## 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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