Step By Step Calculus » 2.1 - Number System

show_hiddenExpand All [+]
Synopsis
You can not do science or mathematics without numbers, so that will be our first topic. This is in large part review, since you have been studying various types of number since your first year of school. In this section, we will briefly describe the different types of numbers, and then discuss each type in detail.
  • Natural numbers: The natural numbers are the numbers generally used for counting, written \mathbb{N}=\left\{1,2,3,4,...\right\} \mathbb{N}=\left\{1,2,3,4,...\right\} .
  • Integers: The integers are the natural numbers, their negatives, and the number 0, written \mathbb{Z}=\{ ...,-2,-1,0,1,2,...\} \mathbb{Z}=\{ ...,-2,-1,0,1,2,...\} . The positive integers are written as \mathbb{Z^+}=\{1,2,3,4,...\}\mathbb{Z^+}=\{1,2,3,4,...\} so \mathbb{Z^+}\mathbb{Z^+} is the same as \mathbb{N}\mathbb{N}.
  • Whole numbers: The whole numbers are composed of the natural numbers and 0, written \mathbb{N}_0=\{0,1,2,3,4,...\}\mathbb{N}_0=\{0,1,2,3,4,...\}. This is also called the set of non-negative integers.
    Above we used set notation. In a later section, we will present a detailed treatment of set notation. However, for now, we simply describe the following ways to denote a set
    Listing elements Given a finite set for which we know all the elements, we write the set listing the elements within curly braces separated by commas. For example, \{1.2, 3, 6.5\}\{1.2, 3, 6.5\} is a set containing three elements 1.2, 31.2, 3 and 6.56.5. If the set is infinite, we write out enough elements/members to make the membership obvious, and put in dots to show the set is infinite. For example, the infinite set of even natural numbers can be written
    \displaystyle \{ 2, 4, 6, 8, \ldots \}
    \displaystyle \{ 2, 4, 6, 8, \ldots \}
    and the set of even integers can be written
    \displaystyle \{ \ldots -8, -6, -4, -2, 0, 2, 4, 6, \dots \} \, .
    \displaystyle \{ \ldots -8, -6, -4, -2, 0, 2, 4, 6, \dots \} \, .
    Set-builder notation Sometimes it is more convenient to indicate the rule by which a set is created. For example, to describe the set of even natural numbers, we can write
    \displaystyle \{ 2k \;| \; k \displaystyle \{ 2k \;| \; k is a natural number \displaystyle \}. \displaystyle \}.
    This notation says “the set consists of all numbers of the form 2k2k, where kk is a natural number" . This way of representing a set is called set-builder notation.
  • Rational numbers: The rational numbers, \mathbb{Q}\mathbb{Q}, consists of numbers of the form \dfrac{n}{d}\dfrac{n}{d}, where nn and dd are integers, with d \neq 0 d \neq 0.
    Notice that there is a problem here: The rational numbers \dfrac{1}{2}\dfrac{1}{2} and \dfrac{4}{8} \dfrac{4}{8}, represent the same rational number. In fact infinitely many ratios reduce in lowest terms to \dfrac{1}{2}\dfrac{1}{2} . The solution to this problem is based on common sense: a rational number is in fact the set of ratios of integers which all reduce to the same result in lowest terms. For example, the rational number \dfrac{1}{2}\dfrac{1}{2} is the entire set of ratios
    \displaystyle \left\{ \frac{m}{2m} \; | \; m\; \textrm{is a non-zero integer } \right\} \, = \left\{ \ldots \frac{-4}{-8} , \frac{-3}{-6}, \frac{-2}{-4}, \frac{-1}{-2}, \frac{1}{2} , \frac{2}{4}, \frac{3}{6}, \ldots \right\} \, .
    \displaystyle \left\{ \frac{m}{2m} \; | \; m\; \textrm{is a non-zero integer } \right\} \, = \left\{ \ldots \frac{-4}{-8} , \frac{-3}{-6}, \frac{-2}{-4}, \frac{-1}{-2}, \frac{1}{2} , \frac{2}{4}, \frac{3}{6}, \ldots \right\} \, .
    As intimidating as this seems, you have had so much practice in school with these numbers that they will not give you any difficulty. Every real number has a decimal expansion. The decimal expansion of a rational number will eventually consist of a repeating block of digits.
  • Irrational numbers: Irrational numbers, are those numbers which cannot be expressed as a ratio of integers. However, they can be approximated arbitrarily closely by rational numbers. They correspond to the numbers that have non-repeating decimal expansions. Examples are \sqrt{2}\sqrt{2} or \pi \pi , and you can see that it is not obvious that these two numbers are irrational. In fact we will prove that \sqrt{2}\sqrt{2} is irrational, but the proof that \pi\pi is irrational takes mathematical techniques you haven’t yet seen.
  • Real numbers: The set of real numbers, \mathbb{R},\mathbb{R}, is the collection of all rational and irrational numbers.