4 is the smallest squared prime (p2) and the only even number in this form. 4 is also the only square one more than a prime number.
A number is a multiple of 4 if its last two digits are a multiple of 4. For example, 1092 is a multiple of 4 because 92 = 4 × 23.
In addition, 2 + 2 = 2 × 2 = 22 = 4. Continuing the pattern in Knuth's up-arrow notation, 2 ↑↑ 2 = 2 ↑↑↑ 2 = 4, and so on, for any number of up arrows. (That is, 2 [n] 2 = 4 for every positive integer n, where a [n] b is the hyperoperation.)
A four-sided plane figure is a quadrilateral (quadrangle), sometimes also called a tetragon. It can be further classified as a rectangle, oblong, square, kite, or rhombus.
A solid figure with four faces as well as four vertices is a tetrahedron, and 4 is the smallest possible number of faces (as well as vertices) of a polyhedron. The regular tetrahedron is the simplest Platonic solid. A tetrahedron, which can also be called a 3-simplex, has four triangular faces and four vertices. It is the only self-dual regular polyhedron.
Four-dimensional space is the highest-dimensional space featuring more than three convex regular figures:
Four is the only integer n for which the (non trivial) alternating group An is not simple.
The four-color theorem states that a planar graph (or, equivalently, a flat map of two-dimensional regions such as countries) can be colored using four colors, so that adjacent vertices (or regions) are always different colors. Three colors are not, in general, sufficient to guarantee this. The largest planar complete graph has four vertices.
Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four square numbers. Three are not always sufficient; 7 for instance cannot be written as the sum of three squares.
Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. 4x = y2 − z2.
Brahmic numerals represented 1, 2 and 3 with as many lines. 4 was simplified by joining its four lines into a cross that looks like the modern plus sign. The Shunga would add a horizontal line on top of the digit, and the Kshatrapa and Pallava evolved the digit to a point where the speed of writing was a secondary concern. The Arabs' 4 still had the early concept of the cross, but for the sake of efficiency, was made in one stroke by connecting the "western" end to the "northern" end; the "eastern" end was finished off with a curve. The Europeans dropped the finishing curve and gradually made the digit less cursive, ending up with a digit very close to the original Brahmin cross.
Television stations that operate on channel 4 have occasionally made use of another variation of the "open 4", with the open portion being on the side, rather than the top. This version resembles the Canadian Aboriginal syllabics letter ᔦ. The magnetic ink character recognition "CMC-7" font also uses this variety of "4".