1. Let’s start a train travel. (Refer the pic below)
2. You can always tell which station is after which one as you know in what sequence they will appear.
3. The same thing is with Natural Numbers (Counting numbers), you know in what sequence the numbers should appear.
4. And what is that define this sequence of numbers? In case with our train, it is the direction of the train in which it is travelling. In case of numbers, this train is the less than equal to relation (Yes the type of relation we saw during sets)!
5. Now, suppose the train authorities perform some particular transactions at each station.
6. You can consider these transactions as some formulas or rules on the natural numbers.
7. How can you, with surety know about what these transactions are?
8. If you see those transactions at any one of the stations, you are not sure if it was carried at that particular station only or on all the stations.
9. To establish that those transactions are performed on every station, you need to see those transactions at different stations. But you can’t see them on all the stations!
10. First thing to establish this rule is to see whether those transactions are performed at the starting station (Station 1).
11. If you don’t see those transactions at the starting station itself, you know that the transactions are not standard rules.
12. If you find that those transactions are performed at the starting station, then you should randomly select any station on the train’s route and see if those transactions are performed there. Suppose you find that those transactions are indeed performed there as well.
13. But, you can’t travel faster than the train to reach the next station to see by yourself if the same transactions are performed on the next station as well.
14. By the time you reach the next station, you see the train has already gone. But, you can still look out for the effects those transactions may have left at the next station.
15. If you can find sufficient proof that logically can conclude that those transactions have been performed on the next station as well and you know that these proofs can be found on all the stations (may be because of law or any other rules by government).
16. So, you will certainly know that these transactions are performed at all the stations and you can establish that as a fact!
17. This is how Mathematical Induction works!
18. This simple story has a tricky part as well. You can very well see that there is different information that you need to know at different times.
19. First you should know the starting station. (This indicates, you should know from which number the particular formula starts applying to natural numbers)
20. You should be well versed with the schedule of the train. (This amounts to have a deep knowledge about the counting of numbers itself, the less than equal to relation on numbers, etc. etc…)
21. You should know the effects of operations, properties of operations on natural numbers, what are other relationships that exist with numbers using these operations (amount to knowing about the laws and rules made by the government).
22. This is the philosophy of Mathematical Induction in Mathematics!
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