Units, constants, and estimation

Why this matters

In physical science, an answer that is off by a factor of a thousand often looks just as convincing as the right one. A wavelength typed in nanometres instead of metres, or grams instead of kilograms, silently wrecks a calculation. Two habits catch most of these errors before they reach a lab notebook or a paper: carry units through every step, and estimate the answer before you calculate it.

The SI system and its constants

Science uses the International System of Units (SIThe International System of Units, with seven base units such as the metre, kilogram and second.). It has seven base units: the metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol) and candela (cd). Every other unit is built from them. A newton is kg·m/s², a joule is a newton-metre, and a watt is a joule per second.

Since 2019, the base units are defined by fixing the exact values of physical constants. For example:

Look values up from the official sources instead of from memory: the BIPM SI pages at bipm.org and the NIST constants database at physics.nist.gov. Checked on 2026-10-23.

Carry units through the calculation

Treat units like algebra symbols. They multiply, divide and cancel. If the units of your final answer are wrong, the formula or a conversion is wrong.

Worked example: the energy of a green laser photon. A photon's energy is E = hc / λ, where λ is the wavelength. A green laser pointer emits at 532 nm.

1. Convert the wavelength to SI: 532 nm = 532 × 10⁻⁹ m = 5.32 × 10⁻⁷ m.
2. Multiply the constants: hc = (6.626 × 10⁻³⁴ J·s)(2.998 × 10⁸ m/s) = 1.986 × 10⁻²⁵ J·m. The seconds cancel.
3. Divide: E = 1.986 × 10⁻²⁵ J·m / 5.32 × 10⁻⁷ m = 3.73 × 10⁻¹⁹ J. The metres cancel, leaving joules, which is an energy. Good.
4. Convert to electronvoltsAn energy unit used for photons and electrons; 1 eV is about 1.602 x 10^-19 J., the unit used for photons and electrons: 1 eV = 1.602 × 10⁻¹⁹ J, so E = 3.73 / 1.602 = 2.33 eV.

If you had left λ in nanometres, you would get an energy a billion times too small, still with a plausible-looking number of digits.

Estimate first

An order-of-magnitude estimateA rough answer within about a factor of ten, made with rounded numbers to check a detailed result. gets the answer to within a factor of about ten using rounded numbers. It takes seconds and tells you whether a detailed result is believable.

For photons there is a famous shortcut: hc is about 1240 eV·nm. So E (in eV) is roughly 1240 divided by the wavelength in nm. For 532 nm: 1240 / 532 ≈ 2.3 eV. That matches the detailed calculation. Since visible light runs from about 380 to 750 nm, visible photons carry roughly 1.7 to 3.3 eV, so any visible-light answer outside about 1 to 4 eV is almost certainly a mistake.

Another estimate: how many atoms are in a 1 cm³ copper cube? Copper's density is about 9 g/cm³ and its molar mass about 64 g/mol. So 9 g is about 9/64 ≈ 0.14 mol, and 0.14 × 6 × 10²³ ≈ 8 × 10²² atoms. Roughly 10²³ atoms, which is the right scale for a small piece of anything solid.

Significant figures and sensible precision

Report as many digits as your inputs justify. If the wavelength is known to three figures (532 nm), report 2.33 eV, not 2.330496 eV. Exact constants do not limit your precision; measured inputs do.

Common mistakes

• Mixing prefixes. nm, µm and mm differ by factors of 1000. Convert to base SI units before multiplying.
• Dropping units halfway. Write them on every line until the end.
• Using a remembered constant with the wrong exponent. Look it up, and check the estimate.
• Trusting a calculator over an estimate. When they disagree by more than a factor of ten, recheck the calculation.

Quick recap

• Seven SI base units; since 2019 they are defined by exact constants such as c, h and e.
• Units multiply and cancel like algebra; wrong final units mean a wrong calculation.
• Estimate first; for photons, E in eV ≈ 1240 / λ in nm.

Try it

Open the Do tab and finish a short Python function that turns a wavelength into a photon energy, then compare it with your 1240 / λ estimate.