How many integers are there from 1 to 9 inclusive? Well, even if you are one of those who admit “not to be good in math” you can relax and confidently say 9, because this is not a trick question. You can also answer the question “what integer comes after 5?” with no hesitation, because the answer is definitely 6.
However, questions such as “How many rational numbers are there between 0 and 1?” and “What rational number comes after ½?” are not really the kind of questions one can answer without giving a serious thought. In fact, there are infinitely many rational numbers between 0 and 1. What is more interesting is no one can decide what rational number comes after ½! This is simply because whatever rational number one proposes to name as “next to ½” another person can take the average of that number and ½ to obtain a rational number that is in the middle of the two and this process can be repeated forever only to see that there is no such thing as “next rational number.”
Fortunately, there is a way to count them. The way we count rational numbers doesn’t go in an increasing manner; it rather jumps back-and-forth because of the absence of the concept of “next rational number.” But at least we are able to “enumerate them.” How about all real numbers between 0 and 1, which includes irrational numbers as well as rational ones? How many are there? Can we count them? It wouldn’t be a wild guess to say “infinitely many” to the first question. However, mathematics tells us that we can’t count them! Yes, we simply “cannot enumerate real numbers between 0 and 1”!
If we can’t count all real numbers between 0 and 1 then it’s evident that we can’t count all real numbers between 70 and 100, either. Why 70 and 100? Amy, a healthy and breathing female who is taller than 100 cm, measured 70 cm high at some point in her life and 100 cm at another time. Therefore, she experienced all the heights between 70 cm and 100 cm once in her life. Can we enumerate all heights she actually reached between those two readings? The answer is a resounding “No,” as we have shown above.
We may also ask: can we count all the breaths she has taken between these two points in time? Though doing so is an almost impossible task, we know the number of breaths is at least, in theory, countable. If she attained many uncountable heights within many countable breaths she has taken, what can we deduce from that? Mathematically speaking we can confidently say that during every single breath she has taken, her height advanced many uncountable values using the so called pigeonhole principle.
“And if you should count the favors of God, you could not enumerate them” reads the Qur’an (14:34,16:18). Even the literal meaning of this verse humbles us without resorting to any mathematical argument, for as the Persian poet Sadi wrote, “Every inhalation of the breath prolongs life and every expiration of it gladdens our nature; wherefore every breath confers two benefits and for every benefit gratitude is due.” A math literate person who is not satisfied with the literal meaning, however, might say, “Simply because we can’t count them doesn’t mean that they are uncountable. After all, there are approximately 1080 atoms in the universe, which is a finite, albeit enormous, number.” To this one can reply with a question: Is gaining height a blessing or not? Assuming that the answer is Yes – as there is nothing to say if it is No – shouldn’t we consider every height we reach a blessing in life? Why should a particular height we reach be less of a blessing than any other? Assuming the answer is Yes again, how many blessings does that make altogether? Well, mathematics tells us that we can’t count how many blessings we are given, even within single breath, let alone the entire life!
Yusuf Ziya Gurtas, Assoc. Prof. of Mathematics at Queensborough CC, Queens, NY