rational numbers Archives – Mathemafia

Class 10 | Chapter 1 | Real Numbers | Exercise 1.4

Sunil Syal — Sun, 20 Jun 2021 03:02:02 +0000

Question 1: Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

Question 2: Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

Question 3: The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form p/q, what can you say about the prime factors of q?

(i) 43.123456789

(ii) 0.120120012000120000. . .

(iii) 43.123456789123456789…

Testimonials

An Excellent Math Teacher

Sunil Sir has been an excellent math teacher for Saiba. His teaching methods are highly informative, engaging, and motivational which have greatly strengthened Saiba’s understanding of maths concepts. He consistently provides support beyond the classroom and has played an important role in improving Saiba’s problem-solving skills and preparation for math competitions. His dedication and professionalism make him an outstanding educator. Saiba is thankful for his continued support and positive influence.

– Saiba’s Parents (USA)

Amazing Math Coach

Sunil is an amazing math coach! My son Rishi loves to learn from him. He makes learning fun and engages Rishi. Sunil is also very patient with kids and understands my son’s learning methods and teaches accordingly with positive reinforcement. Very thankful to him!

– Rishi’s Parents (USA)

Unique, Patient and Good Mentor

Sunil sir has a unique way of explaining complex concepts, making them simple and understandable. You are very patient and always encourages me to ask queries as many times until the concept is clearer. You are a great person besides being a good mentor and always in a light mood and never gets frustrated for anything. My grades have immensely improved and I am much more confident in maths now.

Also, your videos on Mathemafia youtube channel are very informative.

– Lakshay (USA)

WhatsApp for 1-1 Classes

❇️ ❇️ M️aster List – All Maths Formulas, Class 10 ❇️ ❇️

🇮🇳 हर भारतीय के दिल की आवाज़… आइए, इस स्वतंत्रता दिवस मिलकर गाएँ – “भारत माता की जय“।

Class 10 – Mathematics – NCERT Solutions

Real Numbers
- Exercise 1.1
- Exercise 1.2
Polynomials
- Exercise 2.1
- Exercise 2.2
Pair of Linear Equation in Two Variables

Quadratic Equations
Arithmetic Progressions
- Exercise 5.1
- Exercise 5.2
Triangles
- Exercise 6.1
- Exercise 6.2
- Exercise 6.3

Coordinate Geometry
- Exercise 7.1
- Exercise 7.2
Introduction to Trigonometry
Some Applications of Trigonometry
- Exercise 9.1
Circles
- Exercise 10.1
- Exercise 10.2

Areas Related to Circles
- Exercise 11.1
Surface Areas and Volumes
- Exercise 12.1
- Exercise 12.2
Statistics
- Exercise 13.1
- Exercise 13.2
- Exercise 13.3
Probability
- Exercise 14.1

Super Cool Stuff

Calculators
- Quadratic Equation Calculator – Solve any Equation
Super Concepts

Theorems
Quizzes

The post Class 10 | Chapter 1 | Real Numbers | Exercise 1.4 appeared first on Mathemafia.

Class 10 | Chapter 1 | Real Numbers | Exercise 1.3

Sunil Syal — Thu, 10 Jun 2021 03:54:56 +0000

Question 1: Prove that √5 is irrational.

Question 2: Prove that 3 + √5 is irrational.

Question 3: Prove that the following are irrationals:
(i) 1/√2 (ii) 7√5 (iii) 6 + √2

Testimonials

An Excellent Math Teacher

– Saiba’s Parents (USA)

Amazing Math Coach

– Rishi’s Parents (USA)

Unique, Patient and Good Mentor

Also, your videos on Mathemafia youtube channel are very informative.

– Lakshay (USA)

WhatsApp for 1-1 Classes

❇️ ❇️ M️aster List – All Maths Formulas, Class 10 ❇️ ❇️

🇮🇳 हर भारतीय के दिल की आवाज़… आइए, इस स्वतंत्रता दिवस मिलकर गाएँ – “भारत माता की जय“।

Class 10 – Mathematics – NCERT Solutions

Real Numbers
- Exercise 1.1
- Exercise 1.2
Polynomials
- Exercise 2.1
- Exercise 2.2
Pair of Linear Equation in Two Variables

Quadratic Equations
Arithmetic Progressions
- Exercise 5.1
- Exercise 5.2
Triangles
- Exercise 6.1
- Exercise 6.2
- Exercise 6.3

Coordinate Geometry
- Exercise 7.1
- Exercise 7.2
Introduction to Trigonometry
Some Applications of Trigonometry
- Exercise 9.1
Circles
- Exercise 10.1
- Exercise 10.2

Areas Related to Circles
- Exercise 11.1
Surface Areas and Volumes
- Exercise 12.1
- Exercise 12.2
Statistics
- Exercise 13.1
- Exercise 13.2
- Exercise 13.3
Probability
- Exercise 14.1

Super Cool Stuff

Calculators
- Quadratic Equation Calculator – Solve any Equation
Super Concepts

Theorems
Quizzes

The post Class 10 | Chapter 1 | Real Numbers | Exercise 1.3 appeared first on Mathemafia.

Class 10 | Chapter 1 | Real Numbers | Exercise 1.2

Sunil Syal — Mon, 19 Apr 2021 10:06:13 +0000

Question 1: Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

Question 2: Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.

(i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54

Question 3: Find the LCM and HCF of the following integers by applying the prime factorisation method.

(i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25

Question 4: Given that HCF (306, 657) = 9, find LCM (306, 657).

Question 5: Check whether 6ⁿ can end with the digit 0 for any natural number n.

Question 6: Explain why 7×11×13 +13 and 7×6×5×4×3×2×1+5 are composite numbers.

Question 7: There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

Testimonials

An Excellent Math Teacher

– Saiba’s Parents (USA)

Amazing Math Coach

– Rishi’s Parents (USA)

Unique, Patient and Good Mentor

Also, your videos on Mathemafia youtube channel are very informative.

– Lakshay (USA)

WhatsApp for 1-1 Classes

❇️ ❇️ M️aster List – All Maths Formulas, Class 10 ❇️ ❇️

🇮🇳 हर भारतीय के दिल की आवाज़… आइए, इस स्वतंत्रता दिवस मिलकर गाएँ – “भारत माता की जय“।

Class 10 – Mathematics – NCERT Solutions

Real Numbers
- Exercise 1.1
- Exercise 1.2
Polynomials
- Exercise 2.1
- Exercise 2.2
Pair of Linear Equation in Two Variables

Quadratic Equations
Arithmetic Progressions
- Exercise 5.1
- Exercise 5.2
Triangles
- Exercise 6.1
- Exercise 6.2
- Exercise 6.3

Coordinate Geometry
- Exercise 7.1
- Exercise 7.2
Introduction to Trigonometry
Some Applications of Trigonometry
- Exercise 9.1
Circles
- Exercise 10.1
- Exercise 10.2

Areas Related to Circles
- Exercise 11.1
Surface Areas and Volumes
- Exercise 12.1
- Exercise 12.2
Statistics
- Exercise 13.1
- Exercise 13.2
- Exercise 13.3
Probability
- Exercise 14.1

Super Cool Stuff

Calculators
- Quadratic Equation Calculator – Solve any Equation
Super Concepts

Theorems
Quizzes

The post Class 10 | Chapter 1 | Real Numbers | Exercise 1.2 appeared first on Mathemafia.

Class 10 | Chapter 1 | Real Numbers | Exercise 1.1

Sunil Syal — Wed, 17 Mar 2021 03:51:12 +0000

Question 1: Use Euclid’s division algorithm to find the HCF of :
(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

Question 2: Show that any positive odd integer is of the form 6q+1, or 6q+3, or 6q+5, where q is some integer.

Question 3: An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

Question 4: Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

Question 5: Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m+1 or 9m+8.

Testimonials

An Excellent Math Teacher

– Saiba’s Parents (USA)

Amazing Math Coach

– Rishi’s Parents (USA)

Unique, Patient and Good Mentor

Also, your videos on Mathemafia youtube channel are very informative.

– Lakshay (USA)

WhatsApp for 1-1 Classes

❇️ ❇️ M️aster List – All Maths Formulas, Class 10 ❇️ ❇️

🇮🇳 हर भारतीय के दिल की आवाज़… आइए, इस स्वतंत्रता दिवस मिलकर गाएँ – “भारत माता की जय“।

Class 10 – Mathematics – NCERT Solutions

Real Numbers
- Exercise 1.1
- Exercise 1.2
Polynomials
- Exercise 2.1
- Exercise 2.2
Pair of Linear Equation in Two Variables

Quadratic Equations
Arithmetic Progressions
- Exercise 5.1
- Exercise 5.2
Triangles
- Exercise 6.1
- Exercise 6.2
- Exercise 6.3

Coordinate Geometry
- Exercise 7.1
- Exercise 7.2
Introduction to Trigonometry
Some Applications of Trigonometry
- Exercise 9.1
Circles
- Exercise 10.1
- Exercise 10.2

Areas Related to Circles
- Exercise 11.1
Surface Areas and Volumes
- Exercise 12.1
- Exercise 12.2
Statistics
- Exercise 13.1
- Exercise 13.2
- Exercise 13.3
Probability
- Exercise 14.1

Super Cool Stuff

Calculators
- Quadratic Equation Calculator – Solve any Equation
Super Concepts

Theorems
Quizzes

The post Class 10 | Chapter 1 | Real Numbers | Exercise 1.1 appeared first on Mathemafia.

Class 10 | Chapter 1 | Real Numbers | Examples

Sunil Syal — Fri, 05 Mar 2021 05:18:58 +0000

Example 1 : Use Euclid’s algorithm to find the HCF of 4052 and 12576.

Example 2: Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer.

Example 3: Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.

Example 4: A sweet seller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of that can be placed in each stack for this purpose?

Example 5: Consider the numbers 4ⁿ, where n is a natural number. Check whether there is any value of n for which 4ⁿ ends with the digit zero.

Example 6: Find the LCM and HCF of 6 and 20 by the prime factorisation method.

Example 7: Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.

Example 8: Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

Example 9: Prove that √ 3 is irrational.

Example 10: Show that 5 – √ 3 is irrational.

Example 11: Show that 3√2 is irrational.

Testimonials

An Excellent Math Teacher

– Saiba’s Parents (USA)

Amazing Math Coach

– Rishi’s Parents (USA)

Unique, Patient and Good Mentor

Also, your videos on Mathemafia youtube channel are very informative.

– Lakshay (USA)

WhatsApp for 1-1 Classes

❇️ ❇️ M️aster List – All Maths Formulas, Class 10 ❇️ ❇️

🇮🇳 हर भारतीय के दिल की आवाज़… आइए, इस स्वतंत्रता दिवस मिलकर गाएँ – “भारत माता की जय“।

Class 10 – Mathematics – NCERT Solutions

Real Numbers
- Exercise 1.1
- Exercise 1.2
Polynomials
- Exercise 2.1
- Exercise 2.2
Pair of Linear Equation in Two Variables

Quadratic Equations
Arithmetic Progressions
- Exercise 5.1
- Exercise 5.2
Triangles
- Exercise 6.1
- Exercise 6.2
- Exercise 6.3

Coordinate Geometry
- Exercise 7.1
- Exercise 7.2
Introduction to Trigonometry
Some Applications of Trigonometry
- Exercise 9.1
Circles
- Exercise 10.1
- Exercise 10.2

Areas Related to Circles
- Exercise 11.1
Surface Areas and Volumes
- Exercise 12.1
- Exercise 12.2
Statistics
- Exercise 13.1
- Exercise 13.2
- Exercise 13.3
Probability
- Exercise 14.1

Super Cool Stuff

Calculators
- Quadratic Equation Calculator – Solve any Equation
Super Concepts

Theorems
Quizzes

The post Class 10 | Chapter 1 | Real Numbers | Examples appeared first on Mathemafia.

Class 10 | Chapter 1 | Real Numbers | Theorems & Concepts

Sunil Syal — Fri, 05 Mar 2021 04:57:32 +0000

Theorem 1.1 (Euclid’s Division Lemma) : Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.

Euclid’s Division Algorithm : This algorithm is used to find the h.c.f (Highest Common Factor) or g.c.d (Greatest Common Divisor)of two positive integers. It is based on Euclid’s Division Lemma that has been stated and explained above.

Step 1 : Apply Euclid’s division lemma, to a and b to find integers (more precisely whole numbers) q and r such that a = bq + r, 0 ≤ r < b.
Step 2 : If r=0, b is the g.c.d of a and b. If r≠0, apply the division lemma to b and r.
Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be the required GCD (HCF).

Theorem 1.3: Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.

Theorem 1.4: √2 is irrational.

Theorem – Good to know: If a non -ve integer is not a perfect square, then its square root is irrational.

Theorem 1.5: Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form p/q, where p and q are coprime, and the prime factorisation of q is of the form 2ⁿ5^m, where n, m are non-negative integers.

Theorem 1.6: Let x = p/q be a rational number, such that the prime factorisation q of q is of the form 2ⁿ5^m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.

Theorem 1.7: Let x = p/q , where p and q are coprimes, be a rational number, such that the prime factorisation of q is not of the form 2ⁿ5^m, where n, m are non-negative integers. Then, x has a decimal expansion which is non-terminating repeating (recurring).

Testimonials

An Excellent Math Teacher

– Saiba’s Parents (USA)

Amazing Math Coach

– Rishi’s Parents (USA)

Unique, Patient and Good Mentor

Also, your videos on Mathemafia youtube channel are very informative.

– Lakshay (USA)

WhatsApp for 1-1 Classes

❇️ ❇️ M️aster List – All Maths Formulas, Class 10 ❇️ ❇️

🇮🇳 हर भारतीय के दिल की आवाज़… आइए, इस स्वतंत्रता दिवस मिलकर गाएँ – “भारत माता की जय“।

Class 10 – Mathematics – NCERT Solutions

Real Numbers
- Exercise 1.1
- Exercise 1.2
Polynomials
- Exercise 2.1
- Exercise 2.2
Pair of Linear Equation in Two Variables

Quadratic Equations
Arithmetic Progressions
- Exercise 5.1
- Exercise 5.2
Triangles
- Exercise 6.1
- Exercise 6.2
- Exercise 6.3

Coordinate Geometry
- Exercise 7.1
- Exercise 7.2
Introduction to Trigonometry
Some Applications of Trigonometry
- Exercise 9.1
Circles
- Exercise 10.1
- Exercise 10.2

Areas Related to Circles
- Exercise 11.1
Surface Areas and Volumes
- Exercise 12.1
- Exercise 12.2
Statistics
- Exercise 13.1
- Exercise 13.2
- Exercise 13.3
Probability
- Exercise 14.1

Super Cool Stuff

Calculators
- Quadratic Equation Calculator – Solve any Equation
Super Concepts

Theorems
Quizzes

The post Class 10 | Chapter 1 | Real Numbers | Theorems & Concepts appeared first on Mathemafia.

Class 9 | Chapter 1 | Number System | Example 18, 19, 20

Sunil Syal — Sun, 14 Feb 2021 07:22:46 +0000

Example 18: Rationalise the denominator of 1/ (2 + √3) Example 19: Rationalise the denominator of 5 / (√3 – √5) Example 20: Rationalise the denominator of 1/ (7 + 3√2)

Concept: Rational and irrational numbers constitute real numbers. Also, the set of rational and irrational numbers are mutually exclusive that means no number can be both rational and irrational. Symbols Q and Q′ (read it as Q-dash or Q-prime) are used to set of rational and irrational numbers respectively.

Rational Number

A number that can be written in the form of p/q where both p and q are integers and q ≠ 0.

A number that either has a finite decimal part like 2, 4.5 or has an infinite but recurring (repeating) decimal part like 4.6666…, 2.3456456…

Irrational Number

A number that cannot be written in the form of p/q where both p and q are integers and q ≠ 0.

A number that has an infinite and non-recurring (non-repeating) decimal part like

1.01001000100001…., √2, √3, 𝛑

Concept: In this tutorial, you will learn about the very important topic – Number system. It is one of the basic concepts of Mathematics that every math student should learn very well. This is foundation of rest of the mathematics and should be very very clear to you. The tutorial not only covers and explains what different types of numbers are there but also explains what is the need of these numbers in real life.

Chapter 1 – Number system

All Mathemafia YouTube videos of Class 9 chapter 1 (Number System) are embedded on this webpage. You can watch all these videos right here in the same order as they appear in the NCERT textbook. Number system is the base of Mathematics and it is very important for students to have very good knowledge of it. In the first introductory video, Number System has been explained in detail with very easy to understand examples. you are advised to must watch the introductory video before watching any of the chapter’s example or question. Here is the list of all solved video tutorials of this chapter at one place – Examples, concepts, exercises, theorems | Class 9 Chapter 1, Number System.

This playlist covers all questions, concepts and examples of chapter 1 (Number System) of class 9, NCERT textbook. You may watch all these tutorials on YouTube by clicking the link displayed below or click a thumbnail to watch the tutorials on this webpage.

Watch complete chapter on YouTube

Example 1 and 2 Chapter 1 Class 9 | Number System | NCERT Maths

Question 1 Exercise 1.1 Class 9 | Chapter 10 | Number Systems | solution | NCERT Maths

Question 2 Exercise 1.1 Class 9 | Chapter 10 | Number Systems | solution | NCERT Maths

Question 3 Exercise 1.1 Class 9 | Chapter 10 | Number Systems | solution | NCERT Maths

Question 4 (Q4) Exercise 1.1 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Example 3 and 4 Chapter 1 Class 9 | Number System | NCERT Maths

Number System in math – Best explained with real life examples in hindi

Question 1 (Q1) Exercise 1.2 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Question 2 (Q2) Exercise 1.2 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Question 3, Activity 4 (Q3, Q4) Exercise 1.2 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Difference between Rational and Irrational numbers| Number System | Concepts explained with examples

Example 5 and 6 Chapter 1 Class 9 | Number System | NCERT Maths

Example 7, 8 and 9 Chapter 1 Class 9 | Number System | NCERT Maths

Example 10 Chapter 1 class 9 | Find an irrational no between 1/7 and 2/7| NCERT Maths | Solution

Question 1 (Q1) Exercise 1.3 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Question 2 (Q2) Exercise 1.3 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Question 3 (Q3) Exercise 1.3 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Question 4 (Q4) Exercise 1.3 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Question 5 (Q5) Exercise 1.3 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Question 6 (Q6) Exercise 1.3 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Question 7, 8 (Q7, Q8) Exercise 1.3 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Question 9 (Q9) Exercise 1.3 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Example 15 Chapter 1 Class 9 | Number System | Simplify the expressions | NCERT Maths

Example 16 Chapter 1 Class 9 | Number Systems | Rationalise the denominator | NCERT Maths

Example 17, 18, and 19, Chapter 1 Class 9 | Number System | NCERT Maths

Question 1 (Q1) Exercise 1.4 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Question 2 (Q2) Exercise 1.4 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Question 3 (Q3) Exercise 1.4 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Question 5 (Q5) Exercise 1.4 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Example 20, Chapter 1, Class 9 | Number Systems | Simplify using Laws of Exponents | NCERT Maths

Question 1 (Q1) Exercise 1.5 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

Question 2, 3 (Q2, Q3) Exercise 1.5 Chapter 1 Class 9 Maths | Number Systems | NCERT, CBSE

English Version – Number System

The introductory video to understand the Number System has been created in English as well. You may watch this version in case you struggle to understand Hinglish.

Class 9 Chapter 2 | Polynomials

Class 9 | Chapter 1 | Number System | Example 17

Sunil Syal — Sun, 14 Feb 2021 07:21:59 +0000

Example 17: Rationalise the denominator of 1/√2⋅