Rational and Irrational Numbers Concept Explained

 

❓ Question

Show that:

$$5-\sqrt{3}$$

is irrational.

---

🖼️ Solution Image

---

✍️ Short Explanation

We prove this using the contradiction method.

We assume that:

$$5-\sqrt{3}$$

is rational and then arrive at a contradiction 💯

---

🔹 Step 1 — Assume $5-\sqrt{3}$ is Rational

Let:

$$5-\sqrt{3}=\frac{a}{b}$$

where:

• $a$ and $b$ are integers
• $b \ne 0$
  

---

🔹 Step 2 — Rearrange the Equation

$$\sqrt{3}=5-\frac{a}{b}$$

Taking LCM:

$$\sqrt{3}=\frac{5b-a}{b}$$

---

🔹 Step 3 — Check Rationality

Since:

• $a$ and $b$ are integers
• $5b-a$ is also an integer

Therefore,

$$\frac{5b-a}{b}$$

is rational.

This implies:

$$\sqrt{3}$$

is rational.

But this is false because:

$$\sqrt{3}$$

is irrational.

---

🔹 Step 4 — Contradiction

Our assumption is wrong.

Hence,

$$5-\sqrt{3}$$

is irrational.

---

✅ Final Answer

$$\boxed{5-\sqrt{3}\text{ is irrational}}$$

---

⭐ Key Insight

• Rational ± Irrational = Irrational
• Contradiction method is used to prove irrationality

🧠 Memory Line:

Irrational number ko rational number se add/subtract karne par result irrational hi rehta hai

---

📚 Related Topics

• Proof that $\sqrt{2}$ is irrational
• Proof that $\sqrt{3}$ is irrational
• Rational and irrational numbers