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IMAT Physics · Measurement and Units

SI Units, Prefixes and Significant Figures

Every quantity is a number times a unit; SI builds all units from seven base units, and prefixes and powers of ten keep the numbers readable.

All Measurement and Units notes

Why this matters

The 2018 paper asked students to put three tiny lengths in order when each was written with a different prefix. Every other physics question also needs SI units and powers of ten handled without slips.

Concept 1 of 3: SI base units and the common derived units

Scientists agreed on seven base units, and every other unit is built by multiplying and dividing them. A newton is not a new idea: it is the force that gives 1 kg an acceleration of 1 m/s², so it is a kilogram metre per second squared. Knowing what each derived unit is made of lets you check any formula.

Definition

The SI base units are the metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol) and candela (cd).

  • Every other SI unit is a derived unit, a product or quotient of base units.
  • The base unit of mass is the kilogram, not the gram.
  • Temperature in SI is in kelvin: T=t+273T = t + 273, with TT in kelvin and tt in degrees Celsius.
QuantityUnitSymbolIn SI base units
Lengthmetrembase unit
Masskilogramkgbase unit
Timesecondsbase unit
Electric currentampereAbase unit
TemperaturekelvinKbase unit
Amount of substancemolemolbase unit
ForcenewtonNkg m s−2\text{kg m s}^{-2}
Energy, workjouleJ = N mkg m2 s−2\text{kg m}^2\text{ s}^{-2}
PowerwattW = J/skg m2 s−3\text{kg m}^2\text{ s}^{-3}
PressurepascalPa = N/m²kg m−1 s−2\text{kg m}^{-1}\text{ s}^{-2}
ChargecoulombC = A sA s\text{A s}
Potential differencevoltV = J/Ckg m2 s−3 A−1\text{kg m}^2\text{ s}^{-3}\text{ A}^{-1}
ResistanceohmΩ = V/Akg m2 s−3 A−2\text{kg m}^2\text{ s}^{-3}\text{ A}^{-2}
FrequencyhertzHzs−1\text{s}^{-1}
The last column is what you reach when you break a derived unit all the way down.
Practice this conceptself-check · 4 quick reps

The base unit of mass is the kilogram

The kilogram is the only base unit with a prefix in its name. A density in g/cm³ or an energy worked out with a mass in grams is not in SI, and an answer computed that way is out by a factor of 1000.

Concept 2 of 3: Prefixes, powers of ten and scientific notation

Physics deals with sizes from atoms to planets, so we write numbers as a value between 1 and 10 times a power of ten. A prefix is just a short name for one of those powers. To compare two quantities, first write both in the same unit; then compare the powers of ten before the leading numbers.

Definition

Scientific notation writes a number as a×10na \times 10^n with 1≤a<101 \le a < 10 and nn a whole number.

  • Large prefixes: kilo (k) 10310^3, mega (M) 10610^6, giga (G) 10910^9, tera (T) 101210^{12}.
  • Small prefixes: centi (c) 10−210^{-2}, milli (m) 10−310^{-3}, micro (μ\mu) 10−610^{-6}, nano (n) 10−910^{-9}, pico (p) 10−1210^{-12}.
  • A prefix on a squared or cubed unit is squared or cubed too: 1 cm3=(10−2 m)3=10−6 m31\ \text{cm}^3 = (10^{-2}\ \text{m})^3 = 10^{-6}\ \text{m}^3.
  • Volume: 1 L=1 dm3=10−3 m31\ \text{L} = 1\ \text{dm}^3 = 10^{-3}\ \text{m}^3, and 1 mL=1 cm31\ \text{mL} = 1\ \text{cm}^3.

Changing a prefix

1 nm=10−9 m=10−3 μm=103 pm1\ \text{nm} = 10^{-9}\ \text{m} = 10^{-3}\ \mu\text{m} = 10^{3}\ \text{pm}
  • nm\text{nm}nanometre, 10−910^{-9} m
  • μm\mu\text{m}micrometre, 10−610^{-6} m
  • pm\text{pm}picometre, 10−1210^{-12} m

Worked example

A red blood cell is about 7.5 μ\mum across and a virus is about 90 nm across. Write both sizes in metres in scientific notation, and find roughly how many times wider the cell is.
Practice this conceptself-check · 4 quick reps

A prefix on a squared or cubed unit is raised to that power

1 cm31\ \text{cm}^3 is 10−6 m310^{-6}\ \text{m}^3, not 10−2 m310^{-2}\ \text{m}^3, and 1 mm21\ \text{mm}^2 is 10−6 m210^{-6}\ \text{m}^2, not 10−3 m210^{-3}\ \text{m}^2. Convert the length first, then square or cube the whole factor.

Milli and micro are a thousand apart

milli (m) is 10−310^{-3} and micro (μ\mu) is 10−610^{-6}. A list of sizes in mm, μ\mum, nm and pm is easiest to order after rewriting every value with the same prefix.

Concept 3 of 3: Significant figures and rounding a result

A measurement is only as precise as the instrument that made it. Significant figures are the digits that carry real information. A calculated answer cannot be more precise than the least precise value that went into it, so you round the result to match.

Definition

Counting significant figures:

  • All non-zero digits count, and so do zeros between them (4005 has 4).
  • Leading zeros never count (0.0072 has 2).
  • Trailing zeros after a decimal point count (2.50 has 3).
  • Trailing zeros in a whole number are unclear (1500 could be 2, 3 or 4); scientific notation removes the doubt: 1.50×1031.50 \times 10^3 has 3.

Rounding a result:

  • Multiplying or dividing: keep as many significant figures as the value with the fewest.
  • Adding or subtracting: keep as many decimal places as the value with the fewest.

Worked example

Give each result to the right precision: (a) 4.52×2.14.52 \times 2.1; (b) 12.11+0.312.11 + 0.3.
Practice this conceptself-check · 3 quick reps

Leading zeros are not significant

In 0.0072 the zeros only show where the decimal point is; the value has 2 significant figures. Writing it as 7.2×10−37.2 \times 10^{-3} makes this obvious.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Reference tables (1)

SI base units and the common derived units14 rows
QuantityUnitSymbolIn SI base units
Lengthmetrembase unit
Masskilogramkgbase unit
Timesecondsbase unit
Electric currentampereAbase unit
TemperaturekelvinKbase unit
Amount of substancemolemolbase unit
ForcenewtonNkg m s−2\text{kg m s}^{-2}
Energy, workjouleJ = N mkg m2 s−2\text{kg m}^2\text{ s}^{-2}
PowerwattW = J/skg m2 s−3\text{kg m}^2\text{ s}^{-3}
PressurepascalPa = N/m²kg m−1 s−2\text{kg m}^{-1}\text{ s}^{-2}
ChargecoulombC = A sA s\text{A s}
Potential differencevoltV = J/Ckg m2 s−3 A−1\text{kg m}^2\text{ s}^{-3}\text{ A}^{-1}
ResistanceohmΩ = V/Akg m2 s−3 A−2\text{kg m}^2\text{ s}^{-3}\text{ A}^{-2}
FrequencyhertzHzs−1\text{s}^{-1}
The last column is what you reach when you break a derived unit all the way down.

Watch out for (4)

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