Scientific notation, and why the standard form is 1 ≤ x < 10

0:00Three quantities. Nobody can read any of them aloud correctly, and that is the point. The first is a distance in metres: a three, and then twenty zeros. Twenty-one digits. The second is a count of stars: a one, and eleven zeros. Twelve digits. The third is a mass in kilograms: five, nine, seven, six, and then twenty-one more digits. Twenty-five in all. Try to compare those by eye and you are not comparing the numbers.

0:35You are counting the digits. And what you actually want to know about a quantity that size — how big it is — lives in that count. Which is the hardest thing on the page to see. Here is what a miscount is worth. You are owed fifty thousand rupees, and you are handed five thousand. A five and three zeros, against a five and four zeros. One digit. Ten times the money.

1:05Written the other way, that is five times ten cubed against five times ten to the fourth. Same coefficient. Exponent one apart. And one step in the exponent is always exactly ten times. So move that count into an exponent, where it can be read at a glance. That is the whole idea — and it does not work yet. Take five thousand nine hundred. Five hundred and ninety, times ten.

1:36Correct. Fifty-nine, times ten squared. Also correct. Five point nine, times ten cubed. Correct. Nought point five nine, times ten to the fourth. Correct as well. Four coefficients, four exponents, one number, and every one of those lines is true. Watch what they are doing: every step right in the exponent is a step left in the coefficient. They trade against each other. Which means the exponent, on its own, tells you nothing.

2:13I can hand you almost any exponent I like for that number and be arithmetically correct. If two people write one quantity with different exponents, the exponent cannot measure size. So pen the coefficient in. It has to be at least one, and less than ten. Of those four lines, exactly one obeys that. Five point nine times ten cubed. Five hundred and ninety is too big; fifty-nine is too big.

2:44And nought point five nine is too small — the coefficient has dropped below one. Not wrong. Not in the agreed form. Those are different things, and the difference matters. Now the fussy-looking half of the rule, which is not fussy at all. Less than ten. Not up to and including ten. Why exclude the top end? Take a thousand. Under the rule as written it has exactly one form: one times ten cubed.

3:16Let the coefficient reach ten, and it gets a second one: ten times ten squared. Two forms, one number, and the exponent is ambiguous all over again. Sweep a whole pool of numbers under the loose rule and the ones that break are exactly the powers of ten. So the excluded end is not neatness. It is what makes the form unique — and a form that is not unique cannot measure anything.

3:47Here is the form, then. A coefficient, times ten to a power. The coefficient: at least one, and less than ten. The exponent: any whole number at all — positive, negative or zero. It travels under three names — scientific notation, scientific form, standard form — and all three mean this. Five thousand nine hundred is five point nine times ten cubed. Twenty thousand eight hundred is two point zero eight times ten to the fourth.

4:22Eight million is eight times ten to the sixth. And look hard at that middle one. Twenty thousand eight hundred contains three zeros, and its exponent is four. The exponent never counted zeros. It counts the places the point moved. Now, which half of the form is the important one? A city of two crore people: two times ten to the seventh. Change one character. Make the two a three.

4:53Three crore — the city has grown by half. Put the two back, and change the seven to an eight. Twenty crore. One edit multiplied the city by one and a half. The other multiplied it by ten. Same amount of ink. So when you compare two quantities written this way, the exponent goes first, every time. A bigger coefficient can still be the smaller number: nine point nine times ten to the eleventh is smaller than one point one times ten to the twelfth.

5:30The other half of this is a question nobody asks about ordinary numbers. How many digits should the coefficient have? Suppose a town's population is written as one lakh, forty-two thousand, three hundred and ninety-five. Written like that, you have claimed to know it to the last person. If you only know it to about a hundred and forty-two thousand, write one point four two times ten to the fifth.

6:00If you only know it to about a hundred and forty thousand, write one point four times ten to the fifth. Three digits says: I know this to within five hundred either way. Two digits says: to within five thousand. Those digits are not decoration. Each one is a claim. More digits is not more accurate. More digits is a stronger claim about how well the thing is known — and a claim you cannot back is not precision.

6:32There is a joke about this that is really a piece of arithmetic. A visitor asks a museum guide how old a fossil skeleton is. Seventy million and fifteen years, says the guide. It was seventy million when I started here, and I have worked here fifteen years. Seventy million, given to one leading digit, cannot see fifteen years. It cannot see five million. The span a one-digit report cannot tell apart runs from sixty-five million to seventy-five million.

7:08Ten million years wide. You would have to know the age to seven leading digits before fifteen years changed the answer, and that means knowing it to within five years. Nobody knows a fossil to within five years. The fifteen cannot be added. That is arithmetic, not pedantry. Three distances, all in metres. Sun to Saturn: one point four three three five, times ten to the twelfth. Saturn to Uranus: one point four three nine, times ten to the twelfth.

7:45Sun to Earth: one point four nine six, times ten to the eleventh. Which is smallest? All three coefficients start one point four. They look interchangeable. So ignore them, and read the exponents. Twelve, twelve, eleven. The eleven is the smallest, and it is settled — no arithmetic at all. Only the other two need their coefficients: one point four three three five against one point four three nine, so Sun to Saturn is the nearer pair.

8:22Now draw a line: the Sun at the left end, Saturn at the right. Where does the Earth go? Most people reach for somewhere near the middle. The exponents say otherwise. Twelve and eleven — one step apart. And one step is ten times. So the Earth sits about a tenth of the way along. The whole of the Earth's orbit fits inside the first tenth of the distance out to Saturn.

8:49You computed nothing. You read one digit. That is what the coefficient rule bought you. Four to convert, and there is something hiding in one of them. Fifty-nine thousand, eight hundred and fifty-three. Sixty-five thousand, nine hundred and fifty. Thirty-four lakh. And seven thousand and four crore. The answers, in order: five point nine eight five three times ten to the fourth. Six point five nine five times ten to the fourth.

9:24Three point four three times ten to the sixth. And seven point zero zero four times ten to the tenth. Look hard at that last one. The zeros at the end of a number go into the exponent and vanish. The zeros in the middle cannot. Drop them and you have named a different quantity, nearly four thousand million larger. So: the rule. Coefficient at least one and below ten; exponent any whole number.

9:56It looks like a rule about neatness. Without it, one quantity has unlimited forms, and the exponent is just a choice somebody made. With it, every positive quantity has exactly one form. And once the form is unique, the exponent becomes the thing the whole notation was built to give you — one readable number that says how big something is. It is why the smallest of three planetary distances needs no arithmetic.

10:29It is why fifteen years cannot be added to seventy million. The rule is not tidiness. It is what makes the exponent worth trusting.