Real numbers, rational powers and complex arithmetic
Use number-system properties and exponent rules, then extend calculations with i² = −1.
What you will tackle
- Distinguish integer, rational, irrational, real and complex numbers
- Use arithmetic properties and integer/rational exponents
- Add, subtract, multiply and divide complex numbers
Properties determine legal transformations
Every integer is rational; a rational number is a ratio of integers with nonzero denominator. Numbers such as √2 are irrational. Commutative, associative and distributive properties explain rearrangements of sums and products; subtraction and division are not commutative. For positive a, a^(m/n) = (nth root of a)^m. Record restrictions when roots or denominators occur.
Complex arithmetic keeps real and imaginary parts
The imaginary unit satisfies i² = −1. Add real parts and imaginary parts separately. Multiply with distribution, then replace i². To divide by a + bi, multiply numerator and denominator by a − bi; the denominator becomes a² + b², provided the divisor is not zero. Powers of i repeat in a cycle of four.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
Classify −3, 2/7, √2 and 3 + 2i. Evaluate 16^(3/4) and (3 + 2i)(1 − i).
Show a starting hint
Reduce powers of i after expanding.
Compare with the explained answer
- −3 is integer, rational and real; 2/7 is rational and real; √2 is irrational and real; 3 + 2i is nonreal complex.
- The fourth root of 16 is 2, so 16^(3/4) = 8.
- The product is 3 − 3i + 2i − 2i² = 5 − i.
Answer / response criteria: Classifications as stated; 8; 5 − i.
Independent practice
Independent 1
Find (2 − 3i) + (4 + i), i^23, and (1 + i)/(1 − i).
Compare with the explained answer
- Add parts to get 6 − 2i.
- 23 leaves remainder 3 on division by 4, so i^23 = −i.
- Multiply the fraction by (1 + i)/(1 + i); numerator 2i and denominator 2 give i.
Answer / response criteria: 6 − 2i; −i; i.
Application practice
Application 1
A model represents two perpendicular signal components as z = 4 + 3i. A transformation multiplies by i. Find the new complex value and compare √(real² + imaginary²) before and after.
Compare with the explained answer
- i(4 + 3i) = −3 + 4i.
- The original magnitude is √(4² + 3²) = 5.
- The new magnitude is √(9 + 16) = 5; this transformation rotates the representation without changing its magnitude.
Answer / response criteria: −3 + 4i; both magnitudes are 5.
What can you explain now?
A model represents two perpendicular signal components as z = 4 + 3i. A transformation multiplies by i. Find the new complex value and compare √(real² + imaginary²) before and after.
Compare with the explanation
−3 + 4i; both magnitudes are 5.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide