Prove that 3 + 2 root 5 is irrational

Question

Prove that $3 + 2\sqrt{5}$ is irrational.

Solution — Step by Step

We assume as given (or will use) the fact that $\sqrt{5}$ is irrational. (This can be proved using the same method as for $\sqrt{2}$ , and is typically assumed as known.)

We proceed by contradiction.

Assume $3 + 2\sqrt{5}$ is rational.

Then $3 + 2\sqrt{5} = \frac{p}{q}$ for some integers $p$ and $q$ where $q \neq 0$ and $\gcd(p, q) = 1$ (i.e., $\frac{p}{q}$ is in its lowest form).

Rearranging:

2\sqrt{5} = \frac{p}{q} - 3 = \frac{p - 3q}{q}

\sqrt{5} = \frac{p - 3q}{2q}

Now, $p$ and $q$ are integers, so $p - 3q$ and $2q$ are also integers ( $2q \neq 0$ ).

Therefore, $\frac{p - 3q}{2q}$ is a rational number.

But this means $\sqrt{5} =$ rational number.

This contradicts the fact that $\sqrt{5}$ is irrational.

Our assumption was wrong. Therefore, $3 + 2\sqrt{5}$ is irrational. $\square$

Why This Works

The proof relies on the closure properties of rational numbers: if $a$ and $b$ are rational, then $a - b$ , $a \times b$ , and $a \div b$ (when $b \neq 0$ ) are all rational. So if $3 + 2\sqrt{5}$ were rational, we could isolate $\sqrt{5}$ using rational operations, making $\sqrt{5}$ rational too. Since $\sqrt{5}$ is known to be irrational, we have a contradiction.

This “isolate the irrational and contradict” technique works for all numbers of the form $a + b\sqrt{p}$ where $a, b$ are rational ( $b \neq 0$ ) and $\sqrt{p}$ is irrational.

Common Mistake

Many students write the proof as: “Assume $3 + 2\sqrt{5}$ is rational. We know $\sqrt{5}$ is irrational. Therefore $3 + 2\sqrt{5}$ is irrational.” This is circular and earns zero marks. You must algebraically show that assuming $3 + 2\sqrt{5}$ is rational forces $\sqrt{5}$ to be rational. The rearrangement step (isolating $\sqrt{5}$ ) is the heart of the proof and must be explicitly shown.

CBSE Class 10 marking scheme: The proof is typically 3 marks — 1 mark for correct assumption (rational = p/q), 1 mark for the algebraic step showing $\sqrt{5} = \frac{p-3q}{2q}$ , and 1 mark for the correct conclusion citing that $\sqrt{5}$ is irrational, hence contradiction. All three steps must be clearly written.

Question

Solution — Step by Step

Why This Works

Common Mistake

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