A power of ten writes a number as ten to an exponent, and a negative exponent means dividing, not a negative number. In scientific notation there is exactly one figure before the decimal point and the rest goes into the power. The order of magnitude is only that power, with no number in front of it.
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The box of your power bank says 20,000. The hand luggage rule of every aircraft says: up to 100 may come along. Twenty thousand against a hundred does not even look close, and yet it may simply come along.
In five minutes you learn why. You see that an exponent is nothing but counting how often you multiply by ten, that a negative exponent means you divide and not that the number is negative, and you convert the 20,000 on the box yourself into the number aviation actually looks at. After that you learn scientific notation, with exactly one figure before the decimal point, and the order of magnitude, which is only ever a power of ten and never a number in front of it.
For upper secondary physics. Halfway there is a question and a silence: pause it there and work it out yourself first, because the answer only comes afterwards.
In this video
The full explanation, in writing
The problem
0:01On the box it says twenty thousand, and the rule for hand luggage on an aircraft says: up to a hundred may come along. Twenty thousand against a hundred. That does not even look close. And yet this one may simply come along. To see why, we need only one thing: powers of ten.
0:20You can write a thousand in three ways. As a thousand.
The same value, four ways of writing it
0:26As ten times ten times ten. As ten to the third. And that third figure, that three, is called the exponent. Keep dividing by ten, the exponent goes down by one each time. One thousand, a hundred, ten, and then one. One is ten to the zeroth. And you can simply carry on.
0:48One tenth is ten to the minus one, a hundredth ten to the minus two, a thousandth ten to the minus three. So a negative exponent does not mean the number is negative; it means that you divide. Then the first step. On the power bank it says twenty thousand mAh.
From 20,000 mAh to ampere-hours
1:11That m is milli, and milli is ten to the minus three. How many Ah is that, without the prefix? Twenty thousand times ten to the minus three is twenty. Twenty ampere-hours then.
The reckoning: 77 watt-hours
1:25Now the second step. On that slip there is something else: three point eight five volts. That is the voltage of the cell inside. And energy is charge times voltage, so: twenty ampere-hours times three point eight five volts is seventy-seven watt-hours.
1:43Seventy-seven, and that is exactly the number the manufacturer puts on it himself. The limit was a hundred. So this one may come along, with room to spare. And the other one, of ten thousand? At the same voltage that comes to thirty-eight point five watt-hours.
2:02Exactly half, because the voltage is the same. There your first feeling was right after all: at the same voltage, twice as many mAh is also twice as much energy. Only you did not know that when you said it. You were lucky. And if that second power bank had been at another voltage, the answer would have been different.
2:25If you work with very large or very small numbers,
One figure before the point
2:29you agree on one way of writing, and that is called scientific notation. Exactly one figure before the point, and the rest you put into the power of ten. Eight thousand three hundred twelve becomes eight point three one two, times ten to the third.
2:48And zero point zero zero seven nine becomes seven point nine, times ten to the minus three. It works backwards too. Three point six one times ten to the second is three hundred and sixty-one. You shift the point as many places as the exponent says: positive to the right, negative to the left.
3:12Sometimes you do not need to know what a number is exactly,
Only the power, nothing else
3:16but only how big it is roughly. Then you give the order of magnitude, and that is exclusively a power of ten. No number in front, only the power. There is a fixed set of steps. First you put the number in scientific notation. Then you throw away everything after the decimal point.
3:36If a figure is left that is smaller than five, you round to one; if it is five or more, you round to ten. And if you ended up at ten, you write that as a power again. Look: the distance from the earth to the moon is three hundred and eighty-four times ten to the sixth metres.
4:01Scientifically that is three point eight four times ten to the eighth. Figures after the point away: three. Three is smaller than five, so round to one. The order of magnitude is ten to the eighth metres. Notice what is not there. Not four times ten to the eighth. Only the power.
4:26Three things. Negative exponents mean dividing, not negative numbers. In scientific notation there is exactly one figure before the point. And the order of magnitude is only a power of ten, never a number times a power. Two questions. One: what is the order of magnitude of six hundred?
4:49Two: why can two power banks with the same mAh still not be compared? You can now write any number in scientific notation and get the order of magnitude out of it. Once you calculate with those powers, or need a right-angled triangle to resolve forces, this is exactly what you need.
Frequently asked questions
What does a negative exponent mean?
That you divide, not that the number is negative. Ten to the minus three is one divided by ten to the third, so a thousandth. The minus sign says which way you go along the powers of ten, not what sign the number has.
What is the order of magnitude of 600?
Ten to the third. Write it in scientific notation as six times ten to the second, throw away what is after the point, and six is five or more, so you round to ten. Ten times ten to the second is ten to the third.
How do you write a number in scientific notation?
You put exactly one figure before the decimal point and the rest into the power of ten. Eight thousand three hundred and twelve becomes 8.312 times ten to the third. Zero point zero zero seven nine becomes 7.9 times ten to the minus three.
What is the difference between scientific notation and order of magnitude?
Scientific notation keeps all your figures: one before the point and the rest in the power. The order of magnitude keeps only the power, with no number in front of it. Four times ten to the eighth is scientific notation; ten to the eighth is the order of magnitude.
Why may a 20,000 mAh power bank go on a plane?
Because the limit is in watt-hours, not in mAh. Twenty thousand mAh is twenty ampere-hours, and at a cell voltage of 3.85 volts that is seventy-seven watt-hours. The limit is a hundred, so it fits with room to spare.
In the next video you calculate with these powers, and the right-angled triangle joins in, which you need as soon as you resolve forces.
Physics with Thomas makes physics explainers for upper secondary school. No words on screen, so what you see works in any language.
In this series
Quantities and unitswhy 20,000 is not more than 10,0007:30
Ratios and percentagesone table for both3:15
Scientific notationpowers of ten and order of magnitude5:22
Calculating with powers of tenthe four rules3:18
Pythagoras and trigonometrywhich side with which angle3:19
Deriving units from a formulaconverting to base units7:47
Significant figureshow many figures do you write6:24
Line of best fitthe line touches no point5:07
Relationships and the constantbraking twice as fast5:20
Linearising a graphyour measurements never change4:41
About me
My name is Thomas Schuurmans, and I have a passion for physics and for understanding why things work the way they do. What I want is to get young people asking “why is that?” and “how does that work?” more often, and to hand them the tools of physics in a way that is simple and that you can see.
I graduated in applied physics at Delft University of Technology in 2003. After that I spent more than twenty years outside education: first at TNO, the Dutch applied research institute, then at a design agency, and eventually founding Proportion Global, through which I work on innovation questions in Africa, Latin America and South Asia. That work resembles physics more than you would expect: don't start from a solution, first understand what is going on, try something, be wrong, and look again.
Since 2026 I have been teaching physics to both lower and upper secondary classes, and I started a master's at the University of Amsterdam for my full teaching qualification. That is where I learned that a secondary school pupil's real attention for new material lasts about seven minutes. And my own weakness happens to be telling too many side stories.
That is where the idea came from: videos that explain one topic sharply and visually, inside those seven minutes. I make them for my own pupils. Then I publish them, because good explanation should be within reach of anyone who needs it, wherever you live and whatever language you think and speak in.