The Number System
What this domain tests
The Number System covers operations with positive and negative rational numbers-integers, fractions, and decimals-and converting fractions to decimals. Questions test sign rules, absolute value, distance on the number line, and the difference between terminating and repeating decimals.
Core rules
- Adding numbers with the same sign: add absolute values and keep the sign. Different signs: subtract the smaller absolute value from the larger and use the sign of the number with the larger absolute value.
- Subtract by adding the opposite: a − b = a + (−b).
- Multiply or divide: same signs give a positive result; different signs give a negative result.
- A fraction a/b is terminating exactly when b, in lowest terms, has only factors 2 and 5; otherwise it repeats.
- Distance between two points is |a − b|, which is never negative.
Worked example
Compute −3/4 + 5/6. The least common denominator is 12: −3/4 = −9/12 and 5/6 = 10/12. Add: −9/12 + 10/12 = 1/12. Estimate first: −0.75 + 0.833 is small and positive, which matches.
Compute −2/3 ÷ 4/9. Multiply by the reciprocal: −2/3 × 9/4 = −18/12 = −3/2. One negative factor means a negative quotient.
Convert 5/8 by dividing 5 by 8 to get 0.625; it terminates because 8 = 2³. Convert 2/11 to 0.181818…, which repeats because 11 has a factor other than 2 or 5.
Traps that cost marks
- Subtracting a negative and treating it as subtraction: 7 − (−2) = 9, not 5.
- Dropping the negative sign when dividing or when evaluating a power.
- Calling 0.333… exactly equal to 1/3; the repeating decimal is exact, but a rounded decimal is not.
- Reporting distance as a negative number.
- Canceling across a plus sign instead of only across factors.