I don't know much about math, that's why I came here to ask for help. Recently I stumbled in a problem where I have to divide X by Y with specific condition that forbids from doing it directly.
To be exact, while the rules allow to perform any action to X or Y with the exception of P.S; the only action that can be performed between X and Y is addition. The only way I see to do it is an equation, that I'm not even sure exists. Example for multiplication is:
$$X*Y=\frac{(X+Y)^2-X^2-Y^2}{2}$$
If there's a way then I'll be grateful.
P.S. In case that square root cannot be calculated by a formula it cannot be used, also fractions aren't allowed, since they just get cut, while negative numbers are allowed.
If you are allowed to take reciprocals (one divided by a number), then you can substitute $Y = 1/Y$ and you will get
$$\frac{X}{Y} = \frac 1 2 \left(\left(X+\frac{1}{Y}\right)^2 - X^2 - \frac{1}{Y^2}\right)$$
If this is not allowed, you can for example use some iterative method with companion matrices for the polynomial equation $$p(q) = X-Yq=0$$which has root $$q = \frac{X}{Y}$$
You don't even need to do companion matrices, it will suffice to hunt for "multiplicative friends of basis".
For example, using base $2$ and we want to divide $5$ by $3$ and we make the approximation $3^2 = 9 \approx 8$
Then $5/3 = \text{multiply 3 both up and down} = 15/9\approx 15/8 = 1.111$ in binary.
Another $5/3 = \text{multiply 5 both up and down} = 25/15 \approx 25/16 = 1+9/16 = 1.1001$ in binary.
Now what might be more natural for us humans is to use base 10.
Mul with 3 up and down: $5/3 = 15/9 \approx 15/10 = 1.5$
Mul with 33 up and down : $5/3 = 5\cdot 33/99 \approx 165/100 = 1.65$