The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, Math Kangaroo 3-4, Math Kangaroo 5-6, Math Kangaroo 7-8.

An integer number can be written in expanded form, using the values of the digits and the powers of the base of numeration used. In this lesson, we will use base 10.

For example, in base 10, the number 9382746 is written also as:


A rational number written as a terminating decimal can also be written in expanded form. For example,


It is more difficult to write up expansions of non-terminating decimals or of irrational numbers, but it makes sense that they do have one.

Note that this writing depends on the base of numeration.

The expanded form is useful in proving various rules for divisibility. Note that the divisibility rules have the same restrictions as the expanded form: they depend on the base of numeration. The rules of divisibility learned in middle school are valid only in base 10.