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The most common number systems are the following:

The Natural Numbers. These are the counting numbers 1, 2, 3, . . . that are possible answers to the question “how many?” They are abstract concepts that describe sizes of sets.

The Integers. These are abstract concepts that describe, not sizes of sets, but the relative sizes of two sets. They are the possible answers to the question “how many more does A have than B has?” They include both positive numbers (meaning A has more than B) and negative numbers (meaning B has more than A).

The Rational Numbers. These are abstract concepts that describe ratios of sizes of sets. They do not model sizes of sets the way that natural numbers do. If you say “I ate 3/4 of a pie”, you are not saying that the set of things you ate had 3/4 elements. Instead, you are expressing a ratio of two integer quantities: 3, the number of pie-quarters that you ate, and 4, the number of pie-quarters that make up a whole pie.

The Real Numbers. These are abstract concepts that describe measurements of continuous quantities, such as length, weight, quantity of fluid, etc. (Don't let the word “real” fool you; the real numbers are no more “real” in the ordinary English sense of the word than are any other kind of numbers.)

The Complex Numbers. These are pairs of real numbers, with pairs of the form (x,0) behaving the same as ordinary real numbers x, but with other pairs (whose second entry is non-zero) behaving differently. The rule for multiplication is (a, b) times (c, d) = (ac-bd, ad+bc) which means that the pair (0,1), when squared, gives (-1,0) which in this context is considered to be the same as the real number -1 (since a complex number of the form (x,0) is indistinguishable by its arithmetic properties from the real number x, we can consider it to be describing the same underlying concept, much as we consider a fraction of the form x/1 to be describing the same underlying concept as the integer x). Thus, the complex numbers are a number system in which there is an object whose square is -1. This object is denoted by i, and the pair (a,b) is written as “a + bi”. When a=0, such a number is called an “imaginary number”. Complex numbers are extremely important mathematically. They have less direct relevance to real-world situations, since they aren't measurements of single quantities. However, they relate directly to a few real-world situations (such as the strength of an electromagnetic field), and they relate indirectly to many more since many properties of real numbers can be more easily understood in the larger context of complex numbers.

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