The difference between Real numbers and integers is, that integers are whole numbers, like 1,2,3,4,5, and so on. Natural numbers can be positive, negative, or zero. Real numbers encompass integers and decimals, like 3.14, -5.2, and any number you can imagine on a number line. So, all integers are real numbers, but not all real numbers are integers.

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## What is Real Numbers?

The Real Numbers are basically any numbers you can use to measure stuff on a number line. It includes whole numbers like 1,2,3,4, etc., and fractions like ½, and 3.4, and many more, and even decimals that never end like pie(π). So, real numbers are super helpful for representing any value you can imagine, from distances to temperature.

## What is Integers?

Integers are whole numbers, including zero(0), that come in positive and negative numbers. Integers don’t have any decimal or fractional parts or numbers. Think of counting numbers like 1, 2, 3, and so on, but also include negative numbers like ……….-3, -2, -1. Integers are a fundamental part of counting and basic math operations.

## What is the Difference between Real Numbers and Integers?

Numbers are fundamental tools for representing quantities in our everyday lives. But not all numbers are created equal. They might be similar at first glance, but there are key differences that affect how we use them. Let’s explore the difference between Real numbers and Integers in the table below.

Property | Real Numbers | Integers |

Definition | Any number on a number line | Whole numbers (positive, negative, or Zero) |

Examples | 3, 1.5, -2.7, √2, π | 1, 0, -3, 100 |

Decimals | Can include decimals | No decimals |

Properties of Real Numbers | Yes | Yes |

Commutative (order doesn’t affect + or x) | Yes | Yes |

Associative (grouping doesn’t affect + or x) | Yes | Yes |

Additive Identity (0) | Yes | Yes |

Multiplicative Identity (1) | Yes (except for 0) | Not applicable |

Subset Relationship | – | Integers are a subset of Real Numbers |

## Real Numbers and Integers Formulas

Real numbers don’t have specific formulas, but there are formulas related to their properties and operations like addition, subtraction, multiplication, and division. Integers also don’t have their own formulas, but there are concepts related to them that use formulas. Here are a couple of examples that are mentioned below.

For Integers | |

Euclid’s Division Lemma | It is used to find the greatest common factor of two integers. It involves formulas like “a=bq+r” where a and b are the integers, q is the quotient, and r is the real remainder. |

Prime Factorization | It involves finding the prime factors and using their exponents. |

For Real Numbers | |

Commutative Property | It means the order doesn’t affect addition or multiplication (a+b = b+a, axb = bxa). |

Associative Property | Grouping doesn’t affect addition or multiplication ((a+b)+c = a+(b+c)), (axb)xc = ax(bxc). |

**Note: **Integers are a subset of real numbers, so all the real number properties apply to integers as well.

## Application of Real Numbers and Integers in Real Life

Real numbers and integers are all around us, playing essential roles in many aspects of our daily lives. Here’s a breakdown of their applications in real life.

### Real Numbers

**Measurements:**Pretty much any measurement you can imagine uses real numbers like Temperature (25.2 degree celsius), speed (68.5 mph), distance (3.14 meters), and even pi value (3.14159…) for circles all rely on real numbers to represent precise values.**Finance:**Stock prices, interest rates, and percentages all involve real numbers, allowing us to track changes, calculate profits, and express ratios.**Science & Engineering:**Real numbers are the language of science and engineering. From complex physics equations to architectural blueprints, real numbers and vital for calculations and modeling.

### Integers

**Counting:**The most basic application is we use integers to count objects, people, floors in a building, or anything where whole numbers are sufficient.**Scores & Rankings:**Keeping track of points in games, movie ratings, or exam scores all rely on integers.**Money:**While currency often involves decimals, the core unit (dollars, rupees, etc.) is an integer, eg 10 dollars, not 10.5 dollars.- I
**nventory Management:**Businesses track stock using integers, indicating the whole number of items available.

In essence, real numbers provide a comprehensive measurement system for the world around us, while integers act as the building blocks for counting and representing whole quantities. They work together to give us a complete mathematical toolkit for navigating everyday situations.

## Similarities Between Real Numbers and Integers

Here are some of the similarities between real numbers and integers:

- Both real numbers and integers are part of the foundational number system we use in mathematics. This system allows us to represent various quantities and perform operations on them.
- Both real numbers and integers can be used for counting objects and measuring quantities. For instance, you can use integers to count the number of apples in a basket (5 apples) and real numbers (like 2.5 kilograms) to measure their weight.
- Both real numbers and integers can be visualized on a number line. The number line stretches infinitely in both directions (positive and negative) and helps represent the relative magnitudes of numbers.
- Real numbers and integers adhere to the ordering relation ‘<‘ where one number is considered less than another. For example, 3 < 5 and -2 < -1.

## Sample Questions on Real Numbers and Integers

- Which of the following numbers is a real number?

A) 3/0

B) √(-9)

C) 0.5

D) i

Answer: C) 0.5

- Which of the following is not an integer?

A) -5

B) 0

C) 2.5

D) 100

Answer: C) 2.5

- What is the opposite of -7?

A) -7

B) 7

C) -(-7)

D) 0

Answer: B) 7

- Which of the following is not a real number?

A) -3

B) √16

C) π

D) i

Answer: D) i

- What is the result of (-4) × (-6)?

A) -24

B) 24

C) -10

D) 10

Answer: B) 24

- What is the additive inverse of 15?

A) 0

B) 15

C) -15

D) 1

Answer: C) -15

- Which of the following is a whole number but not an integer?

A) 0

B) -3

C) 5

D) -1/2

Answer: A) 0

- What is the absolute value of -12?

A) 12

B) -12

C) 0

D) Undefined

Answer: A) 12

- Which of the following is not a rational number?

A) 1/2

B) √2

C) 0.75

D) -3

Answer: B) √2

- Which of the following is an irrational number?

A) 5

B) -3

C) √5 D) 3/4

Answer: C) √5

## FAQ’s

**How are real numbers and integers different?**

The key difference is that real numbers are a broader category that encompasses all integers. Integers are like a special subset of real numbers that only include the whole numbers on the number line. So, all integers are real numbers, but not all real numbers are integers (like decimals and fractions).

**Are there any real-world examples of the difference between real numbers and integers?**

- When you measure temperature (25.2 degrees Celsius), you’re using a real number (it has a decimal).
- But counting the number of apples in a basket (5 apples) involves integers (whole numbers).

**Is there a formula for real numbers or integers?**

- Real numbers and integers themselves don’t have specific formulas, but we use formulas related to their properties and operations (addition, subtraction, multiplication, division) for real numbers.
- Integers are used in formulas related to number theory, like finding the greatest common factor (GCD) of two integers.

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