MP Board · Class 10 · Mathematics · Real NumbersProve that $\sqrt{5}$ is an irrational number. Also, state and prove the Fundamental Theorem of Arithmetic.

Proof that $\sqrt{5}$ is Irrational

\nTo prove that $\sqrt{5}$ is irrational, we use the method of contradiction.

• Assumption: Let us assume to the contrary that $\sqrt{5}$ is a rational number.
• Fraction form: Therefore, we can find two integers, say $r$ and $s$ ($s \neq 0$), such that $\sqrt{5} = \frac{r}{s}$.
• Co-prime integers: Let $r$ and $s$ have no common factors other than 1. If they do have a common factor, we can divide by that common factor to assume $\frac{r}{s}$ is in its simplest form, where $\text{HCF}(r, s) = 1$.
• Squaring both sides: $$\left(\sqrt{5}\right)^2 = \left(\frac{r}{s}\right)^2$$ $$5 = \frac{r^2}{s^2}$$ $$r^2 = 5s^2$$
• Divisibility: From this equation, it is clear that $5$ divides $r^2$. By Theorem (If a prime number $p$ divides $a^2$, then $p$ divides $a$), $5$ must also divide $r$.
• Substitution: Since $5$ divides $r$, we can write $r = 5c$ for some integer $c$.
• Substituting back: $$(5c)^2 = 5s^2$$ $$25c^2 = 5s^2$$ $$s^2 = 5c^2$$
• Second Divisibility: This shows that $5$ divides $s^2$, which means $5$ also divides $s$.
• Contradiction: Thus, both $r$ and $s$ have at least $5$ as a common factor. This contradicts our initial assumption that $r$ and $s$ are co-prime (having no common factors other than 1).
• Conclusion: This contradiction has arisen because of our incorrect assumption that $\sqrt{5}$ is rational. Hence, we conclude that $\sqrt{5}$ is an irrational number.

Fundamental Theorem of Arithmetic

• Statement: Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.
• Explanation: For example, consider the composite number $210$. It can be factorized as $2 \times 3 \times 5 \times 7$. No matter how we try to break down $210$ using tree diagrams or repeated division, the prime factors will always be $2, 3, 5,$ and $7$, only the order might change (e.g., $7 \times 5 \times 3 \times 2$). This theorem forms the bedrock for finding HCF and LCM of integers using prime factorization.