Is the square root of 7 rational or irrational?
Understand the Problem
The question is asking whether the square root of 7 is a rational or irrational number. To determine this, we need to understand the definitions of rational and irrational numbers and see if the square root of 7 can be expressed as a fraction of two integers.
Answer
$\sqrt{7}$ is irrational.
Answer for screen readers
The square root of 7 is an irrational number.
Steps to Solve
- 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 $\frac{a}{b}$.
- 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.
- 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 $a^2 = 7b^2$.
- 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.
- 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.
The square root of 7 is an irrational number.
More Information
The number $\sqrt{7}$ is often used in various mathematical contexts, such as geometry and algebra. Since it cannot be precisely expressed as a fraction, it has a decimal form that continues indefinitely without repeating.
Tips
- A common mistake is assuming that because $\sqrt{7}$ is approximately 2.6457513110645906, it must be rational; however, the decimal does not determine rationality.
- Another mistake is neglecting the need to prove that both $a$ and $b$ cannot be integers under the condition of them being in lowest terms.
