# Difference Between Real Numbers and Integers

Edited by Diffzy | Updated on: April 30, 2023

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“Numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things,” wrote mathematician and temporary ETH professor Richard Dedekind (1831-1916) in his work The Nature and Meaning of Numbers.

## Introduction

Integers are a part of real numbers that are absent of any fractions or decimals, whereas real numbers contain irrational as well as rational numbers. Real numbers encompass all kinds of numbers, which include integers as well, they share attributes such as the use of whole numbers and can be displayed on a number line. These similarities make it easier for students to understand and apply concepts in both arithmetic and algebra. Understanding the variations between these number types is essential in many domains, including mathematics, physics, and engineering.

## Real Number vs Integers

Integers are real numbers, however not all real numbers are integers. Real numbers, in addition to integers, comprise rational, irrational, whole, and natural numbers. Integers are real numbers that are made up of just positive and negative integers as well as whole numbers. Real numbers can incorporate fractions because of rational and irrational numbers, whereas integers cannot. Integers do not have "least-upper-bound-property," or completeness, as do real numbers.

## What is a Real Number?

Richard Dedekind, a professor of mathematics at the University of Zurich, posed these questions 159 years ago and became the first man to talk about real numbers.

Real numbers are a fundamental concept in mathematics, encompassing all rational and irrational numbers and providing the foundation for solving equations and analyzing mathematical relationships. Real numbers are the foundation for solving equations and analyzing mathematical relationships. They embody all rational and irrational numbers. They can be positive or negative in nature and are represented by the letter "R."

In contrast, real numbers comprise all rational and irrational numbers and are characterized as the set of all numbers that can be written as decimals or fractions. This includes integers as well as decimal figures like 1.4 or -0.45. Real numbers also contain non-fractional numbers such as pi (3.14159...) and the square root of 2. (1.41421...).

### Real numbers are distinguished into four categories

• Whole numbers (positive, negative, or zero): 2, -4, 0
• Fractions: 1/8, -3/4, 2/9
• Decimals (finite or infinite): 0.85, -1.45, 3.1...
• Irrational numbers (cannot be expressed as a fraction): pi (3.14159...), the square root of 2 (1.41421...), e (2.71828...)

### Properties of Real Numbers

• Commutative property
• Associative property
• Distributive property
• Identity property

### Whole Numbers

We rely on counting numbers in our daily lives, such as 1, 2, 3,..... Whole numbers are all of the basic counting numbers plus 0. Natural numbers are counting numbers in mathematics. As a result, the whole number might be regarded as an ensemble of all-natural numbers and 0. Whole numbers encompass all positive integers as well as zero.

The complete wide variety is part of the mathematical machine that protects all of the effective numbers  starting from zero to infinity. These numbers can be found on the number line. As a result, they are all real numbers. We can claim that although all whole numbers are real numbers, not all real numbers are whole numbers. As an outcome, whole numbers can be defined as the set of natural numbers plus 0. It is represented with the capital letter “ W ”

Whole numbers are frequently utilized for counting objects or expressing quantities that are unable to be broken down into smaller parts. For example, if you have three pears, you might write the whole number three.

Examples : 0, 4, 44, 444, 444, 4444,44444.........

### Fraction Numbers

Fractions are numbers that represent a portion of a whole. A fraction is made up of a pair of numbers divided by a line that runs horizontally called the denominator and numerator. The numerator is the number of parts under the account, whereas the denominator is the total number of components in the whole.

The name of the fraction was taken from a Latin term for a fraction that referred to "break".The fractions were initially written in Ancient Rome using phrases to describe a portion of the whole. Fractions were first written in India with one number atop another (numerator and denominator), without the use of a line. It was just the Arabs who introduced the line that divides the numerator and denominator.

Fractions are divided into 2 parts:

• Numerator: The numerator specifies how many fractional elements are displayed or selected. It is positioned above the fractional bar in the upper section of the fraction.
• Denominator: The denominator reflects the number of components in which the whole has been subdivided. It is positioned below the fractional bar in the lower section of the fractional term.

Examples: The fraction 3/4 represents three parts out of a total of four parts. This can be visualized as a pie or a rectangle divided into four equal parts, with three of those parts shaded or highlighted

### Decimal Numbers

A decimal mark (also referred to as a 'decimal separator') is a point or a dot that is used to divide the entire number from the fractional portion of a number. Decimal numbers are numbers that employ a base-ten notation system. The term "decimal" originates from the Latin word "decem," which means "ten."

Each of the digits in an integer indicates a power of 10 in decimal notation. The numbers on the right stand for 100, or one. The next numeral to the left is 10^1, or ten. The digit on the left represents 10^2, or 100, and so on.

10.5, which is a decimal number between 10 and 11, is an example of this. It is greater than 12, but below 13.

Examples: 11.4, 23.58,1.11,100.78……

### Irrational Numbers

Irrational numbers are real numbers that are unable to be expressed in a ratio of numbers. In other words, irrational numbers are real numbers that are not rational numbers. In the 5th century BC, Hippasus, a Pythagorean philosopher, found irrational numbers. Tragically, his theory was mocked, and he was thrown into the ocean.

Irrational numbers can be identified by decimal expansions that do not finish or repeat. After the decimal point, they have an unlimited amount of non-repeating numbers with no obvious pattern. Irrational numbers are not the same as imaginary numbers, which are complex numbers that can be expressed as a+bi, where a and b are real integers and i is the imaginary unit.

In other words, an irrational number is a number that is unable to be expressed in the form of a fraction. If N is an irrational number, it is not equal to p/q, where p/q represents two integers and q is not zero.

Examples:

• √2 (the square root of 2)
• π (pi)
• e (Euler's number)
• φ (the golden ratio)

### Rational Numbers

rational numbers are real numbers that are able to be expressed in a ratio of numbers. In other words, rational numbers are real numbers that are logical numbers. Pythagoras held the opinion that any value, such as 1/4 or 4/12, might be expressed as a combination of two numbers.

Rational numbers are utilized for expressing quantities or values that logical numbers or integers alone cannot, such as mass, length, or time measurement. A rational number is exactly the opposite of an irrational number

A rational number is a value of the form p/q, where p and q are integers and q is not equal to  zero.

Examples :

• 1/3
• -2/5
• 0.75 (which can also be written as 3/4)
• 22/7 (which is an approximation of pi)

## What are Integers?

By allowing fractions for coefficients and solutions, Diophantus was the first Greek mathematician to recognise positive rational numbers as numbers.

Integers include both whole and negative numbers. This means that if we combine negative and whole numbers, we get a collection of integers. Integers are capable of being added, subtracted, multiplied, and divided just like any other form of a number, but they have some crucial qualities that are unique to integers, such as divisibility rules and the idea of parity. (whether a number is odd or even).

Thus, integers are numbers that can be positive, negative, or zero, but not fractions or decimals

### Definition of Integers

An integer is a value that has no decimal or fractional component and can be negative or positive, including zero. Z represents a set of numbers that includes:

### Types of Integers

Positive Numbers: A positive number is one that is greater than zero.

For example 1, 2, 3,4….

Negative Numbers: A negative number is one that is less than zero. For example -1, -2, -3,...

Zero: The number zero is neither a negative nor a positive number. It is an even number.

### Other types of Integers

Natural numbers: These are the positive integers (1, 2, 3, 4, 5, ...). Note that some definitions include zero as a natural number, while others do not.

Whole numbers: These are the non-negative integers (0, 1, 2, 3, 4, ...). In other words, whole numbers include zero as well as natural numbers.

Negative integers: These are integers less than zero (-1, -2, -3, ...).

Even integers: These are the integers that are divisible by 2 (-2, 0, 2, 4, ...).

Odd integers: These are the integers that are not divisible by 2 (-3, -1, 11, 13, ...).

Prime integers: These are the integers that are only divisible by 1 and themselves (2, 3, 5, 7, 11, ...).

Composite integers: These are the integers that are not prime and have more than two factors (4, 6, 8, 9, 10, ...).

Square integers: These are the integers that are the product of an integer and itself (1, 4, 9, 16, 25, ...).

Cube integers: These are the integers that are the product of an integer, itself, and itself again (1, 8, 27, 64, ...).

### Rules of Integers

• When you add, subtract, or multiply two even integers, the result is always an even integer.
• When you add, subtract, or multiply two odd integers, the result is always an odd integer.
• When you add an even integer and an odd integer, the result is always an odd integer.
• When you subtract an odd integer from an even integer, the result is always an odd integer.
• When you subtract an even integer from an odd integer, the result is always an odd integer.
• The product of any integer and zero is zero.
• Any non-zero integer divided by itself is equal to 1.

### Operations in an Integer

They are four different types of operations in integers

Addition of integers: adding the value of two integers and putting the greater number sign.

Example :

(-5) + (+2) = -3

(-6) + (-4) = -10

Subtraction of integers: Subtracting the two values and putting the greater number sign.

Examples :

(-5) - (+2) = 7

(-6) - (-4) = -2

Multiplication of integers

• If both integers have the same sign, then the result is positive.
• If the integers have different signs, then the result is negative.

Division of integers

### Properties of Integers

• Closure Property
• Associative Property
• Commutative Property
• Distributive Property
• Multiplicative Inverse Property
• Identity Property

## Main Difference Between Integers And Real Numbers (In Points)

• All integers are real numbers but not all real numbers are integers
• The symbol of real numbers is “  R ” and symbol of integers is “ Z ”
• Integers is a subtype of real number
• Real number can be combined to produce an uncountable infinite set whereas integers can be used to create a countably infinite set
• Real numbers have positive numbers, negative numbers, zero, fractions, decimals but integers can have only positive numbers, negative numbers and zero.

## Conclusion

To summarize, real numbers are a collection of numbers that contains all irrational and rational numbers, whereas integers are a category of real numbers that only include positive and negative whole numbers, as well as zero. Integers are signified by the symbol Z, while real numbers are denoted by the character R.

Not every real number is an integer, but every integer is an actual number. For example, the number 3 is both an integer and a real number, whereas the number √2 is a real number, not an integer.

In many areas of mathematics, such as algebra, geometry, calculus, and number theory, real numbers, and integers are used. They are also employed in a wide range of real-world applications, including finance, engineering, and science. they are also used to represent quantities like measurements, distances, and probabilities

## References

• https://ethz.ch/staffnet/en/news-and-events/internal-news/archive/2017/11/when-richard-dedekind-invented-real-numbers-at-eth-zurich.html/
• https://mathmonks.com/integer/real-numbers-vs-integers
• https://www.splashlearn.com/math-vocabulary/integers
• https://brainly.ph/question/17779577

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"Difference Between Real Numbers and Integers." Diffzy.com, 2024. Mon. 04 Mar. 2024. <https://www.diffzy.com/article/difference-between-real-numbers-and-integers-1245>.

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