Last year, I had a good discussion with some teachers about Venn diagrams. Several teachers commented that I should change the diagram below because it might give an impression that the number of rational numbers is the same as the number of irrational numbers. Sadly, I was not sure if I was able to explain my point. Anyway, below is an extended explanation to those comments.
Venn diagrams usually represent the logical relationships among sets. They do not concern with their cardinality (the number of elements). For example, if sets A and B have common elements, then their relationship can be represented by any of the three Venn diagrams below. In creating Venn diagrams, you do not represent if there is only one common element, or there are many: you are representing if there are common elements or there are none. Continue reading
A rational number can be expressed in the form where and are integers and . In other words, it is a fraction whose denominator is not zero, and both the denominator and numerator are integers.
Some fractions, however, may contain a numerator or denominator that is not an integer. Some examples of such fractions are
, and .
A rational number may be represented in many ways, but it can always be expressed as a fraction. For instance, is a rational number because we can express it as . Also, the number , a repeating decimal, is a rational number because we can also express it as fraction .
Last night, I received a Facebook message from a Grade 8 student asking why non-terminating repeating decimals are rational. I am posting the answer here for reference.
Rational numbers is closed under addition. That is, if we add two rational numbers, we are guaranteed that the sum is also a rational number. The proof of this is quite easy, so I leave it as an exercise for advanced high school students.
Before discussing non-terminating decimals, let me also note that terminating decimals are rational. I think this is quite obvious because terminating decimals can be converted to fractions (and fractions are rational). For example, can be expressed as
Further, terminating decimals can be expressed as sum of fractions. For example, can be expressed as
Since rational numbers is closed under addition, the sum of any number of fractions is also a fraction. This shows that all terminating decimals are fractions. Continue reading