And what's that supposed to mean?
Here is another proof, that is unarguably true:
You have two cars driving towards each other at 10km/h
When they are exactly 20km apart, a fly flies in front of one of the cars. To escape getting hit by the car, it flies away from the car (towards the other) at 40km/h. When it meets the other car, it instantly turns round and flies away from that car instead, and so on until the cars meet and it is squashed.
If each car is travelling at 10km/h, from 20km away, it will take exactly 1 hour for them to collide, squashing the fly in between.
The question is, how far did the fly travel?
FIRST SOLUTION:
The relative velocity of the fly and the opposite car is 50km/h:
40km/h + 10km/h = 50km/h, so the fly met the car after 24 minutes.
In 24 minutes, the fly would travel 16km
The cars would by then be only 12km apart
The fly would turn, and travel back towards the first car, and the two would have a relative velocity of 50km/h just like with the other car. This time, the fly would meet the car after only 14.4 minutes, and travel only 9.6km
As you can see, each time, the fly only travels 0.6 of its previous distance: 16, 9.6, 5.76, 3.456, etc.
This sequence will go on forever, because you will never reach 0.
If you add up the first few distances, you get 34.816.
2.0736
1.24416
0.746496
0.4478976
0.26873856
0.161243136
0.0967458816
With these next ones, you get 39.8548811776.
As you can see, it approaches 40. If you continue the sequence to infinity, the total will be 39.999... I'm sure you will agree with that.
But wait, what about the second solution???
SECOND SOLUTION:
As we already worked out, the cars will collide after exactly 1 hour. The fly is travelling at exactly 40km/h the whole time (albeit in different directions). 40km/h for one hour means the fly travelled exactly 40km
The two solutions: 39.999...km and 40km. Both are correct. 39.999... has the same value as 40
An easy to understand way of how they can be equal:
The decimal number system is created by us to represent actual numbers. These numbers were not invented by us, but are fundamental laws in the universe. The ACTUAL number which we represent as both 40 and 39.999... is a single number, it's just that the decimal number system provides two ways of representing it.
