Logarithms

The things missed in Mathematics!

1.       You all must have observed this thing if you have tried to lift very heavy weights.

2.       If you want to lift the weight directly, it is not easy.

3.       But if you use a pulley as shown in the pic, the job is very much simpler.

4.       The concept of Logarithms in Mathematics is just the same.

5.       If you are doing a very lengthy calculation, which involves calculation of many powers of numbers, it is not so easy to do it directly.

6.       So, we can take another way to do it and use logarithms.

7.       What makes work easier with Logarithms? Answer is that it converts multiplication into addition and division into subtraction, both addition and subtraction being easier to perform than huge multiplications and divisions.

8.       Logarithms are also a way of asking the same question differently, that makes it easier to arrive at the answer.

9.       Instead of asking “What is a raised to power x?” Logarithms ask the question “If we have y that is a raised to power x, then what is x?

10.   Sometimes, asking the right question is more important in solving a problem, and Logarithms provide one of the ways to ask the right question.

Few formalized representations:

If y = a^x , then

x = log_ay

Mathematical Induction

The things missed in Mathematics!

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!

Binary Operations

The things missed in Mathematics!

1.       I like to travel a lot.

2.       See the map of Bangalore city below (Courtesy: Google Maps).

3.       You can see an intersection of roads circled.

4.       If you are coming from North to South, then this junction is simply a Binary Operation.

5.       The two roads belong to Bangalore City Roads (a Set).

6.       And these two roads combine to give another road that also belong to Bangalore City Roads.

Few formalized representations:

A binary operation on a set A is defined as a function from AxA to A. That is to mean, if you take any two elements from A and apply the binary operation, you will get another element from A.

For example, addition (+) on set of Natural numbers* is a binary operation (addition of two natural numbers gives us another natural number).

But, subtraction (-) on set of Natural numbers is not a binary operation (subtraction of two numbers can be negative, and negative numbers are not Natural numbers.

*Natural numbers are counting numbers, starting from 1,2,3,4,…

Composition of Functions

The things missed in Mathematics!

1.       Here is a process that is very familiar in our day to day life.

2.       It is the process of conversion of same water into ice or vapours.

3.       Here Heating, Refrigeration and Condensation are three different functions. Let us see how these are functions first of all.

4.       Heating changes the water from sets of liquids (water) to sets of gases (vapours) AND Ice to water (sets of solids to sets of liquids) and not vice versa. Hence a function.

5.       Refrigeration changes the water to ice (sets of liquid to sets of gases) and not vice versa. Hence a function.

6.       Condensation changes water vapours to liquid water and not vice versa. Hence a function

7.       Now what will happen if we apply these functions one after another? For example, Lets apply Heating on Ice followed by Heating of water. We get vapours from ice.

8.       Apply Heating of water followed by Condensation. We get water back again.

9.       All these examples are Composition of Functions.

10.   Also note that, the sequence in which these functions are applied does make a difference.

11.   All the sequences may not make sense (e.g. Refrigeration of water and then Condensation).

12.   All the valid sequences may not have an inverse sequence (e.g. Condensation of vapours and then Refrigeration gives us Ice, but Refrigeration of Ice and then Condensation of water is meaningless).

Few formalized representations:

A composition of two functions f and g on a variable x is represented by fog(x) or f(g(x)) and means that first the function g is applied on x and then the function f is applied on the result of g(x).