answersLogoWhite

0


Best Answer

By way of contradiction, suppose sqrt(2) is rational. Then there exist integers n and d such that sqrt(2) = n/d. We can also assume (without loss of generality) that n and d have no common factors, like writing the fraction in lowest terms. Multiplying both sides by d, and then squaring both sides, gives 2d2 = n2. Every integer can be written as a power of 2 times an odd number, so write n = 2ia and d = 2jb, where a and b are odd. Plugging into the previous equation gives 22j+1b2 = 22ia2. Since a2 and b2 must be odd, they must be equal, and hence 2j+1 = 2i. This is impossible; an odd integer cannot equal an even integer. Therefore the original assumption must be false, namely sqrt(2) is an irrational number.

User Avatar

Wiki User

βˆ™ 15y ago
This answer is:
User Avatar

Add your answer:

Earn +20 pts
Q: Why is the square root of 2 an irrational number?
Write your answer...
Submit
Still have questions?
magnify glass
imp
Related questions

Is the square root a 2 an irrational number?

Yes, the square root of 2 is an irrational number.


Is a square root of 2 rational or irrational?

The square root of 2 is an irrational number


Prove that square root of 2 is an irrational number?

The square root of 2 is 1.141..... is an irrational number


Can multiplying two irrational numbers give you an irrational number?

Yes. For example, the square root of 3 (an irrational number) times the square root of 2(an irrational number) gets you the square root of 6(an irrational number)


Is the square root of 2 plus the square root of 8 a rational or irrational number?

irrational


Is the square root of 2 an irrational number or a rational number?

irrational


Is the square root of 2 is rational or irrational?

The square root of (any number that isn't a perfect square) is irrational.


Is the square root of 2 irrational or rational?

The square root of (any number that isn't a perfect square) is irrational.


Is the square root of 2 a irrational number?

Yes, they are irrational.


How is root 2 irrational?

2 is a prime number and its square root is an irrational number that cannot be expressed as a fraction


Is the square root of a positive number an irrational number?

Sometimes the square root of a positive number can be irrational, as in the square root of 2 (which is a non-perfect square number), but sometimes it is a rational number, as in the square root of 25 (which is a perfect square number).


Is root 2 a rational number?

The square root of 2 is irrational. In general, the square root of a positive integer is either an integer (if you take the square root of a perfect square), or it is irrational.