Real numbers, rational powers and complex arithmetic
| English | 中文 | Pinyin |
|---|---|---|
| conjugate/ˈkɒndʒuːɡeɪt/ | 共轭复数 | gòng è fù shù |
| rational exponent/ˈræʃənl ekˈspəʊnənt/ | 有理数指数 | yǒu lǐ shù zhǐ shù |
A decision before an answer
- The sum of two irrational numbers can be rational: √2 + (−√2) = 0. A rule about individual terms need not survive their combination.
- Your goal: Distinguish rational and irrational numbers without assuming their sums retain type.
Read the relationship
- A rational number is a ratio of integers with nonzero denominator; a terminating or repeating decimal is rational. An irrational real number is not such a ratio. Closure of rational numbers under addition and multiplication does not imply closure of irrational numbers. Exact roots may simplify: √18=3√2 remains irrational, whereas √16=4 is rational.
- Apply exponent and radical rules on the permitted domain.
What is (3+i)(3-i)?
Conjugate product is 3²+1²=10.
Use the defining rule
- For a positive base, a^(m/n) combines an nth root and an integer power. A negative exponent forms a reciprocal and requires a nonzero base. Distinguish √(x²)=|x| from x. When an even root is involved, respect the real domain and the principal nonnegative root. Distribute powers across products, not across sums: (a+b)² includes the cross term.
- Calculate with complex numbers using i²=-1 and conjugates.
81^(1/2) equals:
The principal square root is 9.
Check the conditions
- Complex arithmetic uses i²=-1. Add real parts and imaginary parts separately, and multiply by distribution before reducing i². To divide by a+bi, multiply numerator and denominator by a-bi; the resulting denominator is a²+b² when the original is nonzero. Complex roots permit solutions to equations such as x²=-9 that have no real roots.
- Calculate with complex numbers using i²=-1 and conjugates.
16^(3/4)=2³=8. (2+3i)(1-2i)=2-4i+3i-6i²=8-i. For (1+i)/(1-i), multiplying by 1+i gives (1+i)²/2=2i/2=i. The solutions of x²+9=0 are ±3i; neither is real.
The real part of (4+2i)+(1-5i) is ____.
Real parts add to 4+1; imaginary part is -3i.
Apply the task format
- Enhanced ACT still includes selected advanced topics, and no formula sheet is supplied. Build familiarity with the defining rules rather than guessing from a calculator display. Check whether the question asks for real or complex solutions, an exact radical or a decimal approximation. The local checks here are formative, not an official scored form.
- Calculate with complex numbers using i²=-1 and conjugates.
Do not infer that two irrational terms must have an irrational sum, distribute a square over addition, or replace √(x²) by x when x can be negative.
Which answer fits this case?
Distinguish rational and irrational numbers without assuming their sums retain type
The sum of two irrational numbers is always irrational.
A number plus its negative gives rational zero.
Keep the distinctions
- conjugate 共轭复数 — The number a-bi paired with a+bi.
- rational exponent 有理数指数 — An exponent m/n interpreted using powers and roots on the stated domain.
- Distinguish rational and irrational numbers without assuming their sums retain type.
- Apply exponent and radical rules on the permitted domain.
- Calculate with complex numbers using i²=-1 and conjugates.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.