Number and Quantity
Number and Quantity
This number and quantity chapter develops whether you can move fluently among fractions, decimals, percents, integers, exponents, radicals, and scientific notation, and whether you can justify why a number is rational or irrational. Items rarely ask for a bare computation; they ask you to choose the representation or operation that fits a situation.
Core rules to hold
- Rational vs. irrational: a number is rational exactly when it can be written as a ratio of two integers. Terminating and repeating decimals are rational; the square root of 2, the square root of 3, pi, and 0.1010010001... are not. For a nonnegative integer n, its square root is irrational unless n is a perfect square.
- Exponent laws: a^m times a^n = a^(m+n); a^m divided by a^n = a^(m-n); (a^m)^n = a^(mn); a^(-n) = 1/a^n; a^0 = 1 for a not equal to 0. These laws do not distribute over addition: (a + b)^2 is not a^2 + b^2.
- Percent change: percent change = (new - original)/original. A 20% increase followed by a 20% decrease does not return to the start; multiply by 1.2 then 0.8 to get 0.96 of the original.
- Factors and multiples: prime factorization gives the GCF (shared primes, lowest powers) and the LCM (all primes, highest powers). GCF times LCM equals the product of the two numbers.
- Proportional reasoning: set up a ratio with matching units and solve by scaling or cross-multiplying; distinguish a unit rate from a part-to-part ratio.
Worked example
Compute (4.8 x 10^5) divided by (1.6 x 10^(-2)). Regroup the coefficients and the powers separately: 4.8 divided by 1.6 = 3.0, and 10^5 divided by 10^(-2) = 10^(5-(-2)) = 10^7. The result is 3.0 x 10^7, already in scientific notation because 3.0 lies between 1 and 10. Checking magnitude: dividing by a tiny number should enlarge the result, and 10^7 is larger than 10^5, so the answer is reasonable.
Traps that cost marks
- Subtracting exponents in the wrong order when dividing.
- Treating the square root of (a + b) as the square root of a plus the square root of b.
- Using the new value as the base for a percent-change question.
- Forgetting that a negative exponent produces a reciprocal, not a negative value.
- Mixing units inside a proportion (minutes with hours, centimeters with meters) before scaling.