What Is a Decimal Bar Notation?

Decimal Bar Examples

• 0.333... can be written with only one 3 if there is a bar over the first 3. 0.454545... can be written with only one 45 if there is a bar over both the 4 and the 5. 0.812812812... can be written with only one 812 if there is a bar over all three digits. Note that 0.3 with a bar over the 3 is equal to 0.333 with a bar over the last 3. The bar does not have to start at the decimal point. For example, 3.06353535... can be written as 3.0635 if there is a single bar over both the 3 and the 5.

Converting to Fractions

• A repeating decimal can always be written as a fraction. Use this example of converting 0.454545... to a fraction as a model. If x = 0.454545..., then 100x = 45.454545... and 100x - x = 45. So 99x = 45 or x = 45/99 = 5/11. This means that 0.454545... = 5/11. This technique works with any repeating fraction -- anything with a decimal bar -- as long as you make the tails -- the part to the right of the decimal points -- equal, so subtraction removes the part that repeats.

Irrational Numbers

• Non-terminating decimals that do not repeat cannot be written with a bar. The Ancient Greeks did not believe numbers existed that could not be written as fractions -- decimal numbers that did not terminate and could not be written with a bar. This seems to make sense because it is easy to show that between any two fractions there is another fraction. Legend has it that the first Greek who demonstrated that there were numbers that could not be written as a fraction was killed to suppress the information.

Bars in Other Bases

• Numbers that cannot be written as fractions are different from those that can, but numbers that have terminal representation are not different from numbers that are written with a bar. The easiest way to see this is to consider numbers written in other bases. 1/3 is a repeating decimal, while 1/5 is terminal in our standard base 10 numbering system. 1/3 = 0.333... and 1/5 = 0.2. However, in the base 12 numbering system used by many ancient cultures, 1/3 is terminal and 1/5 is a repeating fraction. 1/3 = 0.4 and 1/5 = 0.2444... . This means that repeating decimals exist simply as an aspect of the way in which we write the numbers.