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^2correctly 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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