Prove that 6+√2 is an irrational number
By BYJU'S Exam Prep
Updated on: September 25th, 2023
6+√2 is an irrational number. Rational Numbers can be expressed as fractions or decimals, while irrational numbers have non-repeating, non-terminating decimal representations and cannot be expressed as simple fractions. To prove that 6+√2 is an irrational number, we need to signify that it cannot be expressed as a simple fraction or a ratio of two integers.
Table of content
Difference Between Irrational Number and Rational Number
Rational numbers are numbers that can be expressed as the ratio of two integers (p/q), where q is not equal to zero. These numbers can be written in the form of fractions and can be finite or repeating decimals. For example- 1/2, 3/4, and 0.25 are all rational numbers.
On the other hand, irrational numbers cannot be expressed as a simple fraction or a ratio of two integers. These numbers have non-repeating, non-terminating decimal representations. They are typically represented by square roots (√), cube roots (∛), or other mathematical notations. Examples of irrational numbers include √2, π (pi), and e (Euler’s number).
Solution
Let’s assume that 6 + √2 is a rational number and can be expressed as a fraction a/b,
6 + √2 = a/b
We can rearrange the equation:
√2 = (a/b) – 6
Squaring both sides of the equation to eliminate the square root:
2 = ((a/b) – 6)2
2 = (a2 / b2) – 12(a/b) + 36
Multiplying by b2 to eliminate the denominator:
2b2 = a2 – 12ab + 36b2
a2 = 2b2 + 12ab – 36b2
2c2 = (2d)2 + 12(2d)(2c) – 18(2d)2
2c2 = 4d2 + 96dc – 72d2
c2 = 2d2 + 48dc – 36d2
Therefore 6 + √2 cannot be expressed as a rational number.
Hence it is proved that 6 + √2 is an irrational number.
Summary
Prove that 6+√2 is an irrational number.
6 + √2 is an irrational number. To prove that 6+√2 is an irrational number, it cannot be represented as the quotient of two integers or in the form of a simple fraction.
Related Questions
- A person sitting on the ground floor of a building notices through the window, of height 1.5 m, a ball dropped from the roof of the building and crosses the window in 0.1 s. What is the velocity of the ball when it is at the topmost point of the window?
- Convert the following temperatures to the Celsius scale. (a) 293 K (b) 470 K
- Write one equation each for decomposition reactions where energy is supplied in the form of heat, light, or electricity.
- Let S={1,2,3,4,5,6}. Then the probability that a randomly chosen onto function G from S to S satisfies g(3)=2g(1) is?
- Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 23 seconds. The ratio of their speed is.
- A dealer gets Rs. 56 less if instead of selling a chair at again of 15%, it is sold at a gain of 8%. Find the cost price of the chair.
- What are the raw materials for photosynthesis?
