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=n(Tt0)tnt0t = \frac{n(T - t_0)}{t_{n} - t_0}

Where n0n \neq 0

For example if I wanted a conversion formula between KK and CC using n=20n = 20

20(C0K)20K0K=20(C(273.15))253.15(273.15)=20(C+273.15)20\frac {20(C - 0K)}{20K - 0K} = \frac {20(C - (-273.15))}{-253.15 - (-273.15)} = \frac {20(C + 273.15)}{20}

K=C+273.15K = C + 273.15

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

Formula - Square roots

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

a=a±(xa2)2a = a \pm(x-a^2)^2

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

The second formula allows you to refine aa into an even closer approximation of x\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 xx, these are denoted by zz and yy respectively. The 'If' statement asks whether x>a2x > 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 7\sqrt{7}

77494+4=2.6\sqrt{7} \approx \frac {7 - 4}{9 - 4} + \sqrt{4} = 2.6

±+\pm \rightarrow +

a=2.6+(72.62)2=2.6576a = 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 72.6576\sqrt{7}\approx 2.6576 to 4.d.p. The real value for 7\sqrt{7} to 4.d.p is

7=2.6458\sqrt{7} = 2.6458

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

a=2.6576(72.65762)2=2.6537a = 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
6 years, 5 months ago

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