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Units

The S.I. International System of Units

History

The metric system is based on multiples of ten, which makes sense because we have ten fingers. However, the base ten, or decimal, system has not always been the dominant system in mathematics. About 4000 years ago, the ancient Babylonians had a system which was based on multiples of 60 (base 60 is properly called 'sexagesimal'). You can still see this system in use today, in 360 degrees in a circle, and 60 minutes and 60 seconds for time divisions.

The Roman system of counting had symbols for 5, 50, and 500, as well as multiples of ten, but they had no concept of zero. And more importantly, Roman numerals are not 'positional'. This means their symbols, I, V, X, etc., held their value no matter where in the composite number they were placed. In the Hindu-Arabic system, 12 and 21 have different values because of the order and position of the numerals.

Following the demise of the Greek Empire under the Roman conquest, the rest of the world leaped ahead of Europe in numbering systems. The Indians produced a zero, and the associated infinity. The Arabs invented the numbering system we use today, and with it they developed algebra, which married number theory to Greek geometry.

It was not until the 13th century that Leonardo of Pisa (better known today as Fibonacci, which means 'son of Bonacci') imported the Hindu-Arabic numbering system to Europe. Although invented by the Chinese 1500 years earlier, it was not until the 1500s that the decimal fractional system (using a dot and numerals representing fractions of one, such as 3.142), became widespread in Europe, adding great precision to calculations and measurements.

Seven base units

The story of technology reveals a fascinating struggle to find a system of units and measures which could cross borders and unite commerce and science around the globe. What constitutes the basic measure of quantities such as length and mass is still not totally universal, with a residual three countries [Liberia, Myanmar, and the USA] remaining reluctant to adopt fully either the metric system or its associated system of units. Despite the determination of its people to make life difficult for themselves, the scientific community of the USA is 'bilingual', recognising that the outdated Imperial system of units is not suitable for modern science. As a matter of policy, this website makes no facility for the Imperial system.

During the French Revolution, a search began to find the best means for defining the units of measure for length and mass. The French were the first to officially adopt the metric system, in 1799. By the 1860s, a broader system of base and derived units was being proposed, which today is known by its French initials, S.I. (Système Internationale d’Unités).

In 1875, the BIPM (Bureau international des poids et mesures: International Bureau of Weights and Measures) was founded, and in 1948 the S.I. system of units was officially adopted by the international community as the universal system of units under the metric system. It consists today of 7 base units, from which many other units can be derived.

The base units each describe a fundamental natural phenomenon or dimension. Ideally, these units would be measured directly from nature, and not be dependent on any other units for their definition. In fact, however, only three of the base units can be independently defined. The remaining four have varying degrees of dependence on the other base units for their definition and determination.

Symbol Unit Name Year adopted in S.I. Quantity Dependence on other units
K kelvin 1960 temperature independent
s second 1860s time independent
kg kilogram 1795 mass independent
m metre 1791 length requires the second
A ampere 1900 electrical current requires m, s, kg
cd candela 1960 luminous intensity requires s, m, kg
mol mole 1971 amount of substance requires kg

The kelvin is named after the physicist William Thomson, 1st Baron Kelvin (1824–1907), and the ampere is named after the French mathematician and physicist, André-Marie Ampère (1775–1836).

Derived units

Derived units are units whose definitions are given in terms of other units. For example, the unit for force is the newton. The newton, N, is defined as ${kg⋅m}/{s^2}$. The unit for energy is the joule (J). It is defined as N⋅m, but in base units it is ${kg⋅m^2}/{s^2}$. The unit for power is the watt (W), which is J/s, or in base units ${kg⋅m^2}/{s^3}$.

Prefixes

The International System of Units, the S.I., is entirely metric. It makes use of multiples of ten in reporting quantities of different scales. For example, the 'centimetre' has two bits of information: 'centi' means one-hundredth, or $10^{-2}$, and the metre indicates the S.I. base unit for length. Prefixes may also be used with derived units: e.g. kJ means kilojoule, or one thousand joules of energy.

  •  $10^{-24}$  = yocto (y)
  •  $10^{-21}$  = zepto (z)
  •  $10^{-18}$  = atto (a)
  •  $10^{-15}$  = femto (f)
  •  $10^{-12}$  = pico (p)
  •  $10^{-9}$  = nano (n)
  •  $10^{-6}$  = micro (µ)
  •  $10^{-3}$  = milli (m)
  •  $10^{-2}$  = centi (c)
  •  $10^{-1}$  = deci (d)
  •  $10^{1}$  = deca (da)
  •  $10^{2}$  = hecto (h)
  •  $10^{3}$  = kilo (k)
  •  $10^{6}$  = mega (M)
  •  $10^{9}$  = giga (G)
  •  $10^{12}$  = tera (T)
  •  $10^{15}$  = peta (P)
  •  $10^{18}$  = exa (E)
  •  $10^{21}$  = zetta (Z)
  •  $10^{24}$  = yotta (Y)

e.g. One million tons can be written as 1 Mton (Mt), and the technology of objects one billionth of a metre in size is known as nanotechnology.

The expression $10^{6}$ means the numeral one with 6 zeroes following.

The expression $10^{-3}$ means the numeral one moved three spaces to the right of the decimal point, or 0.001 (careful: it does not mean three zeros between the decimal point and the one).

The prefix system eliminates potential errors in writing very large or very small numbers, and makes calculations easier. Good scientific practice is to record numbers as a single numeral before the decimal place, followed by the correct number of significant figures, followed by the power of ten. e.g. 2,314.7 should be written as 2.3147 x $10^{3}$.

Converting Units

There have been many units systems used throughout history. As trade became more global, and technology accelerated the speed of long-distance interactions, it became increasingly important to have a more efficient and universal system of units and measures.

The history of how this was achieved is in itself a fascinating, and on-going, story. In the 1790s, the French, no doubt in the spirit of revolutionary fervour, began the use of a base-ten (decimal) system, known as the metric system. This system makes it easy to do the mathematics of conversions of units, because the full range of values for a quantity (mass, length, etc.) can be expressed in powers of ten.

However, there was a problem: the unit for time, the second, already existed. This was based on the rotation of the Earth, subdivided into 24 hours, which in turn were subdivided into minutes and seconds, based on the old Sumerian/Babylonian base-60 system. There have been proposals to change this system to a decimal time system, with perhaps ten hours a day, one hundred minutes per hour, and one hundred seconds per minute. However, this was never adopted. There is no other system of time today than the second, minute, hour, day.

The unit for length, on the other hand, did change, as did the unit for mass. The metre and kilogram were therefore the first metric units of what has become the S.I. unit system.

The definition of the metre depends on the second, and is defined by the distance light travels in a certain fraction of a second. Length is measured by units which are powers of ten of the metre: millimetre (mm) = 10-3 m; kilometre (km) = 103 m, etc.

Velocity Units

As a result of these two systems, base-60 for time, and base-10 for length, there is a non-decimal conversion factor for the conversions of m/s to km/h ($1.0 m/s = 3.6 {km}/h$).

The velocity of an object is its displacement per unit time. We can express this as:

v = d/t

If you travel 100km in one hour, you are travelling at 100km per hour. Husain Bolt can run 200m in 19.19 seconds, so his average velocity is:

v = d/t = 200m / 19.19s = 10.4 m/s

How do we compare this to, say, a bicycle, we have measured in km/h? We must convert the units from m/s to km/h in order to compare:

10.4 m/s = 10.4 m/s x (1/1000 km/m ) x (3600 s/hr) = 37.44 km/h

How fast can you run? Time yourself over 100m, and give your answer in m/s and km/h.

Note that 1.0 m/s = 3.6 km/h

And 1.0 km/h = 1/3.6 m/s = 0.28 m/s

Density Units

It is best to give densities in units that are compatible. For example, most substances are a small of number of grams per cubic centimetre, not kilograms. But if the volume is a dm3 (litre), most substances are in the kilogram range.

Expressing a quantity with the most suitable units aids comprehension and comparison.

The density of water can best be expressed as:

1.0 g/cm3 = 1.0 kg/dm3 (L) = 1.0 t/m3

e.g. The mass of a block of copper is 9.0kg. It displaces 1000 cm3. What is the density of copper?

Density = ρ = m/vol = (9.0 x 1000 g)/103 cm3 = 9.0 g/cm3

Imperial Units

The antiquated Imperial unit system of units and measures is still used in three countries: Myamar, Liberia, and the USA. Science students in the USA need to be 'bilingual' with 'those dang foreign units', so to help them out, we provide here a table of conversions:

Imperial UnitS.I. closest equivalentConversion factor
Mass
ounce (oz)gram (g)1 oz = 28.3495g
pound (lb)kilogram (kg)1 lb = 0.45359237 kg
stone (st)kilogram (kg)1 st = 6.35029 kg
hundredweight (cwt)kilogram (kg)1 hw = 50.8023 kg
short ton (USA)tonne (T)1 t (US) = 0.907 T
long ton (ex-UK)tonne (T)1 t = 1.016 T
Length and Area
inch (in)centimetre (cm)1 in = 2.54 cm
foot (ft)decimetre (dm)1 ft = 3.04 dm (30.4 cm)
yard (yd)metre (m)1 yd = 0.9144 m
mile (mi)kilometre (km)1 mi = 1.609.344 km
acrehectare1 acre = 0.4047 ha
1 acre = 4,046.9 $m^2$
Volume
pint (pt)litres (l)1 pt = 0.568.26 l
quart (qt)litres (l)1 pt = 1.136.52 l
gallon (gal)litres (l)1 pt = 4.546.09 l
Energy and Temperature
calorie (cal)joule (J)1 cal = 4.184 J
kiloton of TNTterajoule (TJ)1 kton TNT = 4.184 TJ
ton of oil equivalent (TOE)Gigajoule (GJ)1 TOE = 41.868 GJ
ton of coal equivalent (TCE)Gigajoule (GJ)1 TCE = 29.308 GJ
fahrenheit (°F)celcius (°C)$F = 9/5C + 32$
$C = 5/9(F - 32)$
Pressure
pounds-force/square inch (psi)pascal (Pa = 1$N m^{-2}$)1 psi = $6.8948 ⋅ 10^3$ Pa
atmosphere (atm)bar (bar = 1.0 x $10^{5}$ psi)1 atm = 1.01325 bar (1.01325 x $10^{5}$ Pa)

Content © Andrew Bone. All rights reserved. Created : July 1, 2013 Last updated :March 5, 2016

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