Real Number Systems
The exam tests how well you classify numbers, place them in order on the number line, evaluate expressions with signed values, and use the field properties that justify each manipulation.
The nested sets
- Natural (counting): 1, 2, 3, …
- Whole: naturals plus 0.
- Integers: wholes plus negatives: …, −2, −1, 0, 1, 2, …
- Rational: anything writable as a/b with integers a and b ≠ 0. This includes terminating decimals (0.25 = 1/4) and repeating decimals (0.333… = 1/3), because both can be converted to fractions.
- Irrational: non-terminating, non-repeating decimals that cannot be written as a fraction - √2, √5, π, e.
- Real: every rational together with every irrational; each corresponds to exactly one point on the number line.
The rationals are dense: between any two rationals there is another, so no "next" rational exists. Properties you rely on: commutative and associative laws for + and ×, distributive law a(b + c) = ab + ac, additive identity 0, multiplicative identity 1, opposites −a, and reciprocals 1/a (a ≠ 0).
Absolute value and ordering
|x| is the distance from x to 0, so it is never negative: |−7| = 7. An equation |x| = 5 has two solutions, x = 5 and x = −5. On the number line, the number farther left is smaller, so −8 < −3 even though 8 > 3. To compare fractions, rewrite over a common denominator or cross-multiply; to compare a fraction with a decimal, convert one form: −3/4 = −0.75, and −0.75 < −0.70.
Worked example
Evaluate −2² + (−3)² − |−5|.
- −2² = −(2·2) = −4, because the exponent applies only to the 2, not to the minus sign.
- (−3)² = (−3)(−3) = +9, because the parentheses include the sign in the base.
- |−5| = 5.
- Total: −4 + 9 − 5 = 0.
Traps that cost marks
- −2² versus (−2)²: −4 versus +4. Read parentheses before exponents.
- Subtracting a negative: 6 − (−4) = 6 + 4 = 10, not 2.
- Radicals that are rational: √9 = 3 and √(1/4) = 1/2 are rational; only non-perfect squares under the radical are irrational.
- Zero cases: 0 is an integer and rational (0 = 0/1) but not natural under the standard convention; 5 ÷ 0 is undefined while 0 ÷ 5 = 0.
- Sign of a product: an even number of negative factors gives a positive product.