What is the conjugate of 3 + 4i?

3 − 4i

The conjugate of a + bi is a − bi. So the conjugate of 3 + 4i is 3 − 4i.

Understanding Complex Numbers questions

Complex numbers extend the real line by defining i as the square root of −1. Most introductory questions are ordinary algebra with one extra substitution rule: i² = −1.

How to approach it

1. Treat i as an algebraic symbol and expand normally.
2. Replace every i² with −1 and simplify.
3. Write the result in the standard a + bi form.
4. For division, multiply numerator and denominator by the conjugate.

Common mistakes to avoid

• Leaving i² in the final answer.
• Mishandling powers of i beyond i⁴, which cycles every four steps.
• Forgetting the sign change when using the conjugate.

Where it appears: Studied in CBSE Class 11, IB HL and engineering entrance syllabi.