Chapter 1. Numbers and Operations

Number systems. The natural numbers are 1, 2, 3, …; the whole numbers add 0; the integers add negatives. The rational numbers are all numbers of the form a/b with b ≠ 0; their decimals terminate or repeat. The irrational numbers (such as √2 and π) have non-repeating, non-terminating decimals. Together, rational and irrational numbers make up the real numbers [1][2].

Order of operations. Evaluate in this order: parentheses (and other grouping), exponents, multiplication and division (left to right), addition and subtraction (left to right). Example: 3 + 4 × 2² = 3 + 4 × 4 = 19.

Fractions, decimals and percentages

Add fractions with a common denominator: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.

Multiply fractions straight across; divide by multiplying by the reciprocal.

Percent means “per hundred”: 25% = 25/100 = 0.25.

Percent change = (new − old) / old × 100%.

Divisibility and primes. A prime number has exactly two positive factors, 1 and itself (2, 3, 5, 7, 11, …). Every integer greater than 1 factors uniquely into primes (the Fundamental Theorem of Arithmetic) [8]. The greatest common divisor (GCD) is the largest integer dividing both numbers; the least common multiple (LCM) is the smallest positive common multiple. For positive integers, GCD × LCM = a × b.

Ratios and proportions. If a/b = c/d then ad = bc. Rates compare quantities with different units, for example kilometres per hour.

Review: (a) Find the GCD and LCM of 48 and 180. (Answer: 12 and 720.) (b) A price rises from 80 to 100. What is the percent increase? (Answer: 25%.)