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
No, the square root of an irrational number is not always rational. In fact, the square root of an irrational number is typically also irrational. For example, the square root of 2, which is an irrational number, is itself irrational. However, there are exceptions, such as the square root of a perfect square of an irrational number, which can be rational.
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, 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.