The size and scale of cells

How many times smaller than a dust mite is a ribosome, a tiny part inside a cell? A mite in a bedsheet is about a third of a millimetre long. Both get called small, so a guess of a hundred times sounds generous. It is not even close: that gap is bigger than the one from you down to the mite. Working it out, and writing it down, starts with a ladder of units. The mite and the ribosome come back at the end, once the ladder and its tools are built.

The ladder starts at the millimetre, a thousandth of a metre, roughly the thickness of ten sheets of paper. One step down is the micrometre, a thousandth of a millimetre, so a thousand micrometres make one millimetre. One more step down is the nanometre, a thousandth of a micrometre. That means a thousand nanometres make one micrometre. Below the nanometre sits the picometre, one more rung down. Each prefix is a power of ten. Milli means a thousandth, which is ten to the minus three. Micro is ten to the minus six, nano ten to the minus nine, and pico ten to the minus twelve. How many nanometres, then, make one millimetre? A million. One millimetre is a thousand micrometres, one step down. Each of those micrometres is a thousand nanometres, one more step. Multiply a thousand by a thousand, and that is a million nanometres in one millimetre. Every step on this ladder is a factor of a thousand, never ten. Centimetres to millimetres is a factor of ten, which is why ten feels natural. Use ten on this ladder, though, and the answer comes out a hundred times wrong. Going down the ladder, to a smaller unit, you multiply by a thousand, because more of the small units fit into the same length. Going up, to a bigger unit, fewer fit, so you divide by a thousand. Some real sizes sit on that ladder. A bacterium is about one to two micrometres long, and an animal or plant cell about ten to a hundred micrometres across. A mitochondrion is about one to two micrometres, and a ribosome about twenty to thirty nanometres. Nought point nought five millimetres: five, fifty or five hundred micrometres? Fifty micrometres. Millimetres to micrometres is one step down. Going down means multiply by a thousand. Nought point nought five times a thousand is fifty. Now a different start. Some lengths arrive in centimetres, and the centimetre sits just above the millimetre, off the top of this ladder. A cell is nought point nought nought three centimetres wide. How many micrometres is that? Thirty micrometres. Centimetres to millimetres comes first, a step of ten, and going down means multiply. Nought point nought nought three times ten is nought point nought three millimetres. Then millimetres to micrometres is one step down the ladder, so times a thousand. Nought point nought three times a thousand is thirty. One examiner's report on a Higher paper says many students fail to recognise that they need to convert centimetres to millimetres before converting to micrometres. It adds that many divide by factors of a thousand rather than multiply. In plain words, some skipped the centimetre step, and some went the wrong way. The fix: take the step of ten first, then go a thousand at a time. Before each step, check the direction, because down to a smaller unit means multiply.

An order of magnitude is a factor of ten. One size is an order of magnitude bigger than another when it is about ten times bigger, and two orders when it is about a hundred times bigger. Take an animal cell a hundred micrometres across and a bacterium one micrometre long. Both are in micrometres already, so we divide. A hundred divided by one is a hundred. A hundred is ten times ten, two factors of ten, so the cell is two orders of magnitude bigger. Before you compare two sizes, put them both in the same unit. Bacterium two micrometres, ribosome twenty nanometres: how many orders of magnitude apart? Two orders. Micrometres to nanometres is one step down the ladder, so times a thousand. Two times a thousand gives two thousand nanometres. Now two thousand divided by twenty is a hundred. A hundred is ten times ten, so the gap is two orders of magnitude.

Standard form writes a very big or very small number in three parts. One digit that is not zero, then a decimal point and the rest, then times ten to a power. The number in front always has exactly one digit before its point. The power of ten says how many places the point moved to get there. A negative power means dividing by ten that many times, which makes the number smaller than one. So the sign has a rule you can check in a second: a length smaller than one metre gets a negative power. Take a mitochondrion, nought point nought nought nought nought nought two five metres long. First, the unit conversion, on a line of its own. The metre sits one rung above the millimetre, so we multiply by a thousand. That gives nought point nought nought two five millimetres. Then millimetres to micrometres is one more step down, so times a thousand again. That gives two point five micrometres, a believable size for a mitochondrion. Now the standard form, back in metres. The number in front comes first: two point five, with one digit before the point. Then count how far the point moved, from nought point nought nought nought nought nought two five to two point five. It jumps six places to the right. Is it two point five times ten to the six, or ten to the minus six? Minus six. Remember the sign rule: a mitochondrion is smaller than one metre, so the power has to be negative. That makes it two point five times ten to the minus six metres. That gives four faces of one length. The long decimal is nought point nought nought nought nought nought two five metres. The standard form is two point five times ten to the minus six metres. The third face is two point five micrometres. The fourth face is one more step down the ladder. Two point five times a thousand is two thousand five hundred nanometres. Only the long decimal needs its noughts counted, and a miscounted nought hides well in it. One note on tiers. On some boards this is Foundation and Higher content. On others, calculating with numbers in standard form is marked Higher, and the magnification video covers that part. Either way, everybody should watch this part.

A second examiner's report, on another Higher paper, says some students did not understand the term standard form, and gave a number written in decimal form. It adds that this value was usually an incorrect conversion of picograms to grams. The same report says common incorrect attempts at the standard form included six times ten to the minus six, or six times ten to the minus nine. In plain words, some students wrote the long decimal instead, and some got the power of ten wrong. The habit that fixes it: convert the unit on its own line first, then write the standard form, then check the sign against the size. Now try one like theirs: six picograms in grams. What is the power of ten? Minus twelve. Pico means ten to the minus twelve, so six picograms is six times ten to the minus twelve grams. Six already has one digit before its point, so nothing moves. And six picograms is far less than one gram, so the negative power checks out. Here is a different slip. A student writes the width of a ribosome, twenty-five nanometres, as two point five times ten to the nine metres. It is marked wrong, and there are two faults in it. What are the two faults in two point five times ten to the nine metres? First, the sign. A ribosome is far smaller than a metre, so its power must be negative. Ten to the nine metres would be a million kilometres. Second, the size of the power. Nano means ten to the minus nine, so twenty-five nanometres is twenty-five times ten to the minus nine metres. But twenty-five has two digits before its point, so that is not standard form yet. We turn twenty-five into two point five, which divides it by ten. To keep the length the same, the power of ten has to go up by one. Minus nine, plus one, is minus eight. The ribosome is two point five times ten to the minus eight metres across.

Estimating is the right tool when you need one size compared with another and a rough answer will do, like checking that a worked answer is sensible. On a slide, a plant cell is about forty micrometres across each way, and its nucleus about ten micrometres across. A slide shows a slice through the cell, so this is about area on that slice. Does the nucleus take a quarter, an eighth or a sixteenth of the cell's area? About a sixteenth. Draw the cell on a grid four squares by four, and the nucleus fills about one square of the sixteen. Now the reason: why a sixteenth, when the nucleus is a quarter of the width? Start with the width. Ten divided by forty is a quarter, so the nucleus is a quarter of the cell's width. It is also a quarter of the height. A quarter of a quarter is a sixteenth, so area shrinks much faster than length.

Now the dust mite and the ribosome from the start, with the whole ladder to hand. The mite is a third of a millimetre. One step down the ladder, a third of a thousand micrometres is about three hundred and thirty micrometres. One more step down, times a thousand again, gives about three hundred and thirty thousand nanometres. Three hundred and thirty thousand against twenty-five nanometres: how many orders apart? About four. Both are in nanometres, so we divide. Dividing by twenty-five is the same as dividing by a hundred and then multiplying by four, because four twenty-fives make a hundred. Three hundred and thirty thousand divided by a hundred is three thousand three hundred. Times four, that is thirteen thousand two hundred, about thirteen thousand. That is roughly ten thousand, and ten thousand is ten to the four. That makes the gap four factors of ten, four orders of magnitude. Now the other half of the opening claim. Say you are one and a half metres tall. Metres to millimetres is times a thousand, so that is one thousand five hundred millimetres. Dividing by the mite's third of a millimetre is the same as multiplying by three. So you are four thousand five hundred times the mite's length. The ribosome, then, is about thirteen thousand times smaller than the mite, not a hundred. That beats the four and a half thousand from you down to the mite. The one word small was hiding that gap, and a thousand on every rung, written as powers of ten, is what lets you write it down.

Before the dust mite goes back into the bedsheet and the ribosome back into its cell, three quick ones with sizes this video has not used. Three micrometres: how many nanometres is that? Three thousand nanometres. Micrometres to nanometres is one step down the ladder, so three times a thousand. Now a harder one: nought point nought nought nought nought four metres in standard form? Four times ten to the minus five metres. The point moves five places to give four, and the length is smaller than a metre, so the power is negative. Now an area: a structure a fifth of a cell's width covers what fraction of its area? About a twenty-fifth. It is a fifth of the width and a fifth of the height, and five times five is twenty-five.

If you own this one now, give it a thumbs up, and you will know not to come back to it. If the ladder still wobbles, save the video and write the three rungs out once by hand; written once, it tends to stay.

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