Quotient
In arithmetic, a quotient is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division or a fraction or ratio. For example, when dividing 20 by 3, the quotient is 6 in the first sense and in the second sense.
In metrology, "quotient" refers to the general case with respect to the units of measurement of physical quantities.
Ratios is the special case for dimensionless quotients of two quantities of the same kind.
Quotients with a non-trivial dimension and compound units, especially when the divisor is a duration, are known as rates.
For example, density is said to be a "quotient", whereas mass fraction is a "ratio".
Specific quantities are intensive quantities resulting from the quotient of a physical quantity by mass, volume, or other measures of the system "size".
Notation
The quotient is most frequently encountered as two numbers, or two variables, divided by a horizontal line. The words "dividend" and "divisor" refer to each individual part, while the word "quotient" refers to the whole.Integer part definition
The quotient is also less commonly defined as the greatest whole number of times a divisor may be subtracted from a dividend—before making the remainder negative. For example, the divisor 3 may be subtracted up to 6 times from the dividend 20, before the remainder becomes negative:while
In this sense, a quotient is the integer part of the ratio of two numbers.
Quotient of two integers
A rational number can be defined as the quotient of two integers.A more detailed definition goes as follows:
Or more formally:
The existence of irrational numbers—numbers that are not a quotient of two integers—was first discovered in geometry, in such things as the ratio of the diagonal to the side in a square.