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# 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.

Note by Jack Rawlin
2 years, 6 months ago