数学
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Complex Conjugate Calculator.
Find the complex conjugate of a+bi.
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01
输入数值
输入时结果会更新。
02 / 答案
Real part: 3
Imaginary part: -4
Results update automatically as you type.
Use Cases
Simplifying division of complex numbers
When dividing by a complex number, multiply numerator and denominator by the conjugate of the denominator to get a real denominator.
Example: For (2+3i)/(1-i), multiply by (1+i) to get a real denominator.
Finding the magnitude of a complex number
The magnitude (or modulus) of a+bi is sqrt(a^2+b^2), which can be found by multiplying the number by its conjugate and taking the square root.
Example: For 3+4i, conjugate is 3-4i, product is 25, magnitude is 5.
Frequently Asked Questions
- What is a complex conjugate?
- The complex conjugate of a complex number a+bi is a-bi. It has the same real part but the opposite sign on the imaginary part. For example, the conjugate of 3+4i is 3-4i.
- How do I use this calculator?
- Enter the real part in the 'Real part' field and the imaginary part in the 'Imaginary part' field. The calculator will output the conjugate in the form a-bi.
- Why is the complex conjugate useful?
- Complex conjugates are used to simplify expressions with complex numbers, especially when dividing or finding magnitudes. Multiplying a complex number by its conjugate yields a real number.
Tips & Common Mistakes
Tips
- Remember to change the sign of the imaginary part only; the real part stays the same.
- If the imaginary part is negative, the conjugate will have a positive imaginary part. For example, 5-2i becomes 5+2i.
- Use the conjugate to rationalize denominators in complex fractions.
- Check your input: the imaginary part should be the coefficient of i, not including the i itself.
Common Mistakes to Avoid
- Forgetting to change the sign of the imaginary part.
- Entering the imaginary part with the 'i' included (e.g., typing '4i' instead of '4').
- Confusing the real and imaginary parts when entering values.
Last updated: August 13, 2026