Quotition and partition

In arithmetic, quotition is one of two ways of viewing fractions and division, the other being partition.

In quotition division one asks how many parts there are; in partition division one asks what the size of each part is.

For example, the expression

can be construed in either of two ways:

It is a fact of elementary theoretical mathematics that the numerical answer is always the same either way: 6 ÷ 2 = 3. This is essentially equivalent to the commutativity of multiplication.

Division involves thinking about a whole in terms of its parts. One frequent division notion, a natural number of equal parts, is known as partition to educators. The basic concept behind partition is sharing. In sharing a whole entity becomes an integer number of equal parts. What quotition concerns is explained by removing the word integer in the last sentence. Allow number to be any fraction and you have quotition instead of partition.

Partition

Partition may refer to:

Computing

Mathematics

Natural science

Law and Politics

List of partition topics

Generally, a partition is a division of a whole into non-overlapping parts. Among the kinds of partitions considered in mathematics are

Integer partitions

Set partitions

Partition (number theory)

In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, 4 can be partitioned in five distinct ways:

The order-dependent composition 1 + 3 is the same partition as 3 + 1, while the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition 2 + 1 + 1.

A summand in a partition is also called a part. The number of partitions of n is given by the partition function p(n). So p(4) = 5. The notation λ ⊢ n means that λ is a partition of n.

Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials, the symmetric group and in group representation theory in general.

Examples