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How does the function Around[] work?

Posted 1 month ago
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From the Wolfram Blog (Computing Exact Uncertainties—Physical Constants in the Current and in the New SI):

The arithmetic model of Around[] follows the GUM (Guide to the Expression of Uncertainty in Measurement)—not to be confused with Leibniz’s Plus-Minus-calculus.

I do not get what this has to do with the Plus-Minus-Calculus, perhaps the author meant interval arithmetic? Anyway,

Around[1.2, 0.1]^2

leads to $1.44\pm0.24$ although $[1.2-0.1,1.2+0.1]^2=[1.45-0.24,1.45+0.24]$, so why do I get the value 1.44 instead of 1.45?

Also, does anyone know how CODATA calculates with uncertainties? I tried to reproduce the CODATA 2018 values in SI units of the base Planck units, and indeed, it all fits except for the Planck time.

Sqrt[6.62607015/(2 * Pi) * 10^(-34) * Around[6.67430, 0.00015] * 10^(-11)/299792458^5]

gives $(5.39125\pm 0.00006)\times 10^{-44}$ while the CODATA 2018 value is $5.391\,247(60)\times10^{-44}\,\mathrm s$.

2 Replies
Posted 1 month ago

At https://reference.wolfram.com/language/ref/Around.html under 'Properties & Relations' there is just the same operation you are asking about:

Around[10, 1]^2

According to this: (x+y)^2=x^2+2xy+y^2,

computing (1.2+0.1)^2=1.2^2+21.20.1+0.1^2

Let 0.1=delta, then ( 1.2 + 0.1 )^2= 1.2^2 + 2 * 1.2 * delta + delta^2

Around makes a first order series approximation, i.e. the highest delta power is 1, and you can check that 2 * 1.2 * 0.1=0.24, no 0.1^2 added to it.

I Hope this helps.

Posted 1 month ago
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