Calculator Input
Formula used
For a binomial surd: the conjugate of a + b√n is a − b√n. Their product is (a + b√n)(a − b√n) = a2 − b2n.
For a complex number: the conjugate of a + bi is a − bi. Their product is (a + bi)(a − bi) = a2 + b2.
How to use this calculator
- Choose whether you are working with a surd or a complex number.
- Enter the first part a and second coefficient b.
- For surd mode, enter a positive integer radicand n.
- Select the number of decimal places for approximations.
- Tick worked steps if you want a guided explanation.
- Press Generate Conjugate to show the result above the form.
- Use the CSV or PDF buttons to save the computed output.
Example data table
| Type | Input | Conjugate | Product | Use case |
|---|---|---|---|---|
| Surd | 3 + 2√5 | 3 − 2√5 | −11 | Rationalizing denominators with radicals |
| Surd | 7 + √3 | 7 − √3 | 46 | Difference of squares practice |
| Complex | 4 + 3i | 4 − 3i | 25 | Finding modulus squared |
| Complex | −2 + 5i | −2 − 5i | 29 | Complex division support |
FAQs
1. What is a conjugate in mathematics?
A conjugate is a paired form created by changing one sign. For surds, the radical sign stays. For complex numbers, the imaginary sign changes.
2. Why do students use conjugates?
Conjugates help remove radicals from denominators, simplify products, and support complex division. They also reveal useful identities based on differences of squares.
3. Does the calculator work for decimals?
Yes. The first term and coefficient accept decimals. In surd mode, the radicand remains a positive integer because radical simplification depends on integer factorization.
4. What product appears after multiplying conjugates?
For surds, the middle radical terms cancel and leave a2 − b2n. For complex numbers, the imaginary parts cancel and leave a2 + b2.
5. Can this help with rationalizing denominators?
Yes. The result includes a reciprocal form using the conjugate. This shows the factor commonly multiplied to remove radicals or imaginary parts from denominators.
6. What does the simplified radical component show?
It shows whether the radicand contains a perfect-square factor. For example, √12 simplifies to 2√3, which is useful during manual checking.
7. Is the output suitable for homework checking?
Yes. It displays the conjugate, the product formula, the value, approximations, and optional worked steps. That makes self-checking easier and faster.
8. Can I save the result for reports or revision?
Yes. Use the CSV button for spreadsheet-friendly output or the PDF button for a clean printable summary of the generated conjugate result.