Tag Archives: notation

What if we spoke about division as we do logarithms?

Definition: The quotient, factor a, of c is b if and only if ab is c. We introduce the following notation:


When we read this notation aloud, we say (for the second example), The quotient, factor 5, of 10 is 2.

The important thing to remember about a quotient is that a quotient is a factor. When we see this equation:

we ask ourselves, What factor goes with 3 to make 12?

Properties of the quotient

The following properties of quotient are derived from corresponding properties of multiplication.

laws of the quotient

There are some important laws pertaining to the quotient. The next section will include algebraic proofs of these laws.

These properties will be important when solving equations involving quotients.

factors for the quotient function

Standard quotient tables and calculator functions involve one of two factors: 10 or u, where u, the factor of the natural quotient, is defined by the series below:

Quotients using this factor can be notated one of two ways:

Quotients factor 10 are so common that they also have special notation, so that the following two equations are equivalent in meaning:

One other convention for the factor 10 quotient is this: If a factor is not indicated, we assume the quotient is factor 10. More commonly, we capitalize the Q in Quot in order to indicate the factor 10 quotient but either notation is accepted.

Many times, we will be able to find a quotient by inspection and use of the associated known multiplication facts. In almost all cases involving relatively small numbers, we can approximate the value of a quotient by use of multiplication facts.

When we require a greater level of precision, we will need the change of factor formula.

Change of factor formula

Applications of the quotient

The quotient function appears so often in computations that some important measures are defined in terms of it. Of these, perhaps the most important is speed.

The speed (r) of an object is defined in terms of the distance (d) it travels over the time period (t) in following way: