Class 10 Exam > Class 10 Questions > prove that root 2 plus root5 is an irrational...
Start Learning for Free
prove that root 2 plus root5 is an irrational number
?
Most Upvoted Answer
prove that root 2 plus root5 is an irrational number Related: Chapter...
Community Answer
prove that root 2 plus root5 is an irrational number Related: Chapter...
Proof that √2 + √5 is an irrational number
Assume that √2 + √5 is a rational number.
Step 1: Definition of Rational Number
A rational number is a number that can be expressed as the ratio of two integers.
Step 2: Expressing √2 + √5 as a Ratio
Let √2 + √5 = p/q, where p and q are positive integers with no common factors.
Step 3: Squaring Both Sides
Squaring both sides of the equation (√2 + √5 = p/q), we get:
2 + 2√10 + 5 = p²/q²
7 + 2√10 = p²/q²
Step 4: Isolating √10
Isolating √10, we get:
√10 = (p² - 7q²)/2q²
Step 5: Checking for Even and Odd
Since p² - 7q² is even, it follows that √10 is also even.
Step 6: Division by 2
Dividing both sides of the equation (√10 = (p² - 7q²)/2q²) by 2, we get:
√10/2 = (p² - 7q²)/4q²
Step 7: Checking for Even and Odd
Since p² - 7q² is even, it follows that √10/2 is also even.
Step 8: Contradiction
This contradicts the fact that √10 is irrational, which means that our initial assumption (√2 + √5 is a rational number) is false.
Conclusion:
Therefore, we have proved that √2 + √5 is an irrational number.
Assume that √2 + √5 is a rational number.
Step 1: Definition of Rational Number
A rational number is a number that can be expressed as the ratio of two integers.
Step 2: Expressing √2 + √5 as a Ratio
Let √2 + √5 = p/q, where p and q are positive integers with no common factors.
Step 3: Squaring Both Sides
Squaring both sides of the equation (√2 + √5 = p/q), we get:
2 + 2√10 + 5 = p²/q²
7 + 2√10 = p²/q²
Step 4: Isolating √10
Isolating √10, we get:
√10 = (p² - 7q²)/2q²
Step 5: Checking for Even and Odd
Since p² - 7q² is even, it follows that √10 is also even.
Step 6: Division by 2
Dividing both sides of the equation (√10 = (p² - 7q²)/2q²) by 2, we get:
√10/2 = (p² - 7q²)/4q²
Step 7: Checking for Even and Odd
Since p² - 7q² is even, it follows that √10/2 is also even.
Step 8: Contradiction
This contradicts the fact that √10 is irrational, which means that our initial assumption (√2 + √5 is a rational number) is false.
Conclusion:
Therefore, we have proved that √2 + √5 is an irrational number.
Attention Class 10 Students!
To make sure you are not studying endlessly, EduRev has designed Class 10 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 10.
|
Explore Courses for Class 10 exam
|
|
Similar Class 10 Doubts
Top Courses for Class 10View all
prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT?
Question Description
prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT?.
prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT?.
Solutions for prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT? in English & in Hindi are available as part of our courses for Class 10.
Download more important topics, notes, lectures and mock test series for Class 10 Exam by signing up for free.
Here you can find the meaning of prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT? defined & explained in the simplest way possible. Besides giving the explanation of
prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT?, a detailed solution for prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT? has been provided alongside types of prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT? theory, EduRev gives you an
ample number of questions to practice prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT? tests, examples and also practice Class 10 tests.
|
|
Explore Courses for Class 10 exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup