Is the square root of 3 rational?
Understand the Problem
The question is asking whether the square root of 3 is a rational number, which means determining if it can be expressed as a fraction of two integers.
Answer
$\sqrt{3}$ is not a rational number.
Answer for screen readers
$\sqrt{3}$ is not a rational number.
Steps to Solve
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Definition of Rational Numbers A rational number can be expressed as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers, and $b \neq 0$.
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Assume Square Root of 3 is Rational Assume $\sqrt{3}$ is a rational number. If it is, we can write it as $\sqrt{3} = \frac{a}{b}$, where $a$ and $b$ are integers with no common factors (i.e., the fraction is in simplest form).
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Square Both Sides To eliminate the square root, we square both sides of the equation: $$ 3 = \frac{a^2}{b^2} $$ This implies $$ a^2 = 3b^2 $$
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Analyzing the Equation From the equation $a^2 = 3b^2$, we can conclude that $a^2$ is divisible by 3. This means that $a$ must also be divisible by 3 (because the square of a number can only be divisible by 3 if the number itself is divisible by 3).
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Let a be Divisible by 3 Let $a = 3k$ for some integer $k$. Substitute this back into the equation: $$ (3k)^2 = 3b^2 \implies 9k^2 = 3b^2 \implies b^2 = 3k^2 $$ This tells us that $b^2$ is also divisible by 3, meaning $b$ must also be divisible by 3.
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Conclusion on Common Factors Since both $a$ and $b$ are divisible by 3, this contradicts the assumption that $\frac{a}{b}$ is in simplest form. Therefore, our assumption that $\sqrt{3}$ is rational must be false.
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Final Statement Thus, we conclude that $\sqrt{3}$ is not a rational number.
$\sqrt{3}$ is not a rational number.
More Information
The number $\sqrt{3}$ is classified as an irrational number, which means it cannot be expressed as a simple fraction. Irrational numbers have non-repeating, non-terminating decimal expansions. The proof method used here is a common technique called proof by contradiction.
Tips
- Confusing rational numbers with irrational numbers. Remember that rational numbers can be expressed as fractions, while irrational numbers cannot.
- Not checking if both integers share common factors. A fraction must be in simplest form to verify its rationality.
