Have you ever found a task difficult, and adding fractions can certainly be that, but then had the insight: “If I simply think about repeating this an infinite number of times, it will be easy”? This may sound absurd, but stick with me and you will see.
Our infinite fraction sum
We want to perform the addition:
The three dots tell us that we will keep going with no end point, always making the denominator twice as large! Yet, I am claiming that it is easier to do this than to stop at 
Visualising the sum
Let’s visualise the fractions. We have a square, as Figure 1 below shows, and we want to fill part of it (a fraction of it) at a time. If we stop at one half, the other half of the square will remain uncovered. When we add one quarter, then one quarter of the square will remain uncovered. As you can see from Figure 1, after adding one thirty-second, we were left with a gap equal to one thirty-second of the square’s area.
In fact, each fraction we add will leave that same fraction of the square’s area uncovered.
Are we doomed to keep trying to fill the square, only to be frustrated by a tiny gap that remains uncovered?
First answer: A philosopher says “Yea, doomed thou art!”
Zeno of Elea, a Greek philosopher from the fifth century BC, proposed that we could never really reach a destination, because our journey would consist of an infinite covering of half of the remaining distance! No matter how close you got, you would have to get to the half-way mark of the distance separating you from your target. That knee pain I feel from banging it against the table leg is an illusion, since my knee could never quite reach that leg!
Zeno argued that, since the journey would require an infinite number of steps, no one would be able to complete it and no destinations were reachable.
Second answer: Mathematicians say “No, we are not doomed”
Thankfully, mathematicians became comfortable with the idea of infinity. Their insight was that, if you had an infinity of fractions, each worth half of its predecessor, you wouldn’t have to stop at a stage when any part of the square remained uncovered. Therefore, all those parts of the square that we have covered, if we kept going, would amount to the entire area of the square:
In conclusion, all we have to do is conduct a thought experiment, an act of the imagination: “Say I have a bag of infinitely many fractions, each worth half of its predecessor”. This can get us to the answer without adding any fractions. What a payoff, right? The infinite sum has a finite result, hence my knee did reach the table leg and my pain is real!
If you liked this post and want to soothe my knee pain, leave me a comment or drop me a line at ziad [at] numbersforwordspeople [dot] com and let me know if you want to see more about fractions, infinity or any other mathematical topic.
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