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class 10 real numbers

real numbers definition

Real numbers are all the numbers that can be found on the number line. This includes all positive and negative numbers, zero, and all the decimal numbers in between. Basically, real numbers are a combination of rational and irrational numbers.

Real Numbers Examples

1. Rational Numbers: These are numbers that can be written as a fraction, like 5, -8, or 7/2. The fraction can also be converted into a decimal or percentage.

   • Example: 2 (can be written as 2/1)
   • Example: -4 (can be written as -4/1)
   • Example: 0.75 (can be written as 3/4)

2. Irrational Numbers: These are numbers that cannot be written as simple fractions. They are non-repeating and non-terminating decimals.

   • Example: √2 (Square root of 2)
   • Example: π (Pi, approximately 3.14159)
   • Example: e (Euler's number, approximately 2.71828)

3. Integers: These include all whole numbers and their negative counterparts, including zero.

   • Example: 0, -1, 2, -3, 4

4. Decimals: These are numbers with one or more numbers after the decimal point.

   • Example: 3.14, -0.001, 4.56

Euclid’s Division Lemma

Euclid's Division Lemma says if you have two numbers $�$ and $�$, you can always find unique numbers $�$ (quotient) and $�$ (remainder) such that $�=�\times �+�$, where $�$ is always between 0 and $�$.

Euclid’s Division Algorithm

This is a way to find the Highest Common Factor (HCF) of two numbers $�$ and $�$. You keep dividing $�$ by $�$ and then taking the remainder until the remainder becomes zero. The last divisor you used is the HCF of $�$ and $�$.

Real Numbers

Real numbers include Rational and Irrational numbers. Rational numbers can be written as $\frac{�}{�}$ where both $�$ and $�$ are integers. Irrational numbers can't be written this way. Real numbers include all Rational and Irrational numbers.

Properties of Irrational Numbers

If you add, subtract, multiply or divide two Irrational numbers, the answer might not be an Irrational number. For example, $\sqrt{2}\times \sqrt{2}=2$, which is a Rational number.

Fractions

A terminating fraction is one that ends, like $\frac{1}{2}=0.5$. A recurring fraction goes on forever, like $\frac{1}{3}=\mathrm{0.3333...}$.

Properties Related to Prime Numbers

Prime numbers are numbers that can only be divided by 1 and themselves. For example, 5 and 7 are primes. Some properties include that if $�$ is a prime and divides ${�}^2$, then $�$ also divides $�$.

Decimal Expansion

Decimal numbers can be either terminating or recurring. If a number's decimal expansion terminates, it can be written as $\frac{�}{�}$, where $�$ and $�$ are co-prime and $�$ is of the form $2^{�}\times 5^{�}$.

Fundamental Theorem of Arithmetic

Every number greater than 1 is either a prime number or can be written as a product of prime numbers in a unique way. For example, $12=2\times 2\times 3$.

Finding HCF and LCM

To find the HCF or LCM of two numbers, you first find their prime factors. The HCF is made up of the smallest powers of common prime factors, and the LCM is made up of the greatest powers of all the prime factors.

Relationship Between HCF and LCM

If $�$ and $�$ are two numbers, then $\text{HCF}(�,�)\times \text{LCM}(�,�)=�\times �$.

Method of Finding LCM

You can find the LCM of numbers by listing multiples, using prime factorization, or using the division method.

Real Numbers Class 10 Worksheet with Answers

Worksheets are a great way for students to practice what they learn in class. A real numbers class 10 worksheet with answers can help students test their skills in working with real numbers, which include both rational and irrational numbers. The worksheet often comes with a variety of questions ranging from easy to challenging, and having the answers handy can help in understanding where a mistake was made. This is useful for both students and teachers for review purposes, and for parents to guide their kids at home.

Real Numbers Class 10 Extra Questions

Sometimes, classwork and homework aren’t enough for students to grasp a topic fully. Extra questions on real numbers in class 10 can provide that additional practice needed for mastery. These questions can include problems that test various aspects like finding square roots, identifying rational numbers, or even classifying numbers. Teachers can use these extra questions to challenge students, and parents can use them to help kids practice more at home.

Real Numbers Class 10 Notes

Having good notes can make studying easier and more effective. Real numbers class 10 notes usually summarize key points and formulas about real numbers. They serve as a quick reference guide for students when revising for exams. Teachers can hand out these notes to help students focus on the main ideas, and parents can use them to understand what their child is learning, aiding in more effective home study.

Real Numbers MCQ Class 10

Multiple-choice questions (MCQs) are often part of exams and quizzes. A real numbers MCQ class 10 set tests students on their quick recall and understanding of the topic. MCQs are beneficial for quick assessment and are easy to grade. They are useful tools for teachers to gauge class comprehension and for parents to quickly assess their child's understanding of the subject.

Real Numbers Mind Map

A mind map for real numbers can help visualize how different concepts within the topic are connected. It could show the relationship between rational and irrational numbers, square roots, and other related topics. Mind maps make it easier for students to remember key points and are also a handy tool for teachers during lessons. Parents can use these mind maps to get an overview of what their kids are learning.

Real Numbers Class 10 Question Bank

A question bank on real numbers for class 10 is a collection of various types of questions covering all aspects of the topic. It usually includes short-answer questions, long-answer questions, and sometimes even MCQs. Teachers can pull questions from this bank for tests and quizzes, while students can use it for extra practice. Parents can use the question bank to help their kids prepare for exams.