Prefixes and physical scales

I can use the prescribed prefixes from nano to tera.Syllabus
Syllabus K326 / K327, 1(c). Use the prescribed prefixes from nano to tera.
I can compare physical sizes on scales from atoms to Earth.Syllabus
Syllabus K326 / K327, 1(d). Compare physical sizes on scales from atoms to Earth.

A prefix changes the size of a unit. Milli means one thousandth, so a millimetre is one thousandth of a metre.

The numerical value can change while the physical quantity stays the same. See quantities and units if that distinction is unfamiliar.

Powers of ten make very large and small numbers easier to write. 10³ = 1000, while 10^-3 = 1 / 1000 = 0.001. A negative exponent means a fraction here; it does not mean a negative length or current.

Prefix factors multiply the unit that follows
Prefix	Symbol	Factor
nano	n	10^-9
micro	µ	10^-6
milli	m	10^-3
centi	c	10^-2
deci	d	10^-1
kilo	k	10³
mega	M	10⁶
giga	G	10⁹
tera	T	10¹²

Capital letters matter: mA means milliamperes, but MA means megaamperes. The prefix kilo is lower-case k; the upper-case K is the unit kelvin.

Worked examples

Convert by replacing the prefixed unit

Millimetres to metres: 1 mm = 0.001 m, so 3.2 mm = 3.2 x 0.001 m = 0.0032 m.
Milliamperes to amperes: 1 mA = 0.001 A, so 250 mA = 250 x 0.001 A = 0.250 A.
Metres to millimetres: 1 m = 1000 mm, so 0.0045 m = 0.0045 x 1000 mm = 4.5 mm.

Check the direction: expressing a length in a smaller unit needs a larger number of those units. There are 1000 millimetres in each metre.

Try it step by step

Work it out in steps: Length in m, then Width in m, then Area in m².

Problem 1. A rectangular plate is 25 cm long and 40 mm wide. Find its area in m². (1 cm = 0.01 m, 1 mm = 0.001 m.)
Problem 2. A tile is 60 cm long and 25 mm wide. Find its area in m².
Problem 3. A strip of metal is 4.0 cm long and 15 mm wide. Find its area in m².

Show the answer

Problem 1.

Length in m: Length = 25 x 0.01 = 0.25 m

Width in m: Width = 40 x 0.001 = 0.040 m

Area in m²: Area = 0.25 x 0.040 = 0.010 m² (2 sf)

Problem 2.

Length in m: Length = 60 x 0.01 = 0.60 m

Width in m: Width = 25 x 0.001 = 0.025 m

Area in m²: Area = 0.60 x 0.025 = 0.015 m² (2 sf)

Problem 3.

Length in m: Length = 4.0 x 0.01 = 0.040 m

Width in m: Width = 15 x 0.001 = 0.015 m

Area in m²: Area = 0.040 x 0.015 = 0.00060 m² (2 sf)

Area and volume need more than one conversion

A square measuring 1 cm along each side has an area of 1 cm². Each side is 0.01 m, so convert both dimensions when calculating its area.

1 cm² = (0.01 m) x (0.01 m) = 0.0001 m²The length conversion factor is squared for area.

Both sides change unit. Multiplying only the area value by 0.01 would miss the conversion of the second dimension.

The same reasoning applies to a cube: 1 cm³ = (0.01 m)³ = 0.000001 m³. Apply the length conversion to all three dimensions.

Does the size of the answer make sense?

An order of magnitude gives the approximate scale of a quantity as a power of ten. For example, 10⁰ m is 1 m. The objects below vary in size; these values are useful reference scales, not exact measurements of every example.

Approximate orders of size, from an atom to Earth
Object and dimension	Approximate scale
Diameter of an atom	10^-10 m
Width of a bacterium	10^-6 m
Width of a human hair	10^-4 m
Diameter of a coin	10^-2 m
Height of a person	10⁰ m
Diameter of Earth	10⁷ m

If a calculation gives 0.02 m for the width of a human hair, pause: 0.02 m is 2 cm, about the width of a small coin. A hair is much thinner, with a width of roughly 10^-4 m. Check the original reading and conversion before accepting the answer.

Check your understanding

A current is 250 mA. Which conversion to amperes is correct?

0.250 A
250 000 A
0.0250 A

Show the answer

0.250 A

1 mA = 0.001 A, so 250 mA = 250 x 0.001 A = 0.250 A. The current has not changed; only its unit has.