Is the square root of 7 rational or irrational?

Steps to Solve

1. Definition of Rational Numbers

A rational number can be expressed as the quotient of two integers, $a$ and $b$, where $b \neq 0$. In simpler terms, it can be written as $\frac{a}{b}$.

2. Definition of Irrational Numbers

An irrational number cannot be expressed as a fraction of two integers. Instead, its decimal representation is non-terminating and non-repeating.

3. Establish if $\sqrt{7}$ is Rational or Irrational

To determine whether $\sqrt{7}$ is rational or irrational, we can check if there are integers $a$ and $b$ such that $\sqrt{7} = \frac{a}{b}$. Squaring both sides gives us:

$$ 7 = \frac{a^2}{b^2} $$

This means $a^2 = 7b^2$.

4. Analyzing Integer Conditions

Since we know that $a^2$ is equal to $7b^2$, it indicates that $a^2$ is divisible by 7. Therefore, $a$ must also be divisible by 7. Let’s say $a = 7k$ for some integer $k$. Now substituting back gives:

$$ (7k)^2 = 7b^2 $$ $$ 49k^2 = 7b^2 $$ $$ 7k^2 = b^2 $$

This means $b^2$ must also be divisible by 7, which implies that $b$ is also divisible by 7.

5. Conclusion About $\sqrt{7}$

If both $a$ and $b$ have 7 as a common factor, this contradicts the assumption that $\frac{a}{b}$ is in its lowest terms. Therefore, there cannot be integers $a$ and $b$ such that $\sqrt{7} = \frac{a}{b}$.

Thus, $\sqrt{7}$ is an irrational number.