Chapter 1: Algebra

Numbers & Arithmetic

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Sets of Numbers

The sets of numbers you may work with in your calculus course are:

• the natural numbers $\mathbb{N} = \left\{1, 2, 3, 4, \ldots \right\}$
• the integers $\mathbb{Z} = \left\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \right\}$
• the rational numbers $\mathbb{Q} = \left\{\frac{a}{b} \mid a,b \in \mathbb{Z} \textrm{ and } b \neq 0 \right\}$
• the real numbers $\mathbb{R}$, which is the union of the rational numbers and the irrational numbers. An irrational number is a number that cannot be written as a ratio of two integers, for example, $\pi$ or $\sqrt{2}$.
  
  Irrational numbers have the special property that they have a non-terminating, non-repeating decimal expansion.

Fraction Arithmetic

• Addition/Subtraction: find a common denominator, then add the numerators. Always use the lowest common denominator.
• Multiplication: cancel common factors, then multiply the numerators and multiply the denominators.
• Division: multiply the first fraction by the reciprocal of the second.