Real and Complex Numbers
Key Concepts
- Properties of integers, rational, and irrational numbers
- Order of operations (PEMDAS)
- Absolute value, square roots, and radicals
- Operations with complex numbers (e.g., i^2 = -1)
Skills Tested
- Simplifying expressions with real or complex numbers
- Performing operations like addition, subtraction, multiplication, and division on complex numbers
- Understanding number classifications (e.g., identifying if a result is rational or irrational)
Example Question
Simplify: (3 + 4i)(2 - i) A. 6 - 3i + 8i - 4i^2 B. 6 + 5i + 4 C. Answer: 10 + 5i (Because -4i^2 = +4)
Ratios, Rates, and Proportions
Key Concepts
- Setting up and solving proportions
- Unit rates (e.g., miles per hour, cost per item)
- Scaling (e.g., similar figures, map scale)
Skills Tested
- Translating real-world scenarios into mathematical ratios
- Solving proportions using cross-multiplication
- Interpreting unit conversions
Example Question
A recipe that serves 4 people calls for 3 cups of flour. How many cups are needed to serve 10 people? A. 6 B. 7.5 C. Answer: 7.5 (Set up ratio: \frac{3}{4} = \frac{x}{10} \Rightarrow x = 7.5)
Scientific Notation and Significant Figures
Key Concepts
- Converting between standard and scientific notation
- Multiplying/dividing numbers in scientific notation
- Estimating using powers of ten
Skills Tested
- Performing operations with scientific notation
- Recognizing equivalent expressions
- Estimating large/small quantities
Example Question
(3 \times 10^4) \times (2 \times 10^3) = ? A. 6 \times 10^7 B. Answer: 6 \times 10^7 (Multiply coefficients: 3 \times 2 = 6; add exponents: 4 + 3 = 7)
Quantities and Units
Key Concepts
- Unit conversions (e.g., inches to feet, minutes to hours)
- Dimensional analysis (canceling units)
- Understanding and using compound units (e.g., mi/hr, kg/m²)
Skills Tested
- Converting between different measurement systems
- Setting up and interpreting unit-based problems
- Making sense of units in formulas
Example Question
If a car travels 180 miles in 3 hours, what is its speed in feet per second? (1 mile = 5280 feet) A. 88 ft/s B. Answer: 88 ft/s (180 miles ÷ 3 hours = 60 mi/hr → convert: 60 \times \frac{5280}{3600} = 88)
Rational and Irrational Numbers
Key Concepts
- Rational numbers: can be written as a fraction
- Irrational numbers: non-repeating, non-terminating decimals (e.g., \pi, \sqrt{2})
- Properties of number types under operations
Skills Tested
- Identifying number types
- Predicting results of operations (e.g., rational × irrational = irrational?)
- Working with square roots
Example Question
Which of the following is irrational? A. \frac{7}{3} B. \sqrt{49} C. \sqrt{2} Answer: C. \sqrt{2} (It is non-repeating, non-terminating)