5 Misconceptions About Rational Numbers

High School Mathematics, Pedagogy and Teaching

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).

Misconception 2: All fractions are rational numbers.

Truth: NO. One example is \frac{\pi}{2}. It is a fraction, but it is not a rational number. However, the opposite is true. All rational numbers can be expressed as fractions (see definition above).

 

Misconception 3: Rational Numbers can only be expressed as a terminating decimal.

Truth: All rational numbers can be expressed as infinite decimals, not just \frac{1}{3} and the like. This follows from the fact that 1 is equal to the infinite decimal 0.999 \cdots where the strings of 9’s never ends. This means that

\frac{1}{2} = 0.4 + 0.1 = 0.49999 \cdots (Why?*)

and

\frac{1}{8} = 0.125 = 0.124 + 0.001 = 0.124999 \cdots.

Therefore, it can be concluded that all terminating decimals representing rational numbers can be expressed as a non-terminating decimal ending in an infinite number of 9’s.

Misconception 4: Multiply rational numbers always gives a product less than the two.

All integers are rational numbers (from Misconception 1), so it is clear that it can be smaller or larger depending on the number you multiply. For example \frac{1}{5} \times 5 = 1. The product is larger . However, \frac{1}{5} \times -5 = -1 which is smaller.

Misconception 5: There are more fractions than integers.

The number of fractions and the number of integers are just the same. You can pair every fraction to the positive integers without leaving any fraction behind.

 *Since  The equality 1 = 0.999… means 0.1 = 0.09999 and 0.001 = 0.0009999 and so on.  That’s it. If you know some more common misconceptions, please use the comment box below.

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