There are many proofs of this fact. I think one way to see it it let n be a number that is not a perfect square
Assume √n is rational.
In other words assume that
n = p2/q2
where p and q are integers and p/q is in lowest terms. So
n q2 = p2
So p2 is divisible by n. That means that p must be as well, so p2 is
divisible by n2.
So
q2 = p2/n
and q^2 is divisible by n.
But that means that p and q are both divisible by n, so they weren't
in lowest terms, which is a contradiction.
Here is another way to look at it.
If √n is rational, then it can be expressed by some number a/b (in lowest terms). This would mean: (a/b)² = n. Squaring, a² / b² = n Multiplying by b², a² = nb². If a and b are in lowest terms which we assumed, then their squares would each have an even number of prime factors. nb² has one more prime factor than b², meaning it would have an odd number of prime factors. Every composite has a unique prime factorization so I cannot have both an even and odd number of prime factors. This contradiction violates are assumption that a√n is rational. It is therefore irrational.
The square root of (any number that isn't a perfect square) is irrational.
Yes. The square root of any whole number that is not a perfect square is irrational.
No because its square root is an irrational number
Because its square root is an irrational number
The square root of any non-perfect square is an irrational number. Since sqrt(1000) = 10 sqrt(10), it is irrational.
Nice question! The square root of (any number that isn't a perfect square) is irrational. No prime number is a perfect square. So the square root of any prime number is irrational.
The square root of any number which is not a perfect square;The cube root of any number which is not a perfect cube;Pi, the circular constant.e, the natural logarithm base number.
The square root of (any number that isn't a perfect square) is irrational.
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 ANY positive integer is either a whole number, or an irrational number. Only the perfect squares have whole number square roots; all the rest are irrational. 97 is not a perfect square therefore √97 is 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.
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. The square root of any whole number that is not a perfect square is irrational.
Yes. The square root of any number that is not a perfect square (like 9 or 36) is irrational.
If the square root of a natural number is not an integer, then it is irrational. Another way to look at it: if a natural number N falls between two perfect squares, then the square root of N is irrational.So in this case, 12 is between 9 (3 squared) and 16 (4 squared). So the square root of 12 is between 3 and 4, and also the square root of 12 is irrational.