Algebra/Chapter 1/Real Numbers

Decimals

Algebra Chapter 1: Elementary Arithmetic Section 8: Real Numbers

Order of Operations

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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 $2=\frac{2}{1}$. 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 $\frac{1}{3}=0.333\dots$. 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 $\sqrt{2}=1.41421356237\dots$. 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.