Real Numbers and Subsets

Children learn about numbers quickly. Youngsters are taught to hold up their fingers when asked their age. They learn quickly to notice whether they received the same number of cookies as their siblings during snack time. Counting comes naturally. The natural numbers (or counting numbers) are 1, 2, 3, 4, …. These are the basic values from which we start.

Children learn about zero when they do not get what they asked for. Including zero with the set of natural numbers gives you the set of whole numbers. Students, especially those from colder climates, learn about the negative numbers when “it is too cold to play outside.”

When combined, the whole numbers and their negatives make up the set of integers. Children encounter fractions when they break their cookies apart or share their toys with a playmate. The formal name for fractions, rational numbers, comes from the idea that a fraction is the ratio of two integers. You can see that children have been exposed to a fair amount of mathematics before they have even entered their first classroom!

Having learned about square roots of the “nice” numbers (1, 4, 9, 16, 25, and so on), students are often confused when their teacher asks about the square root of 3. There is some relief in the class when they learn that numbers such as the square root of 3 are irrational. For the record, the numbers are irrational merely because they are not rational. That is, irrational numbers cannot be written as the ratio of two integers.

There is a hierarchy for these sets of numbers. All of the natural numbers are included in the set of whole numbers; all of the whole numbers are included in the set of integers; and all of the integers are included in the set of rational numbers. Of course, there are no numbers that are common to both the set of rational numbers and the set of irrational numbers. However, when these two sets are combined (this is called taking the union of the sets), a new set of numbers is formed, the set of real numbers. A simple explanation for the set of real numbers is that it represents all the numbers that can be placed on a number line. A graphical representation for the relationship among these sets of numbers follows.

Set of Real Numbers and Its Subsets

The number 8, for example, is a counting number (N), a whole number (W), an integer (Z), a rational number (Q), and a real number (R).

is a rational number (Q) and a real number (R), but it belongs to none of the other categories.
is an irrational number (I) and a real number (R).

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