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.
Yes, the square root of 2 is an irrational number.
The square root of 2 is 1.141..... is 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)
irrational
Yes, they are irrational.
Yes, the square root of 2 is an irrational number.
The square root of 2 is an irrational number
The square root of 2 is 1.141..... is 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)
irrational
irrational
The square root of (any number that isn't a perfect square) is irrational.
The square root of (any number that isn't a perfect square) is irrational.
Yes, they are irrational.
2 is a prime number and its square root is an irrational number that cannot be expressed as a fraction
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).
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.