Observations in Maths

5 POINTS :

POINT 1 :

The minus sign is not an EQUIVALENT counterpart (opposite) to the plus sign. Why?

1. The minus sign is both an operation as well as a signing operator (sign) to a number. That is, it is used in (8-3) as well as to portray (-5). 

2. Suppose you have +(+(+(+(+(+(+2))))), it doesnt matter how many pluses are there. The final result is a +2. But if in place  of the plus sign, there are minus signs like in that case, the final answer depends upon the number of minus signs (the parity of it).

POINT 2 : 

Why do we want answers in terms of SINGLE numbers? Why cant I say something is 8+4+3 - thats it. The problem is that you wont know if different representations are the same (i.e. if 7+ 5 + 3 is the same), since  a number can be divided into (even the same number of) parts in various ways.

Anyways, what lesson does this teach us? It is not possible to identify the whole from the parts unless you actually join them! (just like jigsaw puzzles).

Another less strong point is that - if 2 wholes are the same, they (the 2 entities) are the same. But if 2 sets of parts are the same, it doesnt mean that the individual parts are correspondingly the same. 

POINT 3  -

Sense of a number - 

How do you “understand” a number given to you, say 5?

The very meaning of a number is size - indicates HOW MUCH? So the very cognition or understanding of a number is getting a sense of its magnitude, its size.

A number is a size-indicator. 

So, cognizing a number is cognizing its size, which is, comparing it with anything whose (i.e. whose size) you have an understanding of.

You dont cognize anything isolatedly; you compare it with whats in your mind. 

This I guess is where the fundamental genesis of case-based reasoning is. Speaking in a sense, you wouldnt have understood the first thing you saw in life!

POINT 4 - 

B + 5 = 7

Something plus 5 is equal to 7. 

Something is a number. How? Something is added to 5. YOU CAN ONLY ADD NUMBERS, TO NUMBERS. So B i.e. something (here), cannot be a movie or a poem. 

Linguistic commonsensical reversal - Going up takes you close to the cupboard. So, going down takes you away from the cupboard (reversed 2 concepts). 

If something plus 5 is 7, then 7 minus 5 is that something (reversed the order and inverted 1 concept)! Hence B = 2 by mere language! This is just like - The fan is above the floor. Reversal of order and inversion of one concept preserves the truth - The floor is below the fan.

POINT 5 -

Vectors - 

AB + BC = AC

An alternate proof of the triangle rule of addition of vectors - 

Displacement is a vector. The third vector (AC) is clearly the resultant displacement of the 2 steps of earlier displacements - from point A to B (indicated by vector AB) and from point B to C (indicated by vector BC)!