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Integers

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Understanding Integers: Definition and Examples

Comprehensive Definition, Description, Examples & Rules 

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Introduction

In mathematics, integers play a critical role, forming the basis for numerical understanding. An integer is a whole number encompassing all positive and negative numbers and zero. It extends beyond the realm of counting, encompassing positive numbers, their opposites, and zero. Generally, the letter Z is used to denote Integers.

The word “INTEGER” originates in Latin, originally meaning “Whole.”

Integers represent diverse real-world scenarios, such as temperature variations, financial transactions, and even simple arithmetic operations. Through the structured curriculum, students learn about the significance of integers, honing their skills in numeric operations like addition, subtraction, multiplication, and division. The foundational knowledge of integers lays the groundwork for advanced mathematical concepts in subsequent grades, fostering a comprehensive understanding of the numerical systems.

Definition of Integers

All whole numbers that include zero, positive, and negative whole numbers are called Integers, i.e., a set of numbers including zero, positive, and negative. The letter Z generally represents it.”

Integers example:

Z = {…, -2,-1, 0, 1, 2…}”

In mathematics, integers encompass zero, positive, and negative values. Positive integers are numbers greater than zero, negative integers are numbers less than zero, and zero is also considered an integer. On the other hand, fractions, decimals, and percents do not count as an integer.

Integers find extensive application in various mathematical concepts and operations, providing a foundation for understanding the number system better.  By grasping the distinct characteristics of positive, negative, and zero integers, students can enhance their problem-solving skills, and that will lay the groundwork for more advanced mathematical concepts in their academic journey.

Usage in Mathematics

Integers are mostly employed in arithmetic operations, algebraic expressions, and real-life problem-solving. They are essential in representing profits, losses, temperatures, and positions. A number line is a fundamental tool in visualizing integers and aids in comprehending their relative magnitudes.

In algebra, integers feature essentially in equations and inequalities, fostering a deeper understanding of mathematical relationships and operations. Students learn to use integers to facilitate equation-solving and analyzing patterns.

Integers serve as the building blocks of mathematical reasoning, empowering students to navigate complex problems and cultivate a solid mathematical foundation for future academic pursuits.

Types of Integers

Integers are of three basic types:

  • Zero
  • Positive Numbers (Natural) and
  • Negative Numbers (Negatives of Natural Numbers)

Positive Numbers

Positive numbers are numbers having a plus sign (+). But mostly, positive numbers are written simply as a whole number without the plus sign.

We know that every positive number is larger than zero, as well as negative numbers. On a number line, positive numbers are represented on the right side of zero.

Example: 1, 6, 333, 5656, 5555555, etc.

Negative Numbers

Meanwhile, negative numbers are symbolized with a minus sign (-). Negative numbers are always written along with a minus sign (-) . Negative numbers are represented to the left of zero on a number line.

Example: -7776, -51, -13, -1.

Zero

Zero (0) is neither a positive nor a negative integer, i.e., a neutral number with no sign (+ or -). It is considered the center of the number line.

Representation of Integers on Number Line

Integers are represented using a number line. Numberline is the representation of numbers visually, on a straight line. It represents all the numbers, including positive, negative, and zero”. Zero (0) is considered the center of the number line. The positive numbers are shown on the right side of zero, usually without the plus sign (+). Meanwhile, the negative numbers are presented to the left of zero; they always have a minus (-) sign. The essential points are:

  • Zero is the fundamental center of the number line.
  • The numbers on the right of Zero are Positive Integers.
  • The numbers to the left of Zero are always Negative Integers and must have a minus sign (-). 

Representation of Integers on a number line fosters a clear comprehension of numerical relationships. This visual representation aids the students in a better and cleaner understanding of Numbers and their value. The representation is:

-10, -9, -8, -7, -6, -5, -4, -3, -2, -1,  0  1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Properties of Integers

The properties of integers are used to make calculations simple and quick. We will now learn about the different properties of integers.

Addition of Integers

The addition of integers involves the combination of positive and negative numbers. When we add integers of the same sign, the result is the sum of the numbers with the common sign. Conversely, when integers of similar signs are added, the subtraction of the smaller magnitude from the larger is performed, and the sign of the result will be that of the integer with the greater magnitude.

Subtraction of Integers

The subtraction of integers is closely linked to the addition of integers. It is essential to understand subtraction as the addition of the minuend and the additive inverse of the subtrahend. 

Multiplication of Integers

The multiplication of integers does introduce new rules but follows a pattern that aligns with students’ prior knowledge. When multiplying integers of the like sign, the result is positive, while the multiplication of integers with unlike signs yields a negative result.  

Division of Integers

The division of two integers with the same sign results in a positive quotient, and two integers with opposite signs yield a negative quotient. 

Examples of Integers

Now, we will look at integers and their use in real life. Integers are defined as Whole numbers with no fractional part. We can count forward and backward from one integer to the next in a simple jump of one unit, with no stopping part of the way there. We get integers by rounding off to the nearest whole number if the answer is a fraction.

Integers could be positive {1, 2, 3, 4, 5, 6 …}, negative {−1, −2, −3, −4, -5, -6. ….}, or zero {0}

We can showcase all integers examples numbers together as:

  • Integers (Z) = {…6, -5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6… } 

[Dots are representing the infinite numbers.]

Integers are more than mere numbers and have numerous uses in real life. The effect of positive and negative integers in the real world is different. Mostly, they are used to depict two contradictory situations.

For example, when the temperature is above 0 degrees Celsius, positive numbers denote the temperature, whereas negative numbers indicate the temperature below zero. They are also used to compare and measure if two numbers are big or small or to recognize the entities’ magnitude.

Examples of real-life situations where integers are used are scores of players in cricket, hockey, or other sports tournaments and ratings of movies or songs; in banks, credit and debit are written as positive and negative numbers.

Operations With Integers

Integers are used in diverse fields of our life. Knowing how to do basic mathematical operations of integers is crucial for our day-to-day lives. So, we’ll now learn how to do basic numerical operations with integers.

  • Addition: When combining integers, students should understand like and unlike signs. The sum of integers with like signs is obtained by adding their absolute values and preserving the common sign. Adding two positive integers always results in positive integers. On the other hand, adding two negative integers will give the sum of both integers with a negative sign. But we should remember that adding two different signed integers will always result in subtraction only, and the sign of the resultant integer will be the same as the larger number. A few examples are mentioned below:
  • 2+5 = 7
  • 3 + (-3) = 0
  • -2 + (-3) = -5
  • -4 – (-4) = 0

Integers are a special set of whole numbers that are positive, negative, and zero, which are not fractions.

  • Subtraction: Subtraction of integers involves transforming the problem into an addition operation. Understanding the rules for subtracting integers with like and unlike signs is crucial for accurate solutions. The rules for subtraction and addition are similar. The sign will remain negative when we subtract two negative integers and add both numbers. Meanwhile, if we subtract two positive integers, the sign will be that of the greater integer. Moreover, we add both integers if a minus sign is in front of a negative integer.
  • Multiplication: The answer will be positive if you multiply two numbers with the same sign. On the other hand, if the numbers have different signs, the answer will be­ negative. Reme­mber, different signs always make a negative product.
  • Division: The quotient of two positive integers will always be a positive integer, and two negative integers will always be a positive integer. Moreover, the quotient of integers with unlike will always be a negative integer.

Application of Integers

We e­ncounter integers daily, and showing students real-life examples can boost their comprehension. Inte­gers are part of everyday activities, from buying things to noting temperature­ shifts. Infusing tangible instances into learning can make it more interesting and pe­rtinent for pupils.

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Key Takeaways

  1. Integers are whole numbers that can be positive, negative, or zero.
  2. The primary Properties of the integers are addition, subtraction, multiplication, and division. 
  3. The decimal values are not considered to be Integers.

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Frequently Asked Questions

Integers are whole numbers that have an essential role in mathematical calculation. The positive integers are the numbers which are greater than zero. These numbers are represented with a plus sign on the number line. Negative integers are negative numbers less than zero and are denoted with a minus sign. 

There are four major properties of integers, and these are:

  • Addition: It is when you add two or more integers among each other, whether they are positive or negative. 
  • Subtraction: It is when you subtract one integer from another integer. 
  • Multiplication: The multiplication process is when you multiply two or more integers among these others from anywhere on the number line. 
  • Division: The division process is mostly similar to the multiplying process. Here, you divide one integer with the other.

Primary operations that you do with the integers are:

  • Addition
  • Subtraction
  • Multiplication
  • Division

The most common mistake you might commit while using the integers is mixing up the plus sign and the minus sign. While performing the integers’ operations, understanding the integer’s sign is very important for perfect calculation. Students commit a lot of mistakes by mixing up the signs. Decimal numbers are not Integers as they are not whole numbers. 

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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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