Fractions with Terminating Decimals

Late last month, we have talked about fractions with terminating decimals as well fractions with non-terminating decimals. We ended up with a conjecture that a fraction is a terminating decimal if its denominator has only the following factors: 2 (or its powers), 5 (or its powers) or both.  In this post,we refine this conjecture. This conjecture is the same as saying

A rational fraction \frac{a}{b} in the lowest terms has a terminating decimal if and only if the integer b has no prime factor other than 2 and 5.

Note that we have already explained the only if part in the preceding post. It remains to show that if part which is

if \frac{a}{b} is in lowest terms and b contains at most 2 and 5 as factors, then the fraction is a terminating decimal. » Read more

The Mathematics of Tenure, Car Lights, and Logic Gates

In a state office, an employee maybe tenured if (1) he has passed the Career Service Examination, or (2) he has worked as a probationary employee for more than 10 years.

Notice that the employee maybe tenured if either one of the conditions above is satisfied. He may also be tenured if both conditions are satisfied. If we create a table and rephrase the statements above to question form, we can see that the only reason for an employee not to be tenured is if the answer to both questions is NO.


The first table is also the same as the table shown below.  In the second table,  the situation describes the relationship between a car’s ceiling light and its two doors. Unless the ceiling light is turned on or off manually, it will always be ON if both doors or one of the two doors is open. The only time that the light will be OFF is if the two doors are shut. » Read more

Cryptography: The Science of Encoding and Decoding Messages

Sending secret messages using codes has been a practice since the ancient time. In the Roman Empire, military officers would let messengers cut their hair, tattoo a secret message on their head, and send them to another place to relay the message once the hair had grown.


In Sparta, a device called  scytale consisting of a cylindrical tool with a strip of parchment was used to encode and decode messages.  Julius Caesar was known to perform letter substitutions to messages to make them unreadable. The method is now known as the Caesar cipher. » Read more

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