Number Systems and Operations
Number Systems and Operations
This domain tests how fluently you classify numbers, apply operation properties, and move among fractions, decimals, percents, exponents, and radicals. Expect items that combine order of operations with rational numbers rather than isolated arithmetic.
Core rules
- The real numbers contain rational numbers (integers, terminating decimals, repeating decimals, and fractions of integers) and irrational numbers (non-repeating, non-terminating decimals such as √2 and π). Every integer is rational; √16 = 4 is rational, but √2 is not.
- Use prime factorization to find GCF and LCM: take shared prime factors to the lowest power for GCF, and all prime factors to the highest power for LCM.
- For fractions, a/b ÷ c/d = a/b × d/c, and a/b + c/d requires a common denominator. A percent is a ratio out of 100.
- Integer exponent rules: a^m a^n = a^(m+n), a^m/a^n = a^(m−n), (a^m)^n = a^(mn), a^−n = 1/a^n (a ≠ 0), and a^0 = 1.
- Follow order of operations: grouping, exponents/radicals, multiplication/division left to right, addition/subtraction left to right.
Worked example
Evaluate (2/3 + 5/6) ÷ 7/12. First add inside parentheses: 2/3 = 4/6, so 4/6 + 5/6 = 9/6 = 3/2. Then divide: 3/2 ÷ 7/12 = 3/2 × 12/7 = 36/14 = 18/7. The answer is 18/7, or about 2.57.
Traps
Do not “cancel” across addition; common denominators are required. A negative exponent does not make the result negative-it indicates a reciprocal. Division by zero is undefined, not zero. In expressions such as 12 ÷ 3 × 2, work left to right to get 8, not 2. Also check whether a radical simplifies to a rational number before labeling it irrational.