Prove that the square root of 5 is irrational.
Understand the Problem
The question is asking for a proof that the square root of 5 cannot be expressed as a fraction of two integers, which is the definition of an irrational number. To solve it, we will assume the opposite (that it is rational) and derive a contradiction.
Answer
$\sqrt{5}$ is irrational.
Answer for screen readers
The square root of 5 cannot be expressed as a fraction of two integers, proving that $\sqrt{5}$ is irrational.
Steps to Solve
- Assume $\sqrt{5}$ is rational
Assume that $\sqrt{5}$ can be expressed as a fraction in simplest form, that is, $\sqrt{5} = \frac{a}{b}$, where $a$ and $b$ are integers with no common factors (i.e., the fraction is in lowest terms).
- Square both sides
To eliminate the square root, square both sides of the equation:
$$ 5 = \frac{a^2}{b^2} $$
This implies:
$$ a^2 = 5b^2 $$
- Analyze the implications of $a^2 = 5b^2$
Since $a^2 = 5b^2$, it follows that $a^2$ is divisible by 5. This means that $a$ must also be divisible by 5 (as the square of a non-multiple of 5 cannot be a multiple of 5).
- Let $a = 5k$ for some integer $k$
Substituting $a$ into the previous equation gives:
$$ (5k)^2 = 5b^2 $$
This simplifies to:
$$ 25k^2 = 5b^2 $$
which can be simplified further to:
$$ 5k^2 = b^2 $$
- Conclusion about $b$ being divisible by 5
Now, we see that $b^2$ is also divisible by 5, which means $b$ must also be divisible by 5.
- Contradiction arises
Since both $a$ and $b$ are divisible by 5, this contradicts our original assumption that $a/b$ is in simplest form (no common factors). Thus, our assumption that $\sqrt{5}$ is rational must be false.
The square root of 5 cannot be expressed as a fraction of two integers, proving that $\sqrt{5}$ is irrational.
More Information
The proof demonstrates that assuming the square root of 5 is rational leads to a contradiction, thus confirming its status as an irrational number. It follows the method of proof by contradiction, a common technique in mathematics.
Tips
- Many may incorrectly assume that the conclusion can be drawn without showing all steps; ensure each logical step is clearly stated.
- Some might confuse the definitions of rational and irrational numbers. Remember that a rational number can be expressed as a fraction of two integers, whereas an irrational number cannot.
