# Real Numbers Formulas for Class 10

A combination of rational and irrational numbers form real numbers. Real Numbers formulas for class 10 are applied in many important concepts. However, if the understanding of the real numbers is not clear then moving ahead with real numbers formulas can be difficult. This article provides a concise list of formulas along with some examples which will help students in learning and memorizing these formulas.

## List of Real Numbers Formulas for Class 10

The list of real numbers formulas class 10 is provided below for students to memorize.

- Euclid’s division lemma: For positive integers a and b, there are whole numbers m and n satisfying a = bm + n, 0 ≤ n < b.

- Euclid’s division algorithm is based on Euclid’s division lemma. It states that the HCF of any two positive integers ‘a’ and ‘b’, with a > b, is found out as follows:

- a) Apply the division lemma to find ‘m’ and ‘n’ where a = bm + n, 0 ≤ n < b.
- b) If n = 0, the HCF is ‘b’. If n ≠ 0, apply Euclid’s lemma to ‘b’ and ‘n’.
- c) Carry on with the process till the remainder’s value is zero. The divisor will be HCF of (a, b). Also, HCF of (a, b) = HCF(b, n).

- If ‘x’ is a prime number and ‘x’ divides a
^{2}, then ‘x’ divides a, where ‘x’ is a positive integer.

## Applications of Real Numbers Formulas for Class 10

Real numbers formulas for class 10 are widely used in arithmetic. Some of the applications of these formulas are mentioned below:

- Real numbers formulas of finding L.C.M is applied to find the points of common occurrence such as time at which two trains running at different speeds will cross.
- In physics real numbers formulas are used to model most physical constants such as temprature, mass, energy, velocity, and many more.

## Real Numbers Formulas for Class 10 Examples

**Example 1: **Verify if √2 is an irrational number or not.

**Solution:** Let us start by assuming that √2 is a rational number.

Let us find integers ‘x’ and ‘y’ (not equal to zero) such that √2 = x/y

If x and y have a common factor apart from 1. Then, we will divide by the common factor to get

√2 = m/n where ‘m’ and ‘n’ are co-primes.

Hence, n√2 = m.

Squaring both of the sides, we get, 2n^{2} = m^{2}

Considering the theorem: If ‘x’ is a prime number and ‘x’ divides a 2, then ‘x’ divides a, where ‘x’ is a positive integer, it can be said that 2 divides m2 in this case.

Hence, 2 divides ‘m’.

It can be written that m = 2d for some integer ‘d’.

Substituting for ‘m’, we get 2n^{2} = 4d^{2} , that is, n^{2} = 2d^{2}.

This suggests that 2 divides n^{2}, and so 2 divides n.

Therefore, ‘m’ and ‘n’ have at least 2 as a common factor.

But at the same time this contradicts the fact that ‘m’ and ‘n’ have no common factors other than 1.

This contradiction has arisen because of our incorrect assumption that √2 is rational.

Hence, we conclude that √2 is an irrational number.

**Example 2:** Mohan is arranging chocolates in his shop. He has 400 dark chocolates and 220 milk chocolates. How many chocolates should be in a given stack considering Mohan wants an equal number of chocolates in each stack?

**Solution:** HCF of (400, 220) which is equal to 20. Thus, Mohan should make stacks of 20 for both kinds of chocolates.

## Tips to Memorize Real Numbers Formulas for Class 10

Learning the real numbers formulas for class 10 can be challenging due to various factors such as the lack of time to revise. To ease out the learning process the students can follow the below tips that can help them with memorizing the formulas effectively:

- Students must ensure to read the important notes provided in textbook to understand the concept behind the formula and get their doubts cleared immediately from teachers or friends.
- Once the logic behind the formula is clear, students can write it down several times to get thorough in it. This will also help them to grasp the logic that they've read.
- Images of formulas can be saved as wallpaper on mobile phones and computers. This will benefit students in a quick revision throughout the day.

Students can download the printable **Maths Formulas Class 10** sheet from below.

## FAQs on Area Related to Real Numbers Formulas for Class 10

### What are the Important Real Numbers Formulas for Class 10?

The important Real Numbers Formulas for Class 10 are:

- Euclid’s division lemma: For positive integers a and b, there are whole numbers m and n satisfying a = bm + n, 0 ≤ n < b.
- Euclid’s division algorithm is based on Euclid’s division lemma. It states that the HCF of any two positive integers a and b, with a > b, is found out as follows:
- a) Apply the division lemma to find m and n where a = bm + n, 0 ≤ n < b.
- b) If n = 0, the HCF is b. If n ≠ 0, apply Euclid’s lemma to b and n.
- c) Carry on with the process till the remainder’s value is zero. The divisor will be HCF of (a, b). Also, HCF of (a, b) = HCF(b, n).
- If ‘x’ is a prime number and ‘x’ divides a
^{2}, then ‘x’ divides a, where ‘x’ is a positive integer

### Why is it Important to Solve Problems Related to Real Numbers Formulas for Class 10?

Solving problems related to basic real numbers formulas for class 10 will help students in covering all necessary ideas of real numbers, that have important applications in arithmetic, such as how to apply the Fundamental Theorem of Arithmetic to prove the irrationality of numerous numbers and how to determine when a rational number's decimal expansion is terminating and when it is non-terminating.

### What are Important Concepts Covered under Real Numbers Formulas for Class 10?

The real numbers formulas for class 10 explain the important concepts related to properties of real numbers, the equation for Euclid's Lemma, and its proof, Fundamental Theorem of Arithmetic, and how to find the HCF and LCM via Prime Factorization Method.

### How Many Formulas are There in Real Numbers Formulas for Class 10?

There are around three primary formulas in real numbers formulas for class 10 encompassing the properties of real numbers and their theorems. These formulas will not only help the students solve questions quickly but will also be useful in higher-level mathematics as well.

### How can I Memorize Real Numbers Formulas for Class 10?

Students can use the following methods to help them memorize formulas successfully making the learning process easier:

- Students must ensure that they read notes in the textbook to grasp the concept behind the formula.
- Writing down the formula multiple times with proper explanations will help in understanding the theory behind the formula better thus making the brain remember it thoroughly.
- On mobile phones and laptops, images of formulas can be saved as wallpaper. This will also help in a quick revision as and when the students use these gadgets.

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