Yesterday I was working on a puzzle - How to make a 1000 using 8 exactly eight times ? So far I discovered 4 different solutions to this and I believe there are more ways to do it. However, while working on this puzzle, I invented a new way of handling numbers. And a new operator in mathematics.
We know about Addition (+), Subtraction (-), Multiplication (* or X) and Division (/). These are four basic operators in mathematics. Then there are others like Factorial (!), Percentage (%), Power (^) etc. What if we have one special operator that just glues two numbers together side-by-side ? This will be a completely new operation on numbers. And this is very much different from simple Addition. Let me give you an example first.
Example
Say A = 1234 and B = 56, then C = A # B = (1234) # (56) = 123456 !
Or, say X = 123.45 and Y = 67, then Z = X # Y = (123.45) # (67) = 123.4567 !! How's that ?
Another, say P = 123 and Q = 4.5678, then R = P # Q = (123) # (4.5678) = 1234.5678 !!!
I have used the symbol "#" to represent this new mathematical operation. Let us give it a name too. We will call it 'Concatenation'. Similarly there must be an opposite operation called 'Truncation'.
We have to use another symbol for Truncation... we will use '~#' which is a combination of 2 symbols '~' and '#'. Logically, Truncation is a negation of the Concatenation '#' operation.
Example of Truncation could be:
Say A = 123456 and B = 56, then C = A ~# B = (123456) ~# (56) = 1234
Or, say X = 123.4567 and Y = 67, then Z = X ~# Y = (123.4567) ~# (67) = 123.45 !.
I was checking in Wikipedia and I saw that some literature is already there regarding Concatenation of integer numbers. But this is considered mainly as part of Recreational Mathematics, not under mainstream. Moreover, there is no concept of Truncation yet in place. This new angle of numerical operations seems to be under study but there is still not sufficient work done here. We have scopes to improve this further. I am particularly interested to apply this operation for complex calculations where Concatenation and Truncation can help in reducing the calculation effort to a great extent.
My entire thought on this is much more vast. I am thinking about some formula like below:
Let us not forget that this is only a "tentative" formula. It is not yet established and we should not expect it to be correct. We need to do more research to establish such a formula in real:
Say (Tentative Formula), f(A # B) = f(A) # f(B) + (A/B)*10
If we can invent such a formula, our calculation effort for a long number like 123456789 will be reduced when we split it into 2 smaller numbers like 12345 and 6789. In case of fractions, the possible error will be reduced when we split a fraction like 1.23456789 into 2 numbers like 1.2 and 3456789. That will be the usefulness of this whole concept of Concatenation and Truncation.
Once this theory is established, we can extend it even further for more than 2 numbers. Examples:
A # B # C # D = ABCD like (12 # 34 # 56 # 78) = 12345678
Or, X # Y ~# Z = M like (1234567 # 89 ~# 789) = 123456
Limitations
Probably the biggest limitation here is that everything is now at imaginary level :) Nevertheless, the beauty of the proposed numerical operation is very attractive. One day someone will probably establish this in a firm way. One known limitation of our concept is, we cannot Concatenate two or more fractions together - because we cannot have more than one decimal points in a number. Another limitation could be, when we try to Truncate a bigger number from a smaller one. This is an erroneous situation and hence we must have some mechanism to handle it. In algebraic calculations, we can never be sure about the absolute values of two variates - so we should have mathematical provisions (like we have Negative Numbers in case of simple addition and subtraction) to handle Truncation of a bigger number from a smaller one.
All the best for the readers who wish to think more on this and do a fundamental research. Thank you !
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