# Quotition and partition

In arithmetic, quotition and partition are two ways of viewing fractions and division.

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

${\displaystyle 6\div 2}$

can be construed in either of two ways:

• "How many parts of size 2 must be added to get 6?" (Quotition division)
One can write
${\displaystyle 6=\underbrace {2+2+2} _{\text{3 parts}}.}$
Since it takes 3 parts, the conclusion is that
${\displaystyle 6\div 2=3.}$
• "What is the size of 2 equal parts whose sum is 6?". (Partition division)
One can write
${\displaystyle 6=\underbrace {3+3} _{\text{2 parts}}.}$
Since the size of each part is 3, the conclusion is that
${\displaystyle 6\div 2=3.}$

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.