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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 ago

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