Counting numbers have an order. You may need to find a number that is missing from a group of numbers. The numbers may be counting up or counting down. A good problem for kindergartners is to see if they can identify a missing number in a sequence.
In mathematics, the natural numbers are those used for counting ("there are six coins on the table") and ordering ("this is the third largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively (see English numerals). A later notion is that of a nominal number, which is used only for naming.
Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partition enumeration, are studied in combinatorics.
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated.
Ordinals were introduced by Georg Cantor in 1883 to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. He derived them by accident while working on a problem concerning trigonometric series—see Georg Cantor.
Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels. The most basic topics in elementary mathematics are arithmetic and geometry. Beginning in the last decades of the 20th century, there has been an increased emphasis on probability and statistics and on problem solving. Elementary mathematics is used in everyday life in such activities as making change, cooking, buying and selling stock, and gambling. It is also an essential first step on the path to understanding science.
In secondary school, the main topics in elementary mathematics are algebra and trigonometry. Calculus, even though it is often taught to advanced secondary school students, is usually considered college level mathematics.
Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. Topically, mathematical logic bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.
Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see logic in computer science for those.