Units, significant figures, and standard procedures

This lesson is a part of the School of Physics

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In order to understand physics and work with physical theory accurately, there must first be a consistent understanding of the conventions and procedures of working out quantities and numbers.

Units

Most quantities that can be measured must be expressed in the correct units. The unit differentiates between whether one is talking about a measure of time, or a measure of distance, or several other fundamental types of observables that have come to light during our study of the universe.

Standard Prefixes

You already use units in everyday conversation. When you want to purchase something, the price of the item is measured in some combination of monetary units. In the U.S., these units are usually dollars and cents. In this system, a dollar is equivalent to 100 cents, although very few people would pay for an item using hundreds of cents. The American dollar is a simple example of a multiplicative unit. When reading the price of an item that costs more than 99 cents, you will most likely read it in terms of dollars, as that is the most convenient term to use. Seldom is the case where one reads an item priced at 10 dollars as being "a thousand cents".

Similarly, scientists must use appropriate types of scale when talking about large and small quantities. You would not want to talk about the mass of the Earth in the same unit as the mass of a few subatomic particles, for example, unless making a pointed comparison. One of the technical reasons for this is simple accuracy. It is just not feasible to be as accurate in talking about the mass of the Earth as one would be in measuring the mass of a few salt crystals. While the salt crystals can easily be measured to the nearest gram, the Earth's mass to the nearest gram is changing constantly (due to atmospheric evaporation and bombardment by radiation, particles, and tiny meteorites).

Scientists have developed a standard for referring to large and small quantities of units by referring to them as multiples of powers of ten. Each power is referred to by appending a prefix to the standard unit. The most common prefixes are listed below:

Multiple	SI Prefix	SI Abbreviation
1012	tera	T
109	giga	G
106	mega	M
103	kilo	k
102	hecto	h
101	deka	da
10-1	deci	d
10-2	centi	c
10-3	milli	m
10-6	micro	µ
10-9	nano	n
10-12	pico	p
10-15	femto	f
10-18	atto	a

In terms of dollars and cents, we would then call a dollar a hectocent and if we used c as the unit abbreviation for cents, we would write 200 c = 2 hc. We generally don't go around talking about dollars and cents in this manner, but these prefixes are great way to quantify the myriad units we will encounter in the physical sciences. You may already be familiar with the kilogram as a unit of mass, and from the list above, you know that each kilogram is 1000 grams. You might purchase a few kilograms of cement or add a few grams of salt to a recipe, but you wouldn't want to add a few kilograms of salt to a recipe. This is again just an example of what physicists call order of magnitude.

Types of Units

The units of length and distance may seem obvious to you from everyday experience, but there are many other important units (ie., how would you measure the color of an object? We will learn more about color when we study electromagnetism), and there are ways of relating what may seem like completely different types of units to each other in a standard manner, including time and distance. Finding out that some units are really just products or multiples of some familiar unit is part of the allure of discovering new physics.

Units, most often the SI Unit Standard signify first and foremost what the quantity is relevant to. All SI units can themselves be expressed in terms of the seven SI base units below:

• length, measured in metres, expressed as the symbol m
• mass, measured in kilogrammes, expressed as the symbol kg
• time, measured in seconds, expressed as the symbol s
• electric current, measured in amperes, expressed as the symbol A
• temperature, measured in kelvin, expressed as the symbol K
• amount of substance, measured in moles, expressed as the symbol mol
• luminous intensity, measured in candela, expressed as the symbol cd

The bold script indicating the symbols above is for emphasis; no special format is required when hand-writing these quantities provided that they are written clearly and in the correct case as shown.

There are many other SI units which have their own symbol, but every one of them is a shorthand for some product of the base units shown above (or equivalently, a product of other SI units which themselves are equivalent to some other product of base units, and so on). SI units are given their own symbol for two reasons; firstly, that the product they represent might be long and awkward to write, and secondly that units relevant to important scientific discoveries may be renamed and given a symbol of their own to honour their creator if the theory which uses them is a particularly significant discovery.

Writing Units

Many units belonging to quantities we can measure in physics are products of other units. As you know from mathematics, a product can include dividing one term by another - and from the Law of Indices we can also show that this is equivalent to multiplying the numerator by a negative power of the denominator. So, as an example, if we take a measurement of the velocity of an object, we express this in metres per second which we can write as:

• $m/s$

Or equivalently as:

• $ms^{-1}$

According to the the BIPM (Bureau Internationale de Poids et Measures), the international authority responsible for standardising systems of measurement, either of these methods is acceptable in scientific literature as long as the form of the unit is clear and unambiguous.

Significant figures and Uncertainty

We will frequently be using measured quantities (which will be expressed as some number or collection of numbers) in equations and formulas in order to relate them to quantities we wish to predict the value of. In all of these cases, it will be very important to pay attention to the precision of the numbers that we have actually measured, as it is very easy to get lots of meaningless numbers from applying basic formulas, all the while forgetting that we did not measure the observables with the precision that we are getting out of blindly applying the formulas.

Instrument Precision

As an example, when you measure the length of something short with a ruler, you may notice that between the centimeter marks, there are ten marks, allowing you to eyeball measurements like 2.64 cm. However, note that the 0.04 cm in the previous measurement is not a precise measure (it is not infallible). The ruler does not give markings down to hundredths of a centimeter; you can really only say that "It looks like the length is 2.64, but my instrument can only report the length as being between 2.60 and 2.70 cm somewhere close to that measurement." You have no hard evidence to support the 0.04; you only hypothesize that its somewhere around the 4 hundredths mark. It could actually be near the 3 hundredths mark or near the 5 hundredths mark, if there were such markings, but it definitely would not be around the 6 hundredths mark. All numerical measurements have this sort of error involved and scientists include the instrumental precision in every measurement by using the notation $2.64\pm 0.01$ cm which means the object you measured is between 2.63 and 2.65 centimeters in length. That is definitely a statement you can support using your ruler. More advanced instruments also have statistical fluctuations in either the property they are measuring or the process/physical properties of the parts being used to measure the phenomenon and they all report (either live or in the manual) statistical uncertainty. Check your scale at home for its uncertainty. While you may expect your scale to tell whether you have sneakers on or not, it may not be able to tell whether you got a haircut today.

Unqualified measurements, those without explicit errors written in, are usually assumed to have a unit error in the most precise digit unless otherwise stated. Ie., if you see a measurement given as 5.4 g, it is usually meant as a shorthand for $5.4\pm 0.1$ g.

Significant Figures

As we have seen above, all instruments have some finite precision. Significant figures are a method of keeping track of the order of magnitude of the error involved in a calculation. For example, You might read 3.26 cm off of a ruler. We say that such a measurement has 3 significant digits, which means all 3 digits present were measured on the ruler, implying the last digit, 6, is uncertain by some amount.

Suppose you measure something to be 0.05 cm using the same ruler. This measurement has only one significant digit (we didn't measure those two zeroes; they are only placeholders, an artifact of our decimal system), and it is uncertain, making it a very weak measurement (you wouldn't use this type of ruler to measure the thickness of paper). If you are ever confused about whether the digits in a measurement are significant and the context does not specify, simply write the measurement in scientific notation. The number of digits in the multiple of the power of ten are the number of significant digits. Above, 0.05 cm becomes 5 x 10^-2 cm, with one significant digit.

Addition and Subtraction

For simple calculations, it is important to recognize that you can't get more information from a formula than the information contained by the weakest measurement you put into the formula. Ie., if you measure one object with a normal centimeter ruler and a larger object with a meter ruler, the smallest marking of which is one centimeter, and then add the lengths together, you should not expect to get a length that is precise down to the hundredth of a centimeter. Why not? Because you don't know how many hundredths of centimeters may be present in the length of the larger object. It could be anywhere from none at all to hundreds, which would make your overly precise sum quite nonsensical. This yields the following rule when dealing with addition: The result of an addition or subtraction of two measurements is only as precise as the least precise of the two measurements.

For example, if we have 3.4 g of water and 5.504 g of salt, we expect to get 8.9 g of salt water when we mix them together, not 8.904 g. We do not know what's going on in the hundredths and thousandths place of the mass of the water, only that it's between 3.3 g and 3.5 g of water that we have. For all we know, we may get 8.813 g of salt water as measured by a more precise instrument. As you can see, the 0.004 g carries no useful information.

Multiplication and Division

For reasons similar to the above, if one is taking the product or quotient of two numbers, the result is always rounded to the precision of the least precise measurement involved. Suppose a car is measured by a radar gun to be travelling at 10 m/s (accurate to the nearest meter per second), and you time the car as taking 10.7 seconds to go between point A and point B at that speed. As we will learn later, we can get the distance travelled by the car by multiplying the speed by the time taken. We do this and get 107 m. Rounding to account for the least precise measurement, which only has 2 significant digits, we get 110 m as the approximate distance between A and B. In this last value, note that the trailing 0 is only a placeholder and is not significant.

Some texts will denote whether the trailing zeroes are significant or not by placing a decimal point after the last zero. This can get rather pedantic, so whenever the trailing zero is significant, we will simply alert the reader to the precision of the measurement, as in the radar gun measurement above.

We round to 110 m in order to give the correct impression of the uncertainty of the result. The value of 107 m has 3 significant digits, misleading one to believe there is only an error in the units place. If one looks at the possible range of distances (from 95.4 m to 118.8 m using the lowest possible and highest possible values for the underlying measurements, respectively), one sees how inaccurate that would be.

References

• The NIST Reference on Constants, Units and Uncertainty, http://physics.nist.gov/cuu/Units/units.html, retrieved 21.08.06
• Significant Figure Rules, http://chemistry.bd.psu.edu/jircitano/sigfigs.html