Units and dimensions

Contents

• Introduction
• Base units and derived units
• The SI system of units.
• Dimensional consistency
• Return to Main Contents Page

Introduction

In science, a type of question often asked is how much? how big? In order to answer such questions it is important to have systems of measurement which are consistent and understood by all.

A dimension is a property that can be measured such as distance, time, temperature, speed.

A unit is a basic division of a measured quantity and it enables to say how much of the quantity we have - 10 miles, 2 hours etc.

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Base units and derived units

Base units are units that are defined by reference to some external standard. This external standard is arbitrary but is a matter of common agreement.

Derived units are units that are defined by reference to combinations of the base units.

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The SI system of units.

The SI system is an internationally agreed system of units based on seven base units. These are listed in table 1 below. Some of the more important derived units are listed in table 2.

Table 1 Base units of the SI system of units

Quantity	Unit	Symbol
Mass	kilogramme	kg
Length	metre	m
Time	second	s
Mole	mole	mol
Temperature	kelvin	K
Electric current	ampere	A
Light intensity	candela	cd

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Table 2 Some derived units in the SI system

Quantity	Unit	Symbol
Volume	cubic metre	m3
Force	Newton = kg m s-2	N
Pressure	Pascal = N m-2	Pa
Work, Energy	Joule = N m	J
Power	Watt = J s-1	W
Molar concentration	Molar = mol dm-3 or mol L-1	M

Multiples of the basic units are used to avoid having to write very large or very small numbers. These are listed in table 3.

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Table 3 Multipliers for SI unit

Large quantities			Small Quantities
Prefix	Symbol	multiplier	Prefix		Symbol	multiplier
deca	D	10	deci		d	10-1
hecta	h	100	centi		c	10-2
kilo	k	103	milli		m	10-3
Mega	M	106	micro		m	10-6
Giga	G	109	nano		n	10-9
Tera	T	1012	pico		p	10-12
Exa	E	1015	femto		f	10-15

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Dimensional consistency

All equations relating physical quantities should be dimensionally consistent. That is when the units on both sides of an equation are worked out they should be identical.

Example 1

consider the ideal gas equation, Pv = nRT

Writing the units for each term in the equation

LHS:

RHS:

i.e. LHS units = RHS units

If an equation contains additive terms, then each term in the equation must be dimensionally consistent.

Example 2

The equation for the distance travelled by a body moving with uniform acceleration is;

s = u.t +¹/₂ a.t²

Writing the units for each term in the equation

i.e. Units of all three terms in the equation are the same.

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