\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\)
Appendix A: Miscellany
5. (Optional)

Formal Scientific Notation

If you've learned about scientific notation before, you may have been taught that there should only be one nonzero number in front of the decimal point in the mantissa, so something like $52.7\times 10^3$ would be incorrect. Sometimes this "formal notation" is useful, especially when making comparisons: when two numbers are in formal notation, and one number has the larger exponent, then that number is definitely the bigger one. On the other hand, $123.\times 10^1 > 4.1\times 10^2$ even though the exponent on the second number is bigger, because $1230>410$.

However, formal notation is not always necessary, or even a good idea. For example, when you add two numbers in scientific notation by hand, you must first match the exponents: $$\begin{align} &2\times 10^3 + 3\times 10^4\\ &=0.2\times 10^4 + 3.2\times 10^4\\ &=3.2\times 10^4\\ \end{align}$$

An example where
formal notation
might serve to confuse:

PlanetDiameter
in meters
(formal)		Diameter
in meters
(consistent)
Mercury \(4.9 \times 10^6 \)\(4.9 \times 10^6\)
Venus \(1.2 \times 10^7 \)\(12 \times 10^6\)
Earth \(1.3 \times 10^7 \)\(13 \times 10^6\)
Mars \(6.8 \times 10^6 \)\(6.8 \times 10^6\)
Jupiter \(1.4 \times 10^8 \)\(140 \times 10^6\)
Saturn \(1.2 \times 10^8 \)\(120 \times 10^6\)
Uranus \(5.1 \times 10^7 \)\(51 \times 10^6\)
Neptune \(4.9 \times 10^7 \)\(49 \times 10^6\)

Second, if you have a bunch of numbers which are close together, then formal notation may serve to confuse, especially those people who aren't as comfortable with it. For example, the table on the right shows the diameters of the major planets in our solar system: the first column is in formal scientific notation, while the second gives all the diameters with the same exponent ($\times 10^6$). I personally find the second column easier to read and compare with each other.

Engineering notation is a variation of scientific notation where the exponent must be a multiple of 3: thus we would write $3\times 10^4$ as $30\times 10^3$, $4\times10^{-4}$ as $0.4\times 10^{-3}$, and so forth. Your calculator may have an "engineering" mode on it in addition to scientific notation mode. This system is closely related to Metric Prefixes.