There are three observations when we think of numbers
- Number lines have thickness (series of squares)
- When we count with fingers we encounter gaps. And it is because of gaps that fingers are distinct. Similarly, trees and gaps between trees.
- In every day encounters we say 4 cows are going inside shed; 2 cows are coming out of shed. In other words positives and negatives.
Complex numbers were invented to take care of all these and more. Euler was not aware of complex numbers but belonged to an era earlier to complex number invention.
#PythaShastri had written about the e series. The fingers and the gaps. #PythaShastri had also written about 1,2,3,4,5,9,10. The hour glass. The folding of paper. The visible sides being 1,2,3,4,5 and we halve by going from 9 to 10. So 1,2,3,4,5,9,10. But 3,2 to 4,5 was not discussed in full.
Here, I am discussing 3,2 to 4, 5.
3,2 to 4,5 if included with 1 make a complete hand. 1,2,3,4,5. Also 2 to 4 is doubling. 2 to 4 is squared. And 1 to 4 is four times. The reverse is one-fourths.
If we take the 2 to 4 squared and 1 to 4 is one-fourth, then we get square series of natural numbers..
Consider squares of even numbers,
4, 16, 36, 64, 100, 144, . . . . .
The one fourth of above is
1, 4, 9, 16 ,25 ,36, . . . . .
or
1 ², 2 ², 3 ², 4 ², 5 ², 6 ², . . . . .
This demonstrates that 3,2 to 4, 5 is a natural phenomenon of square series.
