1.6 — Prefixes and scientific notation

Physics deals with scales that span from the sub-atomic to the astronomical. The mass of an electron is around \( 10^{-30}\ \mathrm{kg} \); the mass of the Sun is around \( 10^{30}\ \mathrm{kg} \) — 60 orders of magnitude apart. Without scientific notation and SI prefixes, every equation would drown in trailing zeros.

Scientific notation — the rule and the reason

Any number can be written as:

\[ N \;=\; a \times 10^{n} \quad\text{where}\quad 1 \le a < 10 \quad\text{and}\quad n \in \mathbb{Z} \]

\( a \) is called the mantissa or coefficient; \( n \) is the exponent. The mantissa is a number between 1.0 and 9.999... (not one, not ten). This constraint is not physically required — you can write 35 × 10^4 — but the SI convention keeps the mantissa in [1, 10) so the representation is canonical (unique).

Examples:

Ordinary form	Scientific notation	With SI prefix
0.000 000 001 5 m	1.5 × 10^-9 m	1.5 nm
299 792 458 m/s	2.998 × 10^8 m/s	≈ 299.8 Mm/s (rare)
9,192,631,770 Hz	9.192 631 770 × 10^9 Hz	9.192 GHz

Engineering notation — like scientific, but with exponents in multiples of 3

Engineering notation is stricter: the exponent must be a multiple of 3 (\( n \in \{..., -6, -3, 0, 3, 6, ...\} \)). Why? Because SI prefixes step every 3 orders of magnitude (kilo, mega, giga, ...). Engineering notation gives each number a one-to-one correspondence with an SI prefix.

Scientific	Engineering	SI
4.5 × 10^-8 m	45 × 10^-9 m	45 nm
1.5 × 10^12 W	1.5 × 10^12 W	1.5 TW
2.7 × 10^5 Pa	270 × 10^3 Pa	270 kPa

SI prefixes — the full table

The official SI prefix set as extended in 2022:

Prefix	Symbol	Factor	Example
quetta	Q	\( 10^{30} \)	Earth's mass ≈ 6 Qg
ronna	R	\( 10^{27} \)	Jupiter's mass ≈ 2 Rg
yotta	Y	\( 10^{24} \)	Moon's mass ≈ 73 Yg
zetta	Z	\( 10^{21} \)	Mass of Earth's oceans ≈ 1.4 Zg
exa	E	\( 10^{18} \)	Transistors produced through 2020 ≈ 13 E
peta	P	\( 10^{15} \)	World's data storage ≈ several hundred PB
tera	T	\( 10^{12} \)	Modern GPU ≈ a few TFLOPS
giga	G	\( 10^{9} \)	CPU frequency ≈ a few GHz
mega	M	\( 10^{6} \)	L2 cache ≈ a few MB
kilo	k	\( 10^{3} \)	m → km
—	—	\( 10^{0} \)	base unit
milli	m	\( 10^{-3} \)	thousandth
micro	μ	\( 10^{-6} \)	biological cell scale
nano	n	\( 10^{-9} \)	atomic and molecular scale
pico	p	\( 10^{-12} \)	time for light to travel 0.3 mm
femto	f	\( 10^{-15} \)	time for one electron orbit in an atom
atto	a	\( 10^{-18} \)	attosecond physics (2023 Nobel Prize)
zepto	z	\( 10^{-21} \)	proton mass ≈ 1.7 zg
yocto	y	\( 10^{-24} \)	electron mass ≈ 0.9 yg
ronto	r	\( 10^{-27} \)	sub-atomic scales
quecto	q	\( 10^{-30} \)	beyond current particle physics

The bold prefixes are the ones you should have memorized cold. ronna/quetta and ronto/quecto were added only in 2022 — driven mainly by astrophysics and digital data volumes.

Note: prefixes are case-sensitive. mm = milli-meter, Mm = mega-meter. Getting the case wrong is a factor of \( 10^9 \) error.

Warnings and common mistakes

1. Don't stack prefixes. microkilogram is not a thing — if you mean mg, write mg. The one exception is kilogram, since kilo is already baked into the SI base unit. mkg (milli-kilogram) is wrong; write g instead.

2. Powers apply to the whole (prefix + unit). mm^2 means \( (\mathrm{mm})^2 = 10^{-6}\ \mathrm{m}^2 \), not \( \mathrm{m}\cdot \mathrm{m}\cdot 10^{-3} \). The exponent applies to the entire expression "prefix + unit."

3. For unknown quantities, do an order-of-magnitude estimate first. Before running a detailed calculation, decide roughly what scale the answer should live at. If you expected \( 10^{-3} \) but got \( 10^{3} \), something is wrong upstream.

Order-of-magnitude estimates

Named after Enrico Fermi: estimate the order of magnitude of an answer without solving it in detail. Useful in physics for:

• Debugging — sanity-check a numerical result
• Intuition — grasp how two things differ by many orders of magnitude
• Fermi problems — how many piano tuners are in Tehran? How far is the Moon from Earth?

Rule: for each variable, guess a value of the form \( 10^n \) without agonizing over accuracy. Combine exponents by multiplication and division.

Example: what is the kinetic energy of a human walking at ordinary speed?

• Human mass ≈ 70 kg ≈ \( 10^2\ \mathrm{kg} \)
• Walking speed ≈ 1 m/s = \( 10^0\ \mathrm{m/s} \)
• \( E = \tfrac{1}{2} m v^2 \approx 10^2 \cdot (10^0)^2 = 10^2\ \mathrm{J} \)

Answer: tens of joules (\( 10^{1} \) to \( 10^{2} \)). The detailed answer is about 35 J. Order of magnitude correct.

What you should be able to do

After this section you should be able to:

• Write any number in scientific and engineering notation
• Recall the 20 SI prefixes (from quecto to quetta) and know they are case-sensitive
• Convert mm^2 → m^2 correctly and avoid common prefix-power mistakes
• Do a Fermi (order-of-magnitude) estimate for an open-ended question
• Feel in your bones the 60 orders of magnitude between the electron mass (\( 10^{-30} \)) and the Sun's mass (\( 10^{30} \))

📚 See also: Halliday Vol 1, Ch 1 §1-7 (scientific notation). 📖 Free reading: BIPM — SI Brochure §5 (official prefix list). 📖 NIST — SI Units and Prefixes.

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