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**Real numbers **is considered one of the most pivotal chapters in mathematics from the point of view of exams. Practically speaking, any number that comes in your mind can be termed as a real number. But believe it or not, real numbers can be rational numbers as well. Real numbers are easy to comprehend and understand. When the length of any number needs to get measured, the real number gets called for action. So let us continue our discussion and explore the world of real numbers together.

Euclid was proclaimed as one of the renowned mathematicians who facilitated the new methodology to study geometry. One of his most profound contributions to number theory was the introduction of **Euclid’s Division Lemma. **

According to **Euclid’s Division Lemma theory**, on the off chance that you have any two positive integers like a and b, in such case, there surfaced two novel integers that fulfil the following condition-

*a = bq + r* = 0* ≤ r < b*

The Euclidean division algorithm gets hinged on **Euclid’s division lemma**. Whenever you have to calculate HCF (Highest Common Factor), Euclid’s division algorithm gets employed. HCF gets called the largest number that gets divided by two positive integers that mean when divided; it results in zero as a reminder.

The **Fundamental Theorem of Arithmetic **gets profoundly known as the unique factorization theorem in number theory. It expresses that each number greater than 1 can either be a prime number itself or can be part of the product of prime numbers. For instance:

**1200 = 24 . 3 . 52**

**Revisiting Rational Numbers and Their Decimal Expansions **means the decimal expansion of any rational number can either be terminating or non-terminating repeating.

**Terminating Decimal Expansion-**When in the process of the division after the finite number of steps, any number that gets ceased is called Terminating Decimal Expansion. For example – 6.25**Non- terminating Decimal Expansion-**When no number gets ceased in the course of division, it is called as Non- terminating Decimal Expansion. For example – 0.333333

The basic** properties of Real Numbers **get recouped to make complex mathematics simple and easy to solve. Here is the rundown of the real number’s properties:

**Associative Property –**This property states that any real number in multiplication or addition, when added or multiplied in various ways, will not change the results.

**Illustration of the associative property of addition **

(2 + 5) + 1=1 + (6 + 1) 7 + 1=1 + 7 8=8both equation results in 8

**Illustration of the associative property of multiplication **

(2 x 4) x 5=2 x (4 x 5) 8 x 5=2 x 20 40=40both equation results in 40

It states that the upshot of the real number (x) that results in the sum of two real numbers ( y+z) claims equal to the sum of each part of real numbers. Here, numbers inside the bracket get added first, and then the expression gets multiplied by the given number.

Example of **Distributive Property:**

*2(1+3) = 1(2) = 8*

* 2(1)+ 2(3) = 2+ 6 = 8*

* 2(1+3) = 2(1)+ 2(3)*

- This property is composed of two subpart- Additive and Multiplicative Identity.
- The additive
**identity property: – a+0 = a, i.e. 4+0 = 4.**Here, the number 0 is called an additive identity element. - The multiplicative
**identity property:- a.1 = 1.a = a, i.e. 3.35 x1 = 3.35.**Here, the number 1 is called a multiplicative identity element.

In a nutshell, real numbers can have integers, whole numbers, and natural numbers but not imaginary numbers because those numbers are hard to express in the number line and complex in nature. You can say various properties of a real number as discussed above can be used to orchestrate any algebraic expression.

**Frequently Asked Questions (FAQs)**

**Is 0 a real number?****What is the set of all real numbers?****What is considered a real number?****What are not real numbers?****How do you identify real numbers?**

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