Base 10 and Other Bases

Students can acquire arithmetic skills without learning the concept of any base. But sooner or later they learn about base 10.

Base 10 is often introduced in connection with the metric system of measurement where the ratio between two consecutive units measuring the same quantity is 10.

Another approach is to introduce exponentiation as shorthand for multiplication, 102 = 10*10, 103 = 10*10*10, and so on.

We prefer to use the concept of geometric progression with quotient 10, with the additional requirement that the number 1 must be a member of such progression. For example: 1000, 100, 10, 1; or 10, 1, 0.1, 0.01.

When students are familiar with the concept of base 10, a counting board such as

can be simplified by using only the digits 1, 2, 3, 4, and 5 in each column and indicating which column contains the number 1.


Both the left and right boards hold 43.25.

Both the left and right boards hold 14.

Other Bases

The values in each column of such a simplified board vary from 0 to 15 (with at most one counter per square). So the boards can be used to represent numbers in any base from 6 to sixteen. But when we use different bases, we must always state or write down which base is used.


Example in Base 6

The values of the columns are 1, 0, 1. So, 1*62 + 1*60 = 37.

Example in Base 12

The values of the columns are 8, 9, 5. So, 8*122 + 9*121 + 5*120 = 1265.

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