 in mathematics, the phrase up to indicates that its grammatical object is some equivalence class, to be regarded as a single entity, or disregarded as a single entity. if this object is a class of transformations (such as "isomorphism" or "permutation"), it implies the equivalence of objects one of which is the image of the other under such a transformation. if x is some property or process, the phrase "up to x" means "disregarding a possible difference in x". for instance we might follow the statement "an integer's prime factorization is unique up to ordering", meaning that the prime factorization is unique if we disregard the order of the factors; or we might say "the solution to an indefinite integral is , up to addition by a constant", meaning that the added constant is not the focus here, the solution  is, and that the addition of a constant is to be regarded as a background, of secondary focus. further examples concerning up to isomorphism, up to permutations and up to rotations are described below. in informal contexts, mathematicians often use the word modulo (or simply "mod") for similar purposes, as in "modulo isomorphism". ==examples== === tetris === thumb|tetris pieces i,j,l,o, s,t,z a simple example is "there are seven reflecting tetrominos, up to rotations", which makes reference to the seven possible contiguous arrangements of tetrominoes (collections of four unit squares arranged to connect on at least one side) which are frequently thought of as the seven tetris pieces (o, i, l, j, t, s, z.) this could also be written "there are five tetrominos, up to reflections and rotations", which would take account of the perspective that l and j could be thought of as the same piece, reflected, as well as that s and z could be seen as the same. the tetris game does not allow reflections, so the former notation is likely to seem more natural. to add in the exhaustive count, there is no formal notation. however, it is common to write "there are seven reflecting tetrominos (= 19 total) up to rotations". in this, tetris provides an excellent example, as a reader might simply count 7 pieces Ã 4 rotations as 28, where some pieces (the 2Ã2 o being the obvious example) have fewer than four rotation states. ===eight queens=== thumb|a solution of the eight queens problem in the eight queens puzzle, if the eight queens are considered to be distinct, there are 3&nbsp;709&nbsp;440 distinct 
Mathematical_terminology