I haven’t yet watched your videos, Robert. But I’ve been playing around with this which might be useful.

I, developing a modified decimal system to represent numbers. Instead of confining each power of ten to a single column with a single digit, they are marked by commas which allows one overflows and also -ve numbers. Breaking free of the tyranny of the columns allows us to keep partial results in their original form as long as we wish and it allows us different ways to represent the same decimal number which can preserve structure that gets flattened by the column system.

This allows the same number to be represented different ways while still retaining a decimal form. In particular we can work with numbers in compliment form.

97 = 9,7 = 1,0,-3 or 10,-3 we are quite happy holding 10 tens in the second position

97^{2} =(10,-3)^{2} = 1,0,0,-6,9 or = 100,-6,9

100,-6,9 =( take 1 from the 100 in the 3rd position 10+(6)=4) 99,49

The answer can be represented either way 9949 == 100,-6,9

Writing 97 as 1,0,-3 makes the calculation easier because there are hardly any carries.

(100+d) where |d| <=5 and d may be -ve : (1,0,d)^{2} = 1,0, 2d, ( d^{2})

d^{2} <= 25 and |2d| <= 10

105^{2} = 1,0,10,25 = 1 which normalizes to 1,1,2,5 = 1125

Two digit numbers with digits <= 5 also make for easy work

10a+b = a,b |a|&|b| <= 5 may be -ve

a,b ^{2} *(commas have a higher precedence than exponentiation)* = (a,b)^{2}

a,b^{2} = (a^{2}), 2ab, (b^{2})

again a^{2}<25 & 2ab<50

43^{2} = 16,24,9 = 18,29 = 1829

57^{2} = 5,7^{2} = 1,-4,-3^{2}

keeping the 43 as a single number 1,-43^{2} = 1,0,-86,0,(43^{2})

1,0,-86,(43^{2}) = 24,0,1849 = 32,0,49 = 32,4,9 = 3249

and it’s clear why both 43^{2} & 57^{2} have the same last digits.

76^{2} & 24^{2} should have the same last digits

24 = 576

76^{2} = 1,0,-24^{2}= 1,0,-48,0,(24^{2}) = 52,0,576

52,0,576 = 5776

With mixed digits:

27^{2} = 3,-3^{2} = 9,-18,9 = 72,9 = 729

73^{2} = 1,-27^{2} = 1,-3,3^{2}

now we have all digits <=5 , keep the last two digits as a single number -3,3

1,-3,3^{2} = 10, (-6,6), 0,(-3,3^{2})

= 46, 0, (-3,3^{2} )

Here it’s easiest to recognize that -3,3 = -27 & that -27^{2} = +27^{2} and we

have 46,729 = 5329

My meds leave my mind a bit cloudy at the moment so I need to check this. But I think this may offer a way to see patterns before they get normalized and flattened into single digit columns.