring

Yes, both zero and natural numbers are considered whole numbers.

Natural numbers are the set of positive integers that begin from 1 and continue indefinitely (1, 2, 3, …).

Whole numbers include all the natural numbers along with zero.

The set of whole numbers is denoted as W={0,1,2,3,…}.

Whole numbers are applied in various mathematical operations like addition, subtraction, multiplication and division.

Exponents (Powers):

  • Whole numbers can be used as exponents.

Example: 23 =2×2×2=8

  •  Representing repeated multiplication.

Roots:

Whole numbers can be used in root operations.

  • Example: √9=3 

Application: Finding the square root or cube root.

Whole numbers are considered fundamental in mathematics for several reasons:

  1. Basic Building Blocks: They serve as the foundation for more complex numbers and mathematical structures.
  2. Elementary Arithmetic Operations: Whole numbers are essential for basic arithmetic operations such as addition, subtraction, multiplication, and division.
  3. Counting and Enumeration: They represent the natural order of counting discrete objects or items.
  4. Foundational for Algebra: Whole numbers are used as variables and constants in algebraic expressions.
  5. Logical Reasoning: Understanding relationships between numbers, solving problems, and making informed decisions often involve reasoning with whole numbers.
  • Addition
  •  Subtraction
  • Multiplication
  • Associative Property of Addition
  • Commutative Property of Addition
  • Identity Element for Addition
  • Associative Property of Multiplication
  • Commutative Property of Multiplication
  • Identity Element for Multiplication

 

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2D Shapes2cosacosb Formula30-60-90 Formulas3D ShapesAbsolute Value FormulaAcute AngleAcute Angle triangleAdditionAlgebra FormulasAlgebra of MatricesAlgebraic EquationsAlgebraic ExpressionsAngle FormulaAnnulusAnova FormulaAnti-derivative FormulaAntiderivative FormulaApplication of DerivativesApplications of IntegrationArc Length FormulaArccot FormulaArctan FormulaArea Formula for QuadrilateralsArea FormulasArea Of A Sector Of A Circle FormulaArea Of An Octagon FormulaArea Of Isosceles TriangleArea Of ShapesArea Under the Curve FormulaArea of RectangleArea of Regular Polygon FormulaArea of TriangleArea of a Circle FormulaArea of a Pentagon FormulaArea of a Square FormulaArea of a Trapezoid FormulaArithmetic Mean FormulaArithmetic ProgressionsArithmetic Sequence Recursive FormulaArithmetic and Geometric ProgressionAscending OrderAssociative Property FormulaAsymptote FormulaAverage Deviation FormulaAverage Rate of Change FormulaAveragesAxioms Of ProbabilityAxis of Symmetry FormulaBasic Math FormulasBasics Of AlgebraBinary FormulaBinomial Probability FormulaBinomial distributionBodmas RuleBoolean AlgebraBusiness MathematicsCalculusCelsius FormulaCentral Angle of a Circle FormulaCentral Limit Theorem FormulaCentroid of a Trapezoid FormulaChain RuleChain Rule FormulaChange of Base FormulaChi Square FormulaCirclesCircumference FormulaCoefficient of Determination FormulaCoefficient of Variation FormulaCofactor FormulaComplex numbersCompound Interest FormulaConditional Probability FormulaConeConfidence Interval FormulaCongruence of TrianglesCorrelation Coefficient FormulaCos Double Angle FormulaCos Square theta FormulaCos Theta FormulaCosec Cot FormulaCosecant FormulaCosine FormulaCovariance FormulaCubeCylinderDiscover the world of MathsEllipseEquilateral triangleEuler’s formulaEven numbersExponentsFractionHeptagonHyperbolaIntegersIntegrationIntegration by partsLinesLocusMatricesNatural numbersNumber lineParallelogramPercentage formulaPerimeterPolygonPolynomialsPrismProbabilityPyramidPythagoras theoremScalene triangleSetsSlope formulaSolid shapesSphereSquareStandard deviation formulaSubtractionSymmetryTimeTrianglesTrigonometry formulaTypes of anglesValue of PiVectorVolume formulasVolume of a coneVolume of sphere formulaWhole numbers
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