Units

Video: Units and results

Units are part of the value. The engine carries them through every operation, converts between them, and stops with an error when a formula does not add up dimensionally.

Attaching units

Write the unit after a number, with or without a space: 15 kg, 12kN/m, 1230000 mm³.

a: 1.23 m + 35 cm + 12 mm -> cm
v: 10 m/s -> km/h
P: 2.5 kN / (50 mm * 200 mm) -> MPa
g0: 9.81 m/s²
m: 70 kg
W: m * g0 -> N

a = 1.23 m + 35 cm + 12 mm = 159.2 cm
v = 10 m/s = 36 km/h
P = 2.5 kN / (50 mm · 200 mm) = 0.25 MPa
g0 = 9.81 m/s²
m = 70 kg
W = m·g0 = 70 kg · 9.81 m/s² = 686.7 N

Without ->, a sum comes out in the first unit it meets. With -> unit the result is converted to the unit you choose. In the Calcpad language the same conversion is a vertical bar: a = 1.23m + 35cm + 12mm | cm.

SI base units

-> base shows a result in the SI base units of its dimensions, so you do not have to spell them out:

F: 5 kN
L: 2 m
M: F * L -> base
P: 3 MPa -> base

The moment comes out as 10000 kg·m²/s² and the pressure as 3000000 kg·m⁻¹·s⁻². The units used are kg, m, s, A, mol and cd. A value with no unit is left as it is, and so is one with a temperature or an angle in it. In the Calcpad language the same thing is M = F*L | base.

Prefixes work on every unit that takes them: kN is 10³ N, MPa is 10⁶ Pa, mm is 10⁻³ m. See “Prefixes” below for the full list.

Prefixes

Any SI prefix goes in front of any SI unit that takes one, so 3 mN, 2 GPa, 5 µm, 40 nF and 1.2 kΩ all work.

Micro can be typed as the Greek mu μ, the micro sign µ (what a keyboard gives) or a plain u; the result shows μ. The units that take a prefix are g, m, s, A, mol, cd, rad, L, N, Nm, Pa, bar, J, W, Wh, VA, VAR, Hz, V, C, F, Ω, S, Wb, T, H, Bq, Gy, Sv, lm, lx, kat, eV, Ah and Da. Everything else stays as it was: min, h, d, ft, in, lb, psi, °C and the other non-SI units take no prefix (kft and mmin are errors), the kilogram is kg (Mg is a megagram, not kkg), and kelvin K takes none either.

A name the unit list already had keeps its meaning, and the list is read first. So min is the minute (not a milli-inch), mm is the millimetre, Pa the pascal, cd the candela, ft the foot, h the hour, d the day, T the tesla and u the atomic mass unit; da alone is not a unit, and dam is the decametre. The prefix spellings that read as another unit are told apart by their case: fT is a femtotesla and ft a foot, mS is a millisiemens and ms a millisecond. One name is shadowed: ks has always been the metric horsepower, so a kilosecond is written 1000 s. A variable you define always wins over a unit of the same name, so a sheet that has a variable called mN still works.

F: 3 mN
F_n: F -> N
E: 2 GPa
E_m: E -> MPa
t: 5 µm
t_n: t -> nm

F_n = F = 3 mN = 0.003 N
E_m = E = 2 GPa = 2000 MPa
t_n = t = 5 μm = 5000 nm

Errors you want

x: 6 m
t: 2 s
speed: x / t

speed = x/t = 6 m / 2 s = 3 m/s

Dividing a length by a time is fine. Adding them is not:

x: 6 m
t: 2 s
speed: x / t
len: x + t

Error in "len = x + t" on line [4]: Inconsistent units: "m + s".

That is the engine doing its job. A unit slip is the most common mistake in hand calculations, and here it stops the sheet and names the line.

Which units exist

The built-in list is long. The groups are:

The Insert tab in the sidebar lists all 400 or so, grouped, and a click inserts one. The Math tab has a Units palette organised by dimension.

Angles

Angle units are accepted by trigonometric functions and override every other setting. sin(30 deg) is 0.5 whatever the angle setting is. A bare number follows the setting: degrees by default, changed on the Math Formatting tab (#deg, #rad, #gra in Calcpad). Inverse trigonometric functions return a plain number unless you define ReturnAngleUnits = 1 (Calcpad line) first:

a = asin(0.5)
ReturnAngleUnits = 1
b = asin(0.5)

a = asin(0.5) = 30
ReturnAngleUnits = 1
b = asin(0.5) = 30°

Temperatures

20 °C is a temperature; 300 K is another, and the difference of two temperatures is a difference:

T1: 20 °C
T2: 300 K
d: T2 - T1

d = T2 − T1 = 300 K − 20 °C = 280 K

US and UK names

A few bare names mean different things in the US and UK systems (gal, ton, pt, qt, bbl, cwt, therm). The setting Non-metric units picks one. Write the explicit suffix (gal_US, gal_UK) when a sheet must not depend on the setting.

Percentages

A ratio has no unit, so it comes out as a plain number. Convert it with -> % to show it as a percentage, and write % after a number to enter one. ‰ (per mille) works the same way.

Ed: 245.25 kN
Rd: 355.1 kN
ratio: Ed / Rd
util: ratio -> %
r: 4%
f: Ed * r
slope: 1 / 40 -> ‰

ratio = Ed / Rd = 245.25 kN / 355.1 kN = 0.691
util = ratio = 0.691 = 69.07 %
r = 4 %
f = Ed · r = 245.25 kN · 4 % = 9.81 kN
slope = 1 / 40 = 25 ‰

The value itself does not change, only the way it is shown: util is the same number as ratio, and r is 0.04 in every formula. A result in percent is rounded to the decimals set on the Math Formatting tab like any other result, and the rounded figure typed back in (69.07%) is the same ratio to that many digits. An input works the same way: rate: ? 12.5 % shows a box holding 12.5 with % after it, and the name is 0.125 in the formulas.

A percentage times a plain number is a plain number (4 % · 2 is 0.08); add -> % to the line to keep it as a percentage.

Your own units

A sheet defines a unit of its own with a #unit line: the name, =, and its value in units the engine knows or in units defined above it. Later lines use it after a number and after ->, like any other unit.

#unit bag = 25 kg | cement bag
#unit pallet = 40 bag
load: 3 pallet -> t
n: 1000 kg -> bag
rate: 6 bag / 1 m³ -> kg/m³

bag = 25 kg   cement bag
pallet = 40 bag
load = 3 pallet = 3 t
n = 1000 kg = 40 bag
rate = 6 bag / 1 m³ = 150 kg/m³

A unit name is letters and _ only (bag, ton_short), with no digits; a name with a digit is an error on its line. The value can use the usual spellings (m², kN·m) and can have a note after |. A built-in unit cannot be redefined: #unit slug = 14.6 kg is an error, because slug is already a unit. In the Calcpad language, and in a >> line of a sheet, the same unit is a line with a dot in front of the name: .bag = 25kg.

Currency and information units are defined as dimensionless units, the first as 1 and the others as multiples:

#unit USD = 1
#unit AUD = 0.65 USD
price: 200 AUD -> USD

USD = 1
AUD = 0.65 USD
price = 200 AUD = 130 USD

Two families of dimensionless custom units in one sheet share a dimension, so do not mix them.

Units and names

A word that follows a number immediately is a unit (2 m). A standalone word is a variable if you defined one with that name, and a unit otherwise. Put a dot in front to force a unit even when a variable has the same name. In a sheet this rarely matters, but it explains one surprise: if you name a variable m and later write 5 m, the m after the number is still the unit.

Removing and reading units

The functions getunits(x), clrunits(x) and setunits(x, u) read, strip and attach units. They are in the function reference.

q: setunits(5, kN)
F: 5 kN
n: F / 1 kN

q = setunits(5; kN) = 5 kN
F = 5 kN
n = F / 1 kN = 5 kN / 1 kN = 5

Use clrunits sparingly. A value without a unit is a value the engine can no longer check.

Units in vectors and matrices

A vector or matrix whose elements all carry the same unit shows the unit once after the bracket. See Vectors and matrices.

Related pages: Calculations with units, Simply supported beam, Pipe flow, Darcy-Weisbach. More pages are in Calculators and guides.