Random formulas

Today I wanted to share with you some random formulas that I found \ made myself to find solutions to problems (or just because I can). So let's get started.

Formula - Personal temperature conversion

$t = \frac{n(T - t_0)}{t_{n} - t_0}$

Where $n \neq 0$

For example if I wanted a conversion formula between $K$ and $C$ using $n = 20$

$\frac {20(C - 0K)}{20K - 0K} = \frac {20(C - (-273.15))}{-253.15 - (-273.15)} = \frac {20(C + 273.15)}{20}$

$K = C + 273.15$

Feel free to make your own temperature scale and use this formula.

Formula - Square roots

$\sqrt {x} \approx a = \frac {x - z}{y - z} + \sqrt {z}$

$a = a \pm(x-a^2)^2$

This one is a little bit more complicated, firstly $a$ is approximately $\sqrt{x}$ but it is also equal to an equation which can be used to give a close (works better with bigger values of $x$) value to $a$.

The second formula allows you to refine $a$ into an even closer approximation of $\sqrt{x}$. This formula also includes an 'If' statement located at the $\pm$ where it is either $+$ or $-$ depending on the outcome.

To use these equations you need to know the closest square before and after $x$, these are denoted by $z$ and $y$ respectively. The 'If' statement asks whether $x > a^2$ is true or not, if it is then the $\pm$ becomes a $+$, if it's false then the $\pm$ becomes a $-$.

For example let's try and find $\sqrt{7}$

$\sqrt{7} \approx \frac {7 - 4}{9 - 4} + \sqrt{4} = 2.6$

$\pm \rightarrow +$

$a = 2.6 + (7 - 2.6^2)^2 = 2.6576$

The second equation can be used endlessly but we're only going to use it once. So according to our formula $\sqrt{7}\approx 2.6576$ to 4.d.p. The real value for $\sqrt{7}$ to 4.d.p is

$\sqrt{7} = 2.6458$

So our equations got close with just one use, so if we use it again

$a = 2.6576 - (7 - 2.6576^2)^2 = 2.6537$

So we are getting closer, just slowly.

That's it for now but I will be updating this when I come up with more formulas.