I will be presenting a work of mine which I developed two years ago, it is regarding the square of integer numbers(will upgrade it later when I have time).
I'm certain that you're all acquainted with the well-known formula: \((a+b)^2\) = \(a^2\) + 2ab + \(b^2\) , and I'm also sure that you all know the proof to this simple formula, however,two years ago, I thought, what about abandoning the traditional math style for a day or two? and this resulted in the following formula, this might not be any important at all, but I feel like sharing it won't hurt.
I came up with two formulas but because it's been a long time since that time, I kinda need time to recap what the second formula is since my worksheet is messy.
Here's the formula which I reached regarding the square of a number(I will be representing the number as X) using a different style which I will introduce soon:
\(X^2\) =\( [29+X+ 4 \times Z+ 6 \times W + Y \times S ] \times 10\) + C
I know I know, there are a lot of constants, but it's only a general formula and they will be substituted soon. This formula was built using the relation between the numbers and their squares only, to be more specific The difference between the integer of " X^2/10", minus the number X.
This formula works for all the integer numbers and special cases can be derived from it, furthermore, this formula divides the numbers from 0 to 9 into three groups:
Group no.1: from 4 to 7.
Group no.2: 8 and 9.
Group no.3: from 0 to 3. These groups were not randomly picked, they depend on the properties of the numbers(I selected them selectively).In addition to that, this formula relates to a sequence with a constant change in differences between bounds.
I will now be introducing the variable "A" which is equal to the last digit of the number X, and the variable "B" refers to the other digits, for example:Given the number 1234, A=4, B=123.
The formula depends on the definition: "X^2 - B " is a bound of a sequence with a constant change in differences between bounds, I can prove you this definition later, just keep it in mind for now, you probably won't find it elsewhere, excuse me if I'm not clear enough, it's quite hard to explain this, it's my first time participating on the internet in such a forum.
I will explain the variables Z,W,S,Y and C which will be substituted later with A and B only:
Z: It's the ratio of increasing with relation to the number of complete cycles from 4 to 7, a cycle from 4 to 7 is considered complete if the sequence mentioned above passed all the bounds which relate to numbers with A from the first group from each 10 numbers(0 to 9).
W: It's the ratio of increasing with relation to the number of complete cycles from 0 to 3 and from 8 to 9, a cycle like that is considered complete if the sequence mentioned above passed all the bounds which relate to numbers with A from the second and third group from each 10 numbers(0 to 9), note: it has to pass them all in order to be counted as a cycle.
Y: It's the current increment(an increment is the difference between a bound and the next bound) in the sequence.
S: It's the number of bounds in the incomplete cycle(incomplete means that the sequence hasn't passed them all, i.e: if the sequence passed 3 bounds from the 4 to 7 cycle: that means it passed the bounds 4,5 and 6.
C=A^2 (mod 10).
According to the definitions I defined above, I derived the following formulas from the original formula:
For the first group the formula is: \(X^2\)=\([29+X+(B-2)\times(10 B+16) +2\times B\times (A-3) ]\times 10 + A^2 (mod 10)\)
For the second group the formula is: \(X^2\)=\([29+X+(B+2)\times (10B-16) +(2 B +1) \times (A-7) ]\times 10 + A^2 (mod 10)\)
For the third group the formula is: \(X^2\)=\([29+X+(B+1)\times (10B-26) +(2 B-1) \times (A+3) ]\times 10 + A^2 (mod 10)\)
To determine which group X belongs to, look at the value of A and match it to the proper group, note that A ranges from 0 to 9.
The sequence I talked about is: ,,,,11,14,17,20,23,26,29,33,37,41,45,50,55,60,65,70,75,81,87,93,99,106......
Note that the differences between bounds can be arranged as following:.... ,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6 ......, I hope you have an idea about what I'm talking about now.