Before, I discuss the misconceptions, let us recall the definition of rational numbers. A rational number is a number that can represented by the fraction where and are integers and not equal to 0. From this definition and other previously learned concepts, let us examine the following misconceptions about rational numbers.
Misconception 1 : Zero is not a rational number.
Truth: YES, it is. Zero, and negative and positive integers are all rational numbers. For example, , , and are all fractions whose numerators and denominators are integers and denominator 1 (which is clearly not equal to 0). » Read more
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 in the lowest terms has a terminating decimal if and only if the integer has no prime factor other than and .
Note that we have already explained the only if part in the preceding post. It remains to show that if part which is
if is in lowest terms and contains at most and as factors, then the fraction is a terminating decimal. » Read more
Another representation of rational numbers aside from fractions is the decimal form. Every fraction has a decimal representation:
, , and .
Notice that some of the fractions above are terminating, while the others are repeating decimals. The fractions and have only one decimal place, while and have infinitely many (Can you see why?). Now, given a fraction, can we determine if it’s a terminating or non-terminating decimal without dividing?
First let us examine the characteristics of terminating decimals, say 0.125. The easiest way to convert this decimal into fraction is by dividing a whole number by a power of 10: » Read more