Units and Measurement
Line by Line — Part 1

You cannot measure something on its own. You can only measure it against something else.

When you say a rope is 4 metres long, you are saying: lay one metre down beside the rope, over and over, and it fits four times. The metre is the thing you compared against. That is the unit.

The word "internationally accepted" matters. Everyone has to agree on the same metre, or two people measuring the same rope would get different answers and both would be right.

Every measurement has two halves, and both are needed.

"4" on its own tells you nothing. Four what? "Metres" on its own tells you nothing either. Only 4 metres is a real measurement.

Write the number and leave off the unit in an exam and the answer is wrong, even if the number is right. This is not a rule invented to be strict — half the information really is missing.

Speed is length divided by time. So once you have a unit for length and a unit for time, you already have a unit for speed — metres per second. You do not have to invent a new one.

The same trick works everywhere. Area is length × length. Volume is length × length × length. Force is mass × length ÷ time ÷ time. Choose a handful of units well, and the rest build themselves.

Base units are the starting pieces. Nothing is used to build them; they are simply agreed on.

Derived units are everything you can build afterwards. The newton, the joule, the volt — all of them are just base units multiplied and divided together in some combination.

Sort a few units below and you will see that most of the familiar ones are derived, not base.

Before everyone agreed, each system picked its own starting units. The names are just the initials of what they picked:

Notice that all three used the second. That is why, later on, converting between them only ever involves length and mass factors — the time factor is always 1.

SI is French for "International System". It won because it is decimal — every step up or down is a power of ten, so converting means moving the decimal point.

Compare that with the older British system, where you had to remember 12 inches in a foot and 5280 feet in a mile. Those numbers have to be memorised. Powers of ten do not.

Two names to keep apart: BIPM is the permanent office that develops the system. CGPM is the conference that votes changes in. The 2018 vote took effect on 20 May 2019.

A plane angle is a flat angle, the kind you measure with a protractor. The book defines it as the curved edge divided by the radius.

Both of those are lengths. A length divided by a length leaves nothing behind — the units cancel completely. That is what dimensionless means here.

A solid angle is the three-dimensional version. Instead of an arc divided by a radius, it is an area divided by the radius squared. Area divided by area again cancels completely.

Careful with one thing: dimensionless does not mean there is no unit. The radian and the steradian are real units. They just do not carry any length, mass or time inside them.

This is the big change of 2019, and it is worth understanding rather than memorising.

The kilogram used to be a metal cylinder kept near Paris. Every kilogram in the world was checked against that one object. But objects can be scratched, or pick up dust, or be lost — and if that cylinder changed, the kilogram itself changed.

So the definitions were flipped around. Instead of measuring the Planck constant using a kilogram, they declared the Planck constant to be an exact number, and let the kilogram be whatever mass makes that number come out right. Constants of nature are the same everywhere and never wear out.

The sizes of the units did not change at all. A kilogram is exactly as heavy as it was before. Only the definition changed.

A mole is just a count. It means 6.022 × 10²³ of something. But you have to say of what.

A mole of oxygen atoms and a mole of oxygen molecules are different amounts of matter, because each molecule contains two atoms. Saying "a mole of oxygen" without saying which is genuinely ambiguous, and the book is warning you about it.

These survive because they are practical. Nobody wants to say a year is 3.156 × 10⁷ seconds in ordinary conversation, and land is easier to describe in hectares than in square metres.

The ones that come up most in questions are worth knowing as numbers:

Watch the difference between the bar and the atmosphere. The bar is a round 10⁵ Pa by definition. The atmosphere is 1.013 × 10⁵ Pa, because it comes from an actual column of mercury 760 mm tall.

Prefixes are the shorthand that make SI easy. Instead of writing 0.000001 metres you write 1 µm.

The main ones step in thousands: kilo (10³), mega (10⁶), giga (10⁹), tera (10¹²) going up, and milli (10⁻³), micro (10⁻⁶), nano (10⁻⁹), pico (10⁻¹²), femto (10⁻¹⁵) going down.

Four of them break the pattern: deca (10¹), hecto (10²), deci (10⁻¹) and centi (10⁻²). Centi is the troublemaker, because it appears constantly and is 10⁻² rather than 10⁻³.

No measurement is ever exact. There is always a last digit you had to judge rather than read.

The clever idea in this section is that you can show how careful a measurement was just by how you write the number, without adding any extra note about error.

This is the definition to hold on to: all the digits you are sure of, plus one you had to estimate. One estimated digit. Not two, not none.

The book's example is a length read as 287.5 cm. The 2, the 8 and the 7 came straight off the scale — those are certain. The 5 was judged by eye between two marks — that is the uncertain one. Four digits altogether, so four significant figures.

Writing 287.53 would be claiming you could see a level of detail the ruler cannot show. Writing 287 would be throwing away a digit you genuinely did work out.

This one sentence is the reason all the counting rules exist, so it is worth pausing on.

You measured the object once. Rewriting the answer in a different unit does not send you back to measure it again, so it cannot possibly make the measurement more or less careful.

But look at what happens to the zeros. Going from 2.308 cm to 0.02308 m added two zeros at the front. Going to 23080 µm added one at the back. If the count has to stay at four, then those new zeros cannot be counting for anything. They are only there to hold the decimal point in place.

That is where the rules come from. They are not arbitrary — they are what you get by insisting the count stays fixed.

Only zeros are ever in doubt. Every other digit always counts. So there are really just three questions to ask about a zero:

Try a few numbers below and the rule being used will be named each time.

This is the fix for the whole problem. Write 4700 mm and nobody can tell whether you measured two, three or four digits. Write 4.700 × 10³ mm and it is settled — four, plainly.

In scientific notation there are never any leading zeros to argue about, and any trailing zero you write must have been deliberate. So the count is simply the number of digits you can see in front.

Sometimes you do not need the exact size, just a rough idea of how big something is. Order of magnitude is that rough idea, expressed as a power of ten.

Two things to be careful about. First, the cut is at 5, not somewhere else. Second, the order of magnitude is not just the exponent you can see. For 6.7 × 10⁴ the mantissa is bigger than 5, so it rounds up to 10, and 10 × 10⁴ is 10⁵ — order 5, not 4.

"17 orders of magnitude larger" means larger by a factor of 10¹⁷.

A calculator does not know where your numbers came from. Divide 4.237 by 2.51 and it will happily give eleven digits. But you only measured three or four, so most of those digits are invented.

The book calls recording all of them "absurd and irrelevant", which is fair. The rules below exist to decide where to stop.

The weakest measurement decides. If one quantity was measured to three figures and another to four, the answer gets three.

The book's example: mass 4.237 g (four figures) ÷ volume 2.51 cm³ (three figures) gives 1.68804780876 on a calculator, and should be reported as 1.69 g cm⁻³.

A useful habit: count the significant figures of your inputs before you start calculating. Then you already know where to stop.

Here the rule counts something different. Not significant figures — decimal places.

The book adds 436.32 g, 227.2 g and 0.301 g. The calculator gives 663.821 g. But 227.2 g was only known to one decimal place, so the answer is rounded to one decimal place: 663.8 g.

Why the switch? Because when you add, what matters is where each number's uncertainty sits. If one measurement is already uncertain in the first decimal place, no amount of precision in the others can rescue the second decimal place.

Look at what happened. Both starting numbers had three significant figures. The answer has one.

Subtraction is the only operation that can do this. The two numbers were close together, so almost everything cancelled and only the small difference survived — and that small difference was exactly the part you were least sure about.

The practical lesson: if you need a small difference, measure the difference directly rather than measuring two big things and subtracting.

These two cases are the ordinary rounding you already know. Look at the digit you are throwing away. Bigger than 5, round up. Smaller than 5, leave it alone.

This is different from what most people are taught, so read it twice. When the dropped digit is exactly 5, you look at the digit before it and let that decide.

Even before it → drop the 5. Odd before it → round up.

Why bother? Because "always round 5 up" pushes every borderline case in the same direction. Do a hundred calculations that way and your results drift steadily upward. The even/odd rule sends about half of them up and half down, so the drift cancels out.

Notice that both of the book's examples land on 2.74 — one coming down, one going up. If you remember that coincidence, you have remembered the whole rule.

The book proves this with a neat example. The reciprocal of 9.58, rounded to three figures, is 0.104. Now take the reciprocal of 0.104 to three figures and you get 9.62 — you have lost the number you started with.

But keep 1/9.58 = 0.1044 through the middle step, and taking its reciprocal brings you right back to 9.58.

Rounding early throws away information you still needed. Round once, at the very end.

This is the first time the chapter puts a number on the uncertainty instead of hiding it in the digits. Follow the three moves:

One. Turn each absolute error into a percentage. 0.1 out of 16.2 is about 0.6%. 0.1 out of 10.1 is about 1%. The same 0.1 cm is a bigger deal on the smaller measurement.

Two. For a product, the percentages add. 0.6% + 1% = 1.6%.

Three. Turn the percentage back into a real number. 1.6% of 163.62 is about 2.6 cm².

Then round sensibly. Since the uncertainty is around 3 cm², there is no point quoting the area to two decimal places — hence 164 ± 3 cm².

The same warning as before, in a slightly different form. Look at 12.9 — it stops at one decimal place. So the answer cannot go beyond one decimal place either, whatever the calculator shows.

The book explains why in one phrase: uncertainties in addition and subtraction combine by decimal places, not by significant figures.

Same balance, same ± 0.01 g, very different quality of measurement.

0.01 out of 1.02 is a hundredth of the whole thing. 0.01 out of 9.89 is only a thousandth. The bigger the object, the less that fixed uncertainty matters.

This is a genuinely useful idea in the laboratory. If you need the mass of one small item accurately, weigh a hundred of them together and divide — the balance's uncertainty gets shared out.

The same advice as in §1.3.2, repeated because it matters. Keep a spare digit through the middle of a calculation; round only when you write the final answer.