Base Counting Systems

When we learn to count in our counting system, we use just ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This is a system that developed as a result of the ten digits we have on our hands: two thumbs and eight fingers. Did you know that fingers and thumbs are called digits? Sometimes it is more convenient to count in a different base system. Computers use base two: 0's and 1's. Color codes on websites use base sixteen: 0's, 1's, 2's, 3's, 4's, 5's, 6's, 7's, 8's, 9's, a's, b's, c's, d's, e's, and f's. Base three is used to help describe the Cantor Set. I have used different bases for many other applications when it has been convenient to do so. In addition to all of the practical applications, learning to use different base counting systems can also help you understand our own counting system more deeply since much of the mathematics works the same way in any base system. For these reasons, if you are not familiar with how to work in different bases, I hope you will read on.

Digits and Counting

Following are images of a hand with 8 digits and the digits we will use for a base 8 counting system, along with a hand with 10 digits and our own base 10 counting system, along with a hand with 12 digits and the digits used in a base 12 counting system.  
Of course if were using either base 8 or base 12 all of the time as our primary base, then we would call it base 10.

If you need to see the counting pattern in these bases, you can download them here:

base eight counting chart
base ten counting chart
base twelve counting chart
You can also print your own counting charts in other base counting systems here: Counting Charts

You may also find it useful to print some counting blocks on cardstock paper and cut them out to help you with borrowing when subtracting, carrying when adding, multiplication, division, and converting as well as just visualizing numbers as in the following image.

base eight counting blocks, base ten counting blocks, base twelve counting blocks, pick your own base counting blocks

Place Value, Face Value and Converting

A number expands the same way in any base counting system: 236.13, but the physical quantity that this represents is of course different in different base systems. If we converted this number from base eight to base ten, we would get: 2×82 + 3×81 + 6×80 + 1×8-1 + 3×8-2 = 2×64 + 3×8 + 6 + 1/8 + 3/64 = 128+24+6 + 11/64 = 158.171875. The base twelve version of 236.13 would translate to: 2×122 + 3×12 + 6 + 1/12 + 3/144 = 330.14016 in base ten. As you can see from these two examples, the face value for the digits that the systems have in common is the same, but the place value is what changes since in any base counting system the place value of each digit is bn where b stands for the base of the system you are working in and n stands for the position of the number relative to the decimal place. The first number to the left of the decimal place has n=0. From there you add one for each position to the left and subtract one for each position to the right of the decimal place.

  1. Convert 569.abase 12 to base 10.
  2. Convert 89base 10 to base 8.

Arithmetic

Adding, subtracting, multiplying and dividing, all work the same way that you are used to, and will give you the same physical results, but the symbols used to represent them will be different and if you were to memorize your math facts, those facts would look different. Take a look at the addition chart below and see if you can use it to fill in the multiplication chart for base 8:
    Click on the chart to see the filled in version.

  1. Evaluate in base 12: 8 + 4
  2. Evaluate in base 12:
  3. Evaluate in base 8: 670 - 22

Fractions and More About Decimals

The rules for fractions remain the same: get a common denominator when adding or subtracting, multiply numerators and denominators together when multiplying, multiply by the reciprical of the divisor when dividing. To convert a fraction from one base to another, simply convert the numerator and denominator separately. It might be helpful to fill in an addition and multiplication chart for base 12 before answering these questions.
Examples: (Click on the problem to see the solution.)

  1. Evaluate in base 12:
  2. Rewrite (3/2)base 12 as a decimal in base 12.
  3. Rewrite (1/5)base 8 as a decimal in base 8.
  4. Rewrite (1/7)base 8 as a decimal in base 8.
Notice that a decimal that repeats in base 10, doesn't necessarily repeat in a different base and a decimal that doesn't repeat in base 10, might repeat in another base that you are working in. Alternately it could repeat in both bases or neither base.

Copyright ©2011 Laura Shears, You may print this for personal use or non-profit educational purposes only without permission. You may not reproduce this electronically or in printed form for any other use without express written permission from Laura Shears of Lsquared Math.