Understanding integers with a number line

A number is any arithmetic object or an arithmetic value of numerical. When used with mathematics operations like Addition, Subtraction, Multiplication, Division, or any other mathematical equation and formula provides the desired value in respective unit terms.

These numbers, when represented in a single line with different values, including positives, negatives, and zero, are called number lines, which in simple words means line representation of numerical values.

Types of Numbers:

In mathematics, there are various types of numbers based on different theories.  Some of the commonly known types are:

• Whole Numbers
• Natural Numbers
• Real Numbers and Imaginary Numbers
• Prime and Composite Numbers
• Even Numbers
• Odd numbers
• Integers
• Rational and Irrational Numbers

What are integers?

An integer is a set of numbers excluding fractions and decimal properties, which comprises positive, negative, and zero as its constituent. It is said to be a set of counting numbers denoted by the symbol Z in mathematics.

Z ={….., -5,-4,-3,-2,-1,0,1,2,3,…………}

Types of Integers:

Integers are generally divided into two types:

• Whole numbers 
• Negative Natural Integers

Whole Numbers:

The whole numbers are natural numbers, including zero and varying from 0 to infinity, and are positive integers in nature.

Whole numbers are further classified into

• Zero
• Positive natural Integers

Zero

Zero is considered to be a whole number, as well as an integer that constitutes an important part of the number line. It is a neutral number that is neither positive nor a negative integer. It has no sign as well.

Positive Natural Integers

Positive Natural Integers are denoted by Z+  in mathematical symbols and are the part after Zero in any number line to the right side.

Negative Natural Integers

Negative Natural Integers are denoted by Z- in mathematics and are the part before zero on any number line. It denotes numbers like -5,-4,-3,-2…, etc.

In negative integers, as we approach zero, the number becomes bigger, which means -2 is greater than -4 in negative integers.

What is a number line?

A number line is a line representation with arrows on both sides. All kinds of integers, positive, negative, and zero, are represented together for a graphical representation of integers in a number line.

Rules of Integers

Like every mathematical function and operation, Integers also come with rules to be taken care of during their application.

Some of the commonly practised and known rules of integers are as follows:

• Integers do not include any fractional properties and are not represented as fractions.
• Integers can never be in the form of decimals.
• The sum of any two or more positive integers is always a positive integer.
• The sum of any two negative integers is negative in sign while the operation is Addition.
• The converse of any integer is undefined or results in 0 in return with no definite value.
• The product of the reciprocal of any integer is always one.
• If we multiply any positive integer with a negative zero, the answer always results in a negative sign of the integer.

Mathematical Operations with Integers

Just answering the question, what are integers, isn’t enough. One must also know what are the applications of an integer as well.

The mathematical operations possible with integers are:

• Addition
• Subtraction
• Multiplication
• Division

Rules of Integers: Addition

• (+) + (+) = (+)
• (+) + (-) = (-)
• (-) + (+) = (-) 
• (-) + (-) = (-)

Rule of integers: Subtraction

• (+) – (+) = (+)
• (+) – (-) / (-) – (+) / (-) – (-)

Depends on the numbers attached to the signs.

For example,

1.  9 – 5 = 4
2.  9 – (-11) = 9 + 11 = 20
3.  -9 – 8 = – (9+8) = 17

Rule of integers: Multiplication

• (+) X (+) = (+)
• (+) X (-) = (-)
• (-) X (+) = (-)
• (-) X (-)= (+)

Rule of integers: Division

• (+) ÷ (+) = +
• (-) ÷ (+) = (-)
• (+) ÷ (-) = (-)
• (-) ÷ (-) = (+)

Properties Of Integers

Properties of integers based on which calculations are conducted are as follows:

• Closure Property

The closure property says that the sum of any two whole numbers will always be a whole number when any mathematical operation is conducted.

• Identity Property

As the identity name goes, the property states that any number multiplied by 1 gives the same identity and the fact that when a number is added with 0 or subtracted, it leaves the same value.

• Distributive Property

The distributive property says that two or a combination of numbers, when added or multiplied in any order, give the same answer in return.

• Associative Property

 The term association talks about the association of a group of numbers that can be grouped to derive an answer using mathematical operations and signs depending on the type of integer.

• Commutative Property

The property states that changing the order of numbers, specifically in multiplication, does not lead to a change in the product of that number.

Conclusion

Integers are an integral part of mathematics and are highly used in mathematical operations of addition, division, subtraction, and multiplication.

Apart from that, Integers are a vital part of the school syllabus and Mathematical subjects as it uses alpha-numerical questions and number lines to express themselves. The integers denoted by Z are also used in graphs and economics to study profit, loss, check sea levels, topography by geologists and other relevant terms, and experts to conduct research in subjects like Chemistry, Mathematics, Statistics, Geography, and Physics. This article serves as a guideline for integers rules, properties and mathematical operations needed in our day-to-day lives.