I was investigating some methods of computing square roots by hand and I found a certain script which presents the following method of computing:
NOTE: the following notation represents positional notation of a number, e.g. $$\overline{ab} = 10a + b$$
The algorithm is presented with an example of a square of a 4-digit number:
$$\overline{abcd}^2 = (10^3a + 10^2b + 10c + d)$$ $$= a^210^6 + b^210^4 +c^210^2 + d^2 + 2ab10^5 + 2ac10^4 +2ad10^3 + 2bc10^3 + 2bd10^2 +2cd10$$ $$= a^2 \cdot 10^6 + b \cdot 10^4 \cdot (2a \cdot 10 + b) + c \cdot 10^2(2 \cdot (10a + b) \cdot 10 + c) + d(2 \cdot (a \cdot 10^2 + b \cdot 10 + c) \cdot 10 + d)$$ $$= a^2 \cdot 10^6 + \overline{(2a)b} \cdot b \cdot 10^4 + \overline{(2\overline{ab})c} \cdot c \cdot 10^2 + \overline{(2\overline{abc})d} \cdot d$$
The final equality is used in the algorithm. Now it is shown on the example of $\sqrt{14175225}$:
$$14175225 = 9 \cdot 10^6 + 5175225$$ $$= 9 \cdot 10^6 + 67 \cdot 7 \cdot 10^4 + 485225$$ $$= 9 \cdot 10^6 + 67 \cdot 7 \cdot 10^4 + 746 \cdot 6 \cdot 10^2 + 37625$$ $$= 9 \cdot 10^6 + 67 \cdot 7 \cdot 10^4 + 746 \cdot 6 \cdot 10^2 + 7525 \cdot 5$$ $$= 3765^2$$
So the process basically finds the numbers $a,b,c,d$ so they satisfy the final equality. Now there is no other explanation for this, for example the author just says that $a = 3$ with no further logic behind it. Why couldn't it be $a=2$ for example? How can one know at the start that $a = 3$ and then continue the process by simply knowing it will satisfy the equation? Also how can we know that a number such as $14175225$ will have a 4-digit square root?
None of these things are explained in the script so I would like an explanation if someone has any insight into this.
Pairing from the implied decimal point, the number is "14 17 52 25".
The value of "a" is chosen as the largest digit whose square isn't greater than 14.
4×4 is 16, which is too big, but 3×3 is 9, which fits, so a = 3.
If someone simply asked you what the square root of 14 million is, you'd likely think that 4000 is slightly too big and 3000 is a lot too small, so you'd reply something like "in the high 3000s".