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Saturday, December 09 2023

Properties of Integers – Explanation & Techniques For How To Be Fail-Proof

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We often watch the news in European countries where temperatures fall below 0°C. We also have observed basements and car parking spaces below the floors in malls and other high-rises. 

What are the ways to represent these? 

Situations that demand the need for values that are less than zero introduce negative numbers. If negative values are added to the entire number, then we have integers. 

Integers are the collection of negative and positive numbers as well as zero. Also, these Integers are a variation of natural numbers and complete numbers. Not just that, Integers don’t include decimal numbers or fractions. 

Just like such essential and fun facts about Properties of Integers Operation, we will cherish a lot more in this article.

What are Integers?

Integers are the natural numbers set and their inverses that are additive, including zero. 

The integer set is:

{…..-3,-2,-1,0,1,2,3…. }.

Integers are numbers that may not be fractions.

Integers are a variety of natural and whole numbers. Natural numbers in combination with negative and zero natural numbers form integers. Complete numbers as well as negative natural numbers from integers.

That is,

Total Numbers + negative Natural number = Integers

Natural Numbers + Zero + Negative Natural Numbers = Integers

What is the Symbol of Integers?

You can denote the set of integers byIorZ.I=Z={…..–3,–2,–1,0,1,2,3….}

Representing Integers on a Number Line

A number line is a straight line with numbers placed in equal intervals or segments that run along its length. We can extend the number line infinitely in any direction and is depicted horizontally.

Positive integers are shown on the right-hand edge of the zero numbers line, while negative integers are represented on Zero’s left side. The more the numbers shift to the left from zero on the number line, they are worth more. Numbers rise, and the further the numbers shift to the left of the line of numbers from zero, the value of the numbers decreases.

If we are comparing a pair of numbers along a line, the one that is on the left side of the line will be larger.

What are the Types of Integers?

Integers are composed of three kinds:

  1. Positive integers are also called natural numbers.
  2. Negative integers are the additive inverses of natural numbers.
  3. Zero is neither negative nor positive.

What are the Operations and Rules on Integers?

Similar to natural numbers and whole numbers, all of the mathematical operations can be performed on integers as well.

Addition of Integers

In contrast to natural numbers and total numbers, the main difference between integers is that integers contain negative numbers.

Therefore, integers are to be understood in greater detail in cases where negative and positive numbers are involved.

Addition of Two Positive Integers

Similar to combining whole numbers and natural numbers, If the two positive integers are combined to produce a positive integer.

That is,

Positive + Positive=Positive

Addition of a Positive and Negative Integer

Before we learn about the process of adding an integer with a negative or positive sign, It’s vital first to comprehend the notion of absolute values. 

The absolute value of a value is what it is in terms of numerical values, regardless of its symbol. Absolute value is not positive or negative. It is simply the numerical value of the number.

Addition of Two Negative Integers

A negative addition to two numbers is performed in the same method as adding two positive integers. The only difference is the format of the result. If two negative numbers are combined, the result is a negative integer. When adding two negative integers, multiply the absolute values of both numbers and then add a negative sign to the result.

For example,

-3 + (-2)=-5

Subtraction of Two Positive Integers

If a smaller integer needs to subtract from a larger integer, use the standard subtraction.

For example,


Subtraction of Two Negative Integers

If two negative numbers are subtracted, it’s the sum of a positive and negative integer.

Here’s the example,

-4-(-2)=-4+ 2=-2

Subtraction of a Positive and Negative Integer

When a negative and positive integer is subtracted, determine whether a negative number needs to be subtracted from a positive one or negative numbers subtract a positive value.

Multiplication of Integers

Multiplication of Two Positive Integers

If the two positive integers are multiplied, the result will also be a positive number.

For example,


Multiplication of Two Negative Integers

If two negative integers are multiplied, the result can be described as a positive number.

Here’s the example,


Multiplication of a Positive and Negative Integers

When a positive and negative integer is multiplied, the result can be described as a negative number.

For example,


However, we can see that the moment we multiply two integers, the result is a number. This means that integers become closed when multiplying.

Division of Integers

The integers are not decimals or fractions. Therefore, the division of integers is done only if the number of quotients can be described as an integer. 

The other situations for the division of integers are not defined. Furthermore, division using zero isn’t defined. Thus, the division cannot be closed in the case of the rule of integers.

The sign of the quotient 2 integers divided is comparable with the value of its product.

Division of Two Positive Integers

If 2 positive numbers are divided, the resultant quotient is a positive number.

Division of Two Negative Integers

When two negative numbers are divided into a quotient, it will result in a positive integer.

Division of a Positive and Negative Integer

If a negative and a positive integer is divided by a quotient, the result will be negative.

What are the Properties of Integers?

Integers are the property holders of natural and whole numbers. Integers also have additional properties.

The five main characteristics of the integer are

  1. Closure Property
  2. Distributive Property
  3. Commutative Property
  4. Associative Property
  5. Existence of Identity


What about the conclusion with a question?

What if we ask – Can you explain the difference between the total amount and an integer?

You should answer – Any natural numbers including 0 and 1. 2, 3, 4, 5 and more are all whole numbers. On the other hand, they have negative counterparts, such as -4, -3 1, 0 1 2, 3 etc. However, a and b represent both numbers.

Photo by Markus Krisetya on Unsplash

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