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.
如果根不是实数怎么办？
- 方程 x²+1=0 没有实数根。扩展数系使我们能够表示并计算两个根。
- 本课学习 共轭复数 (complex conjugate)：通过取虚部符号相反得到的数。
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.
选择数学结构
- 设 z=a+bi，其中 i²=-1。其共轭为 a-bi，且 z 乘以其共轭等于 a²+b²。模长为 √(a²+b²)；辐角需确定正确的象限。具有实系数的多项式会出现共轭根。
- 计算前请先明确允许的输入项和单位。方程应表达关系本身，而不仅仅是记录计算器按键过程。
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.
通过验证案例进行推导
- 将结果与初始量进行比对。代入原始关系式，或在适当情况下对比图表与数值答案。
对于 z=3+4i，|z|=5，其共轭为 3-4i。倒数为 (3-4i)/25。对于方程 x²-6x+13=0，其根为 3±2i。乘以共轭可将复数分母化为实数。
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.
检验一个诱人的捷径
- 和的平方根不等于平方根的和。反三角函数单独使用时可能返回错误的象限。不要混淆模长与模长的平方。
- 当捷径失效时，找出其违背的假设。保留精确值直到题目要求的最终舍入步骤。
对于 a+bi，其模长总是 a+b。此说法是错误的。请解释它违背了哪个定义或假设。
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.
解读新情境
- 绘制水平实轴和垂直虚轴。对于乘积，模长相乘，辐角相加；棣莫弗定理将此几何模式推广至幂运算。
- 完整解答需给出数学结果并解释其含义。检查其在所述背景下是否可行。
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 further 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.
将其应用于你的课程
- Edexcel IAL高等数学；官方单元 FP2。其他单元的补充内容已在范围审查中标识；这并非额外的加分要求。
- 在给出最终答案前先展示解题方法，并遵循试卷的计算器及公式使用规则。通过定位第一个无效步骤来复盘错误答案。
通过取虚部符号相反得到的数。选择关系式，展示方法，检查其假设条件并解释结果。