There are different types, or domains, of numbers. Determining the proper domain of a given set of numbers is important because different domains have different mathematical properties and allow you to perform different operations. Numerical domains are nested within one another, from smallest to largest: natural numbers, integers, rational numbers, real numbers and complex numbers. The proper domain of a given set of numbers is the smallest domain that is required to contain all members of that set.

Draw a reference diagram, a series of concentric circles, labeled with the domain names and a representative member or two of the domain. For example, the innermost circle, NATURAL NUMBERS, could include “0, 5;” the next outer circle, INTEGERS, could include “-6, 100;” the next outer circle, RATIONAL NUMBERS, could include “-4/5, 19/5;” the next outer circle, REAL NUMBERS, could include pi and the square root of 3; the outermost circle, COMPLEX NUMBERS, could include the square root of -1, and “4 plus the square root of -8.”

If even one member of the target set falls into a larger domain, the entire set falls into that domain. For example, if the target Set A = {4, 7, pi}, then the set is in the domain of real numbers. Without pi, the set would be in the domain of the natural numbers.

Write down a full list or a definition of the target set of numbers. It may be a comprehensive list—such as Set A = {0, 5}, or Set B = {pi}—or it may be a definition, such as “let Set C equal all the positive multiples of 2.” As an example, consider this target set: {-15, 0, 2/3, the square root of 2, pi, 6, 117, and "200 plus 5 times the square root of -1, also known as 200 + 5i"}.

Determine whether every member of the target set is a natural number. Natural numbers are the “counting” numbers, zero and greater. In order from the smallest value up, the set of natural numbers is {0, 1, 2, 3, 4, ...}. It is infinitely large, but includes no negative numbers. If every member of the target set is a natural number, then the target set belongs to the domain of natural numbers. If not, focus on the members of the target set that are not natural numbers. In our example (listed in Step 1), the numbers 0, 6, and 117 are natural numbers, but -15, 2/3, the square root of 2, pi, and 200 + 5i are not.

Determine whether all of those members are integers. The integers includes all the natural numbers and their values multiplied by -1. In order, the set of integers is {..., -3, -2, -1, 0, 1, 2, 3, ...}. If every member of the target set is an integer, then the target set belongs to the domain of integers. If not, focus on the members of the target set that are not integers. In our example, the number -15 is another integer in addition to the natural numbersin the set, but 2/3, the square root of 2, pi, and 200 + 5i are not.

Determine whether all of those members are rational numbers. The rational numbers include not only the integers, but also all numbers that can be expressed as a ratio of two integers, not including division by zero. Examples of rational numbers include -1/4, 2/3, 7/3, 5/1, and so forth. If every member of the target set is either an integer or a rational number, then the target set belongs to the domain of rational numbers. If not, focus on the members of the target set that are not rational numbers. In our example, 2/3 is another rational number in addition to the integers in the set, but the square root of 2, pi, and 200 + 5i are not.

Determine whether all of those members are real numbers. The real numbers include, not only the rational numbers, but numbers that cannot be represented by integer ratios, even though they exist on the number line between two other rational numbers. For example, no integer ratio represents the square root of 2, but it falls on the number line between 1.1 and 1.2. No integer ratio represents the value of pi, but it falls on the number line between 3.14 and 3.15. The square root of 2 and pi are “irrational numbers.” If every member of the target set is either a rational number or an irrational number, then the target set belongs to the domain of real numbers. If not, focus on the members of the target set that are not real numbers. In our example, the square root of 2 and pi are other real numbers in addition to the rational numbers in the set, but 200 + 5i is not.

Determine whether all of those members are complex numbers. Complex numbers include, not only real numbers, but numbers that have some component that is the square root of a negative number, like the square root of negative one, or “i.” If every member of the target set can be expressed as a real number or a complex number, then the target set belongs to the domain of the complex numbers. If not, then you do not have a set that is composed only of numbers. For example, “Set A: {2, -3, 5/12, pi, the square root of -7, pineapple, a sunny day on Zuma Beach}” is not a set of numbers. In our example, 200 + 5i is a complex number. So, the smallest domain that includes every member of our set is the complex numbers, and this is the domain of our example target set.

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#### References

- “Handbook of Mathematics and Computational Science,” JW Harris & H Stocker, Springer-Verlag, 1998

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- Yagi Studio/Digital Vision/Getty Images