Complex arithmetic and quadratic roots
| English | Português |
|---|---|
| complex conjugate/ˈkɒmpleks ˈkɒndʒuːɡeɪt/ | número complexo conjugado |
What if the root is not real?
- The equation x²+1=0 has no real roots. Extending the number system lets us represent and calculate with both roots.
- This lesson studies complex conjugate 共轭复数: The number obtained by reversing the sign of the imaginary part.
Choose the mathematical structure
- Write z=a+bi with i²=-1. Add components, multiply brackets and use a conjugate to make a division denominator real. Non-real roots of a real-coefficient quadratic occur in conjugate pairs.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines complex conjugate?
The number obtained by reversing the sign of the imaginary part.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For z=3+4i, |z|=5 and z conjugate=3-4i. The reciprocal is (3-4i)/25. For x²-6x+13=0, the roots are 3±2i. Multiplying by the conjugate makes a complex denominator real.
Complex arithmetic and quadratic roots
Write z=a+bi with i²=-1
Compare the model with the worked case and explain one change.
Find |3+4i|.
Modulus=√(3²+4²)=5.
Test a tempting shortcut
- The square root of a sum is not the sum of the square roots. An inverse tangent alone may return the wrong quadrant. Do not confuse modulus with squared modulus.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
For a+bi, its modulus is always a+b. This claim is false. Explain which definition or assumption it violates.
Find the real part of (3+4i)(3-4i).
Conjugate multiplication gives 3²+4²=25, with imaginary parts cancelling.
For a+bi, its modulus is always a+b.
The square root of a sum is not the sum of the square roots. An inverse tangent alone may return the wrong quadrant. Do not confuse modulus with squared modulus.
Interpret a new situation
- FP1 uses complex arithmetic, modulus, Argand representation and quadratic roots. De Moivre powers and roots of unity are reserved for FP2.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the real part of 1/(3+4i).
Multiply numerator and denominator by 3-4i: 1/(3+4i)=(3-4i)/25. Its real part is 3/25=0.12.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- edexcel IAL further mathematics; official unit FP1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The number obtained by reversing the sign of the imaginary part. Choose the relationship, show the method, check its assumptions and interpret the result.