Is the square root of 51 rational or irrational

Square Root of 51

Is It Rational or Irrational?

To determine if the square root of 51 is rational or irrational, we need to recall the definition of rational and irrational numbers.

A rational number is any number that can be expressed as the quotient or fraction of two integers, say $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. On the other hand, an irrational number cannot be expressed as a simple fraction; its decimal form is non-repeating and non-terminating.

Mathematical Representation

Let's denote the square root of 51 as $\sqrt{51}$.

1. Suppose, for contradiction, that $\sqrt{51}$ is rational. This means we could write it as:

   $$\sqrt{51} = \frac{a}{b}$$

   Where:

   $$a, b \in \mathbb{Z} \quad \text{and} \quad b \neq 0$$

2. Squaring both sides, we get:

   $$51 = \frac{a^2}{b^2} \quad \Rightarrow \quad 51b^2 = a^2$$

3. The equation $51b^2 = a^2$ implies that $a^2$ is a multiple of 51.

Prime Factorization

We can factorize 51 as:

From this factorization, any perfect square involving 51 must also have both 3 and 17 as squared factors. However, neither 3 nor 17 is a perfect square.

Conclusion

Since neither 3 nor 17 appears in pairs within $a^2$:

Therefore, the square root of 51 is irrational.