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I'm working through "An introduction to error analysis" by John R. Taylor. It states,

If the quantity $x$ is measured with uncertainty $\delta x$ and is used to compute the product $$q=Bx,$$ where $B$ has no uncertainty, then the uncertainty in $q$ is just $|B|$ times that in $x$, $$\delta q=|B|\delta x$$

Taylor gives the following example of the application of this rule:

Suppose you measure the diamter of a circle as $$d=5.0 \pm 0.1 \ \text{cm}$$ and use this value to calculate the circumference $c=\pi d$. What is your answer, with its uncertainty?

The answer is $$3.141...\times (5.0 \pm 0.1 \ \text{cm}) = 15.707... \pm 0.3141...\ \text{cm}$$ Thus, the final answer (properly rounded) is $$(\text{circumference of circle})=15.7 \pm 0.3 \ \text{cm}$$

However, Taylor then gives the following explanation to an example problem further along in the book that doesn't seem to follow this rule:

Suppose that we measure $g$, the acceleration of gravity, using a simple pendulum. The period of such a pendulum is well known to be $T=2\pi \sqrt{l/g}$, where $l$ is the length of the pendulum. Thus, if $l$ and $T$ are measured, we can find $g$ as $$g=4\pi ^2l/T^2$$ This result gives $g$ as the product or quotient of three factors, $4\pi ^2$, $l$, and $T^2$. If the various uncertainties are independent and random, the fractional uncertainty in our answer is just the quadratic sum of the fractional uncertainties in these factors. The factor $4\pi ^2$ has no uncertainty (emphasis is mine), and the fractional uncertainty in $T^2$ is twice that in $T$: $$\frac{\delta (T^2)}{T^2}=2\frac{\delta T}{T}.$$ Thus, the fractional uncertainty in our answer for $g$ will be $$\frac{\delta g}{g}=\sqrt{\left(\frac{\delta l}{l}\right)^2+\left(2\frac{\delta T}{T}\right)^2}.$$

But shouldn't, based on the previously defined "error in measured quantity times exact number" rule, the uncertainty be calculated as?: $$\frac{\delta g}{g}=4\pi ^2 \times \sqrt{\left(\frac{\delta l}{l}\right)^2+\left(2\frac{\delta T}{T}\right)^2}.$$

If "no", then what are the circumstances when the "error in measured quantity times exact number" rule be used and/or not used?

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    $\begingroup$ There are factors of $4\pi^2$ in both $\delta g$ and $g$ (when expressed in terms of $l$ and $T$), so they cancel in the fraction. $\endgroup$
    – whuber
    Commented Jun 11, 2019 at 16:25
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    $\begingroup$ @whuber Ah, OK. So this is a matter of absolute uncertainty versus fractional uncertainty. Thanks for your clarification! $\endgroup$
    – Armadillo
    Commented Jun 11, 2019 at 17:06

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