Complex numbers and roots · 复数与根
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| complex conjugate/ˈkɒmpleks ˈkɒndʒuːɡeɪt/ | 共轭复数 | gòng è fù shù |
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. The conjugate is a-bi and z times its conjugate=a²+b². The modulus is √(a²+b²); the argument needs the correct quadrant. Conjugate roots occur for polynomials with real coefficients.
- 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? · 哪个描述正确定义了复数共轭？
The number obtained by reversing the sign of the imaginary part. · 通过改变虚部符号而得到的数。
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 numbers and roots · 复数与根
Write z=a+bi with i²=-1 · 将 z=a+bi 写成 i²=-1 的形式
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
Find |3+4i|. · 求 |3+4i|。
Modulus=√(3²+4²)=5. · 模=√(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). · 求 (3+4i)(3-4i) 的实部。
Conjugate multiplication gives 3²+4²=25, with imaginary parts cancelling. · 共轭相乘得 3²+4²=25，虚部相互抵消。
For a+bi, its modulus is always a+b. · 对于 a+bi，其模长恒等于 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. · 和的平方根不等于平方根的和。仅用反正切可能得出错误的象限。切勿混淆模长与模长的平方。
Interpret a new situation
- Draw the real axis horizontally and imaginary axis vertically. For products, moduli multiply and arguments add; powers extend this geometric pattern through de Moivre's theorem.
- 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). · 求 1/(3+4i) 的实部。
Multiply numerator and denominator by 3-4i: 1/(3+4i)=(3-4i)/25. Its real part is 3/25=0.12. · 分子分母同乘 3-4i：1/(3+4i)=(3-4i)/25。其实部为 3/25=0.12。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- edexcel IAL pure mathematics; official unit FP2. 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.