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− | Tables AND math!
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− | From [[2295: Garbage Math]]:
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− | {| class="wikitable"
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− | !Formula as shown
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− | !Resulting uncertainty
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− | !Explanation
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− | |Precise number + Precise number = Slightly less precise number
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− | |<math>\mathop\sigma(X+Y)=\sqrt{\mathop\sigma(X)^2+\mathop\sigma(Y)^2}</math>
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− | |{{Nowrap|If we know absolute error bars, then adding two precise numbers will}} at worst add the sizes of the two error bars. For example, if our precise numbers are 1 (±10<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our sum is 2 (±2·10<sup>-6</sup>). It is possible to lose a lot of relative precision, if the resultant sum is close to zero as a result of adding a number to its approximate negation, a phenomenon known as {{w|catastrophic cancellation}}. Therefore, both of the numbers must be positive for the stated assertion to be true.
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− | |Precise number × Precise number = Slightly less precise number
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− | |<math>\mathop\sigma(X\times Y)\cong</math><br><br><math>\sqrt{\mathop\sigma(X)\times Y^2+\mathop\sigma(Y)\times X^2}</math>
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− | |Here, instead of absolute error, relative error will be added. For example, if our precise numbers are 1 (±10<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our product is 1 (±2·10<sup>-6</sup>).
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− | |Precise number + Garbage = Garbage
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− | |<math>\mathop\sigma(X+Y)=\sqrt{\mathop\sigma(X)^2+\mathop\sigma(Y)^2}</math>
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− | |If one of the numbers has a high absolute error, and the numbers being added are of comparable size, then this error will be propagated to the sum.
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− | |Precise number × Garbage = Garbage
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− | |<math>\mathop\sigma(X\times Y)\cong</math><br><br><math>\sqrt{\mathop\sigma(X)\times Y^2+\mathop\sigma(Y)\times X^2}</math>
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− | |Likewise, if one of the numbers has a high relative error, then this error will be propagated to the product. Here, this is independent of the sizes of the numbers.
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− | From [[1047: Approximations]]
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− | {| class="wikitable"
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− | !align="center"|Thing to be approximated:
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− | !align="center"|Formula proposed
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− | !align="center"|Resulting approximate value
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− | !align="center"|Correct value
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− | !align="center"|Discussion
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− | |align="center"|Avogadro's number
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− | |align="center"|<math>69^{\pi^\sqrt{5}}</math>
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− | |align="center"|6.02191201246329 × 10<sup>23</sup>
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− | |align="center"|6.02214129 × 10<sup>23</sup>
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− | |align="left"|Also called a mole for shorthand, {{w|Avogadro's number}} is (roughly) the number of individual atoms in 12 grams of pure carbon. Used in basically every application of chemistry. In 2019 the constant was redefined to 6.02214076 × 10<sup>23</sup>, making the Approximation slightly more correct.
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− | |align="center"|Gravitational constant ''G''
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− | |align="center"|<math>\frac {1} {e ^ {(\pi-1)^{(\pi+1)}}}</math>
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− | |align="center"|6.6736110685 × 10<sup>−11</sup>
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− | |align="center"|6.67385 × 10<sup>−11</sup>
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− | |align="left"|The universal {{w|gravitational constant}} G is equal to ''Fr''<sup>2</sup>/''Mm'', where ''F'' is the gravitational force between two objects, ''r'' is the distance between them, and ''M'' and ''m'' are their masses.
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− | |align="center"|''R'' (gas constant)
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− | |align="center"|<math>(e + 1) \sqrt 5</math>
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− | |align="center"|8.3143309279
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− | |align="center"|8.3144622
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− | |align="left"|The {{w|gas constant}} relates energy to temperature in physics, as well as a gas's volume, pressure, temperature and {{w|mole (unit)|molar amount}} (hence the name).
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− | |align="center"|Proton–electron mass ratio
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− | |align="center"|<math>6 \pi^5</math>
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− | |align="center"|1836.1181087117
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− | |align="center"|1836.15267246
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− | |align="left"| The {{w|proton-to-electron mass ratio}} is the ratio between the rest mass of the proton divided by the rest mass of the electron.
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| {{Comic discussion}} | | {{Comic discussion}} |