Is root 51 a rational number? This question often arises in mathematics, particularly when discussing the nature of square roots and rational numbers. To answer this question, we must first understand the definitions of rational and irrational numbers and then apply them to the square root of 51.
Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. They can be written in the form of p/q, where p and q are integers and q is not equal to zero. On the other hand, irrational numbers cannot be expressed as a fraction of two integers and have non-terminating, non-repeating decimal expansions.
To determine whether the square root of 51 is rational or irrational, we can use the following approach. If the square root of a number is rational, then it can be expressed as a fraction of two integers. In other words, if √51 = p/q, where p and q are integers and q is not equal to zero, then the square of this fraction should equal 51.
Let’s assume that √51 = p/q. Squaring both sides of the equation, we get:
(√51)^2 = (p/q)^2
51 = p^2/q^2
Now, we can multiply both sides of the equation by q^2 to eliminate the fraction:
51q^2 = p^2
Since 51 is a prime number, it cannot be expressed as a product of two smaller integers. Therefore, if 51q^2 = p^2, then p^2 must be divisible by 51. This implies that p is also divisible by 51, as 51 is a prime number.
Let’s denote p as 51k, where k is an integer. Substituting this into the equation, we get:
51q^2 = (51k)^2
51q^2 = 51^2k^2
Dividing both sides of the equation by 51, we get:
q^2 = 51k^2
Now, we can see that q^2 is divisible by 51, which means that q is also divisible by 51. However, this contradicts our initial assumption that √51 = p/q, as both p and q are now divisible by 51.
Since our assumption leads to a contradiction, we can conclude that the square root of 51 cannot be expressed as a fraction of two integers. Therefore, √51 is an irrational number. In summary, the answer to the question “Is root 51 a rational number?” is no, as the square root of 51 is an irrational number.
