**Class 10th Maths syllabus** aims to impart students with the basic and fundamental concepts of this discipline to further build upon in the upcoming grades. Amongst the most interesting of these concepts are Real Numbers which form the foundation of several imperative Mathematical topics. In your previous classes, you must have studied about multiple types of numbers such as Natural, Whole, Rational, Irrational and Integers. But do you know that all these types of numbers fall under Real Numbers? Amazing! Right? As we are here to walk you through the topic, we will keep on stating some witty facts like these. So, stay tuned and let’s start our discussion of class 10 real numbers.

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## What are Real Numbers?

All numbers, including positive, negative, and zero, come under the broad category of **real numbers**. In general terms, real numbers represent all those numbers which are not imaginary numbers. Therefore, a real number combines the properties of rational and irrational numbers.For example: 4, 43, -34, -5, 3/4, 5/9, π(3.14), etc.

Before we move further with the concept of real numbers, let us quickly recall the major types of numbers which you must have previously studied.

### Natural Numbers

Natural numbers represent all the positive numbers in nature from 1 to infinity. The set of natural numbers is represented as ‘N’ and does not include decimals, fractions, or any negative numbers. N = 1, 2, 3, 4, 5, 6, …

### Whole Numbers

Whole numbers represent all positive numbers, including zero, and are represented as ‘W’. W = 0, 1, 2, 3, 4, 5, 6, …

### Integers

Integers include all positive and negative numbers with zero, not representing any decimal or fractional component. For instance, 2, 3, 51, 0, -1, -5, while 2.45, √2, and 10 ¾ are not integers. They are presented as ‘I’. I = …., -2, -1, 0, 1, 2, …

### Rational Numbers

Rational Numbers are numbers that can be presented as fractions and decimals in the form of a/b of two integers with ‘a’ as the numerator and ‘b’ as a non-zero denominator. **For example**, 2/3, 4/5, 1/2, 10 ¾, etc. They are represented as ‘R’. R = ½ , ¾ , 5.24, 0.5432, etc.

### Irrational Numbers

Irrational Numbers represent all numbers that are not a rational number or can not be expressed in the ratio of any integers. **For example**, √2, √3, √5, √7, and π are irrational numbers.

## Presentation Of Real Number

All types or forms of a real number as per real numbers chapter (integers, fractions, and decimals) can be precisely placed on the Number line. For instance, real number 4 as 4.0, -3 as -3.0, fractions in the form of 2/5 as 0.40, 1/2 as 0.50, and so on.

## Euclid Division Lemma (Algorithm)

Euclid Division Lemma algorithm follows the conceptual formula ** [dividend = (divisor x quotient) + remainder]**. As per this theorem, if there are two positive integers let’s say p and q, then there exists a parallel integers r and s that satisfies the statement as p = qr + s.(where s represents a number equal to more than 0 and less than q).

We can easily find the HCF using Euclid’s algorithm. Let’s study a simple example to get the HCF of two integers 455 and 42, respectively.

*Then, by Euclid’s Lemma, we have: *

455 = 42 × 10 + 35

Similarly, if we use the dividend method (divisor 42, remainder 35) and then as per the Euclid’s Lemma,

42 = 35 × 1 + 7*Now, moving on here the divisor becomes 35 and remainder 7, and using the Euclid lemma to find:*

35 = 7 x 5 + 0

As the remainder becomes zero, we can stop the process to conclude the HCF of two integers 455 and 42 as 7. Thus, using Euclid’s algorithm mentioned in Class 10 Real Numbers, you can calculate the HCF of two numbers in a simple manner.

## Decimal Expansion

While studying class 10 chapter on real numbers, you will also get to know about the expansion of a decimal rational number in two ways. Terminating decimal expansion are those when decimal ends with finite numbers. For instance, ½ as 0.50 in decimals, and vice versa.

**0.375 = 375/1000 = 3 x 5 ^{ 3} / 2^{3} x 5^{3} = ⅜**

Non-terminating decimal expansions are with those when decimals are non-terminating. They can be either repeating (rational numbers) or non-repeating (irrational numbers).

*For Example:***7/12 = 0.58333… = 0.583****9/11 = 0.8181… = 0.81(repeating)**

So, decimal expanses of rational numbers will be either terminating or non-terminating.

## Fundamental Theorem of Arithmetic

The theorem states that any composite number is presentable as factors or products of primes. Therefore, irrelevant of the order, the prime factorization of a natural number will always be unique in itself. For instance, 3 × 5 × 7 x 11 is taken the same as 5 × 7 x 11 x 3 or in any order that they are written in.

## Class 10 Maths Real Numbers Practice Questions

Here are a few practice questions for class 10 maths real numbers.

- The decimal expansion of the rational number 43/ 2
^{4}5^{3 }will terminate after how many places of decimals? - Find the largest number that will divide 398, 436 and 542 leaving remainders 7, 11, and 15 respectively.
- Express 98 as a product of its primes.
- HCF and LCM of two numbers is 9 and 459 respectively. If one of the numbers is 27, find the other number.
- Find HCF and LCM of 13 and 17 by prime factorisation method.
- Find the HCF (865, 255) using Euclid’s division lemma.
- Find the largest number which divides 70 and 125 leaving remainder 5 and 8 respectively. (
- Show that 3√7 is an irrational number.
- Explain why (17 × 5 × 11 × 3 × 2 + 2 × 11) is a composite number?

## Class 10 Maths Real Numbers MCQs with Answers PDF

Thus, we hope that this blog has provided you with useful and informative study notes on class 10 real numbers. Being a class 10th student, it is necessary to sort out your interests and select a stream accordingly. If you are struggling in choosing the right stream, connect with **Leverage Edu** experts and they will help you select the right one as per your career aspirations! Hurry Up! Book an e-meeting now.