Mathematics

Problem

Physics spans the very small and the very large. Consider the following extreme length scales: • Planck length

\ell_{P}=0.000000000000000000000000000000000016\,\text{m}

• Radius of the observable universe

R_{u}=4.4\times10^{26}\,\text{m}

a. (i) Write the Planck length in scientific notation with one non-zero digit before the decimal point. b. (ii) State the order of magnitude of

\ell_{P}

and of

R_{u}

. c. (iii) Determine the number of orders of magnitude between

\ell_{P}

and

R_{u}

. d. (iv) Calculate the ratio

\dfrac{R_{u}}{\ell_{P}}

and express the answer as

a\times10^{n}

with

1\le a<10

. e. (v) On the basis of your result in part (iv), explain briefly what the comparison reveals about the range of length scales encountered in physics.

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