Quantities and units

Twenty thousand or ten thousand: which power bank is better?

0:00I have a power bank here, and on the box it said twenty thousand in big figures. There was also one of ten thousand, and that one was half the price. The question looks easy: which of the two charges my phone more often? Twenty thousand is twice as much as ten thousand, so you already know.

0:25But in that box is a slip of paper that nobody ever reads, and on it, in tiny letters, is something else. And once you read that,

Measuring turns it into a number

0:38Look around you. You see straight away that not everyone is the same height, and you need no ruler for that. That is an observation without a number, and such an observation we call qualitative. Measure how tall someone is and a number comes out, and then you have a quantitative observation.

0:58Counting spectators also gives you a number, but then without a unit: that is the only measurement for which you need no unit.

A number on its own is not a measurement

1:09Two pupils each measure the same length nine times. One writes down: one hundred and seventy-five, one hundred and eighty, one hundred and seventy-two. The other writes down: one point seven nine, one point eight two, one point six four. Bare numbers, both of them.

1:29At first sight those two are measuring something completely different. But they are not: one measured in centimetres, the other in metres. Only you cannot see that on their sheet, because neither wrote that unit after their numbers. The whole rule sits in there.

1:50Measurements are only finished once a unit is written with them: the physical quantity is the number times the unit. And the quantity itself gets a symbol, a slanted letter. The unit gets an upright letter. That difference, slanted or upright, is not styling but information.

Who is taller: 180 or 1.79?

2:10Pause. One wrote one hundred and eighty, the other one point seven nine. Which one is taller? And more importantly: could you answer that question before you knew which unit belonged with it?

Seven quantities the rest follows from

2:28To stop everybody inventing their own unit, one system has been agreed internationally. In it seven quantities are central, each with one fixed unit: length in metres, mass in kilograms, time in seconds, current in amperes, temperature in kelvins, luminous intensity in candela, amount of substance in mole.

2:54Seven, and you need no more than that. Every other unit you meet in physics is built up out of these seven. And there is one oddity in it that you never forget once you have seen it: of all seven there is exactly one that already has a prefix in it. The kilogram.

3:18The kilo belongs to it, it sits in the unit itself.

Everything else is built from these

3:22Take area. You work that out with length times width, and those are both metres. So the unit becomes metre times metre, square metre. Volume is length times width times height: cubic metre. Speed is a distance divided by a time: metre per second.

3:42And density is a mass divided by a volume: kilogram per cubic metre. Do you see what is happening? You do not have to learn those units by heart. They roll out of the formula.

A prefix is a power of ten

3:57Now those tiny letters on the power bank. There is an m in front, and that m is not a letter but a factor. Kilo means a thousand, so a kilometre is a thousand metres. Milli means a thousandth, so a millimetre is a thousandth of a metre. Centi is a hundredth, mega is a million, giga is a billion, micro a millionth and nano a billionth.

4:25It is handiest not to remember them as words but as powers of ten: kilo is ten to the third, milli is ten to the minus three. Then you can no longer mix them up, and that happens mostly with centi and milli. From a large unit to a small one, you multiply. The other way round you divide.

4:50From kilometre to millimetre that is six steps of ten, so a factor of a million.

Do mAh and Wh measure the same thing?

5:00Back to the question from the start. On the box it says twenty thousand mAh. On the hand luggage rule of an aircraft there is a limit in Wh. Two units, two different symbols. Do they measure the same thing? No. The mA in mAh is milliampere, a current, and that times a time gives a charge.

5:25The W in Wh is watt, a power, and that times a time gives an energy. Charge and energy are two different quantities, and that is why you cannot lay those two numbers side by side. You need something more, and you work it out yourself. Three things to remember.

5:48One: no measurement is finished until a unit is given. Quantity is number times unit. Two: every unit is built up out of seven base units, so you can derive the unit from the formula instead of memorising it. Three: a prefix is a power of ten.

6:11Kilo is ten to the third, milli ten to the minus three. Two questions, and try to answer them before the answer appears. One: how many millimetres go into a kilometre? Two: why is the kilogram the odd one out among the seven base units? One: kilo is ten to the third, milli ten to the minus three.

6:42That is six apart, so in one kilometre go ten to the sixth millimetres. One million. And two: the kilogram is the only one of the seven that already carries its prefix in its name. Metre, second, ampere, kelvin, mole and candela have no prefix; in the kilogram the kilo is already there.

7:06You can now tell a quantity from its unit, and you know the seven all the others follow from. As soon as you have to convert a number with a prefix, twenty thousand milliampere-hours to watt-hours for instance, this is exactly what you need.