5 Misconceptions About Rational Numbers

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 \frac{a}{b} where a and b are integers and b 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, 0 = \frac{0}{1}, -5 = \frac{-5}{1}, and 100 = \frac{100}{1} are all fractions whose numerators and denominators are integers and denominator 1 (which is clearly not equal to 0). » Read more

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

Fractions with Terminating and Non-Terminating Decimal Representations

Another representation of rational numbers aside from fractions is the decimal form.  Every fraction has a decimal representation:

\frac{1}{2} = 0.5, \frac{1}{5} = 0.2, \frac{2}{3} = 0.666 \cdots and \frac{1}{11} = 0.0909 \cdots.

Notice that some of the fractions above are terminating, while the others are repeating decimals. The fractions \frac{1}{2}  and \frac{1}{5} have only one decimal place, while \frac{2}{3} and \frac{1}{11} 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?

Delving Deeper

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

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