Suppose you are very happy and would like to clap your hands. Instead of moving your both hands towards each other, you hold your right hand in the same position. So the only hand which is moving is your left hand; it moves towards your right hand. Let’s say that the distance between both hands at the start position is 1 meter. Now, half the distance with your left hand by moving it towards your right hand. The distance between both hands is now 1/2 meters. Now, half the distance again by moving your left hand towards your right hand. The distance between both hands is now 1/4 meters. Continue this process again and again, you will obtain the following series: 1/2, 1/4, 1/8, 1/16, 1/32 … . In other words, according to this procedure, you would never be able to clap your hands since there is always a distance between both of your hands, even when this distance is infinitely small. But as you know (hopefully), you are able to clap your hands! So how does this work?
We know that the distance between both hands at the beginning is 1 meter. So, if we prove that the the series of 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + … goes exactly to 1, we solved the problem. But how can we do this? It seems that the answer of this series will be eventually infinity, since the series goes on forever!
Let s be the distance between both hands. So according to our procedure, we have:
s = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + … .
We need to show that this series equals 1. To show this, let us divide both sides by 2. We obtain:
1/2s = 1/4 + 1/8 + 1/16 + 1/32 + … .
Let us subtract from our original series s, the series of 1/2s. We obtain:
s – 1/2s = 1/2.
This is equivalent to:
1/2s = 1/2
After multiplying both sides by 2, we obtain s = 1. Hence we proved that our series of 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + … is equal to 1. So a series of infinitely many terms can have a finite answer.
In a similar way, we can divide a 1×1 square into infinitely many pieces as follows. Let us cut the square into two equal parts, with both a surface of 1/2. Then divide one of this surfaces into two equal parts again. We now have a square which is divided into 3 parts with the following surfaces: 1/2, 1/4, 1/4. Let us now divide one of the squares with surface 1/4 into two equal parts of surface 1/8. We can repeat this procedure an infinitely amount of times. In other words, we obtain again the series 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + … equals 1, i.e. the total surface of the 1×1 square.
After proving that the series 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + … equals 1 , there is just one thing left to do; clap your hands for this beautiful piece of mathematics!
Dit artikel is geschreven door Mark Woelders