Algebra/Chapter 1/Real Numbers

Decimals

Algebra Chapter 1: Elementary Arithmetic Section 8: Real Numbers

Order of Operations

1.8: Real Numbers

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Real Numbers

We have already talked about the different types of numbers in Section 1. However, in this section, we will be using more sophisticated language to refer to them.

In mathematics there are names for many different types of numbers and you've encountered lots of these types already and some of these types contain the others. For instance we can start with the whole numbers such as 0, 1, 2, 3, etc. Using subtraction we can build negative numbers by subtracting a bigger number from a smaller giving us an answer in the set {... -3, -2, -1, 0}.

Using division we can identify fractions between 0 and 1 by dividing a smaller number by a bigger e.g. {1/2, 2/3, 3/4, ...} or {-1/-2, -2/-3, -3/-4, ....} We can also identify negative fractions between -1 and 0 by dividing a negative number by a positive or a positive number by a negative {-1/2, -2/3, -3/4, ...} or {1/-2, 2/-3, 3/-4, ...}. Every whole number can be written as a fraction, such as $\textstyle 2={\frac {2}{1}}$. The rational numbers are exactly those numbers which can be written as fractions.

Rational numbers are a subset of numbers we call real numbers. Some calculators allow you to differentiate between rational numbers and real numbers by representing the rational number as a fraction. If you use decimal notation the decimals in your rational number may go on forever, for example $\textstyle {\frac {1}{3}}=0.333\ldots$. The real numbers include all of the types of numbers mentioned before (whole numbers, negative numbers, fractions, etc.) and others that require special operations such as roots to represent. These other numbers may not have any recognizable pattern to their digits, such as ${\sqrt {2}}=1.41421356237\ldots$. But, at the end of the day, the real numbers act just like the rational numbers that you're already familiar with. For those readers that are geometrically inclined, one may think of the real numbers as a line (or ruler), where every point on the line corresponds to exactly one number, as in the picture below.

Types of Numbers

Real numbers consist of zero (0), the positive and negative integers (-3, -1, 2, 4), and all the fractional and decimal values in between (0.4, 3.1415927, 1/2). Real numbers are divided into rational and irrational numbers. The set of real numbers is denoted by ℝ.

Rational numbers are numbers that can be expressed as a ratio (that is, a division) of two integers (${2 \over 3}$, $0.6$, $3$, $-4.7$, $0.{\overline {111}}...$). If a number has a terminating decimal, or a decimal that ends ($3.6$, $5.263$) or repeats ($1.{\overline {333}}....$), it is rational. The set of rational numbers is denoted by ℚ.

Irrational numbers have decimal parts that do not terminate or repeat ($2.71828...$, $3.14159...$) and cannot be expressed as a fractional equivalent. For example, the number ${\sqrt {2}}=1.41421356...$ does not have an equivalent ratio or division of two numbers. There are several other different "sets" of rational numbers. The set of irrational numbers is denoted by 𝕀.

Natural numbers, also known as "counting numbers", are the first numbers you learn. The natural numbers include all of the positive whole numbers (1, 24, 6, 2, 357). Note that zero is not included, and fractions or decimals are not included. The set of natural numbers is denoted by ℕ.

Whole numbers are the natural numbers, plus zero. The set of whole numbers is denoted by 𝕎.

Integers are all positive and negative numbers without a decimal part (3, -1, 15, -42). The set of integers is denoted by ℤ.