PrepShorts · Study sheet · Class 8 Mathematics · Chapter 2, Power Play
Chapter 2 · Power Play
Naming the powers of ten: the Lalitavistara list, the million-to-decillion names, and the googol
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Ancient India named the powers of ten far past anything anyone could count. The googol was named by a nine-year-old.
The idea
Every culture that has ever named large numbers has solved the same problem the same way: choose a step in the exponent, then give each step a word. The Indian names step by two — a hundred of one name makes the next — so they climb 10⁵, 10⁷, 10⁹, 10¹¹ and onward. The American and international names step by three — a thousand of one makes the next — so they climb 10⁶, 10⁹, 10¹². The ancient lists choose differently again: the Lalitavistara steps by two, while the Amalasiddhi names every power of ten to 10⁹⁶ and Mahaviracharya's list of 24 terms runs to 10²³ a step at a time. Different step, same device, and each reaches its ceiling with a vocabulary a person could hold. So the naming is not decoration around the mathematics. It is the mathematics — the same addition of exponents the chapter has been doing all along, carried out in words — and the step size is the thing to read off any such list.
What you should be able to do
- Read a naming ladder and state the step it uses, in exponent terms
- Convert between a name and a power of ten, in both systems
- Show why a hundred lakhs is a crore using the addition of exponents
- Compare the Indian and the international ladders and say where they agree and where they part
- Continue either ladder for one or two further names, given its step
- Identify the counting prefix inside an international name and match it to the exponent
- State what a googol and a googolplex are, in powers of ten
- Place a historically named number on a ladder of powers of ten
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| lakh / crore / arab / kharab | Indian names for 10⁵, 10⁷, 10⁹ and 10¹¹ | printed in this chapter (Part I pp.36, 43) |
| neel / padma / shankh | Indian names for 10¹³, 10¹⁵ and 10¹⁷ | printed in this chapter (Part I p.43) |
| maha shankh | the Indian name for 10¹⁹ | printed in this chapter (Part I p.43) |
| million / billion / trillion | international names for 10⁶, 10⁹ and 10¹² | printed in this chapter (Part I p.43) |
| quadrillion / quintillion / sextillion | international names for 10¹⁵, 10¹⁸ and 10²¹ | printed in this chapter (Part I p.43) |
| googol | the number 10¹⁰⁰ | printed in this chapter (Part I p.43) |
| googolplex | 10 raised to a googol | printed in this chapter (Part I p.43) |
| koṭi | a Pali and Sanskrit word for ten million, used as a base for building larger names | printed in this chapter (Part I pp.42–43) |
| sahassa | the Pali word for thousand, used the same way | printed in this chapter (Part I p.43) |
| asaṅkhyeya | the name at the top of the Kāccāyana list, 10¹⁴⁰ | printed in this chapter (Part I p.42) |
| step of the ladder | the constant jump in exponent between one name and the next | an added term; not printed in this chapter, which shows the step without naming it |
Where people slip up
- "Indian and international names are two ways of saying the same thing." They step differently. A hundred of one Indian name gives the next; a thousand of one international name gives the next. Only where the ladders happen to coincide — arab and billion, both 10⁹ — do the two systems name the same number.
- "A billion is a lakh crore, or some fixed multiple I should memorise." Convert through exponents and no memorising is needed: 10⁹ against 10⁷ is two rungs, so an arab is a hundred crore, and an arab is a billion. Do it once with exponents and the relations stop needing storage.
- "These old names are curiosities." They encode the same exponent arithmetic being taught in this chapter, and three of them — kharab, neel, padma — are used as working units on the previous page.
- "A googolplex is a googol times a googol." It is 10 raised to a googol, an exponent tower, not a product. The distinction is the whole reason the name exists.
- "A googol is the biggest number with a name." The chapter prints asaṅkhyeya at 10¹⁴⁰, which is larger, and a googolplex, which is beyond comparison with either.
- "A hundred-trillion-dollar note means the country was rich." It means each dollar was worth very little. The chapter gives the exchange figure — about thirty US dollars — precisely so the point is unmissable.
- "The names go up smoothly." The Lalitavistara list names odd powers only. Different lists choose different steps, and reading the step is the skill.
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Worked answers to this chapter’s exercises
Transcript1,370 words
Two people write down the same number. One writes one, comma, zero zero, comma, zero zero zero. The other groups the same digits in threes. Same digits. Same quantity. Different commas. And they read it aloud differently: ten lakh, or a million. Neither is wrong. It is a story about what a name for a big number actually is. Underneath both systems is a single choice that decides the rest.
See the choice, and you can read any naming system off its own words. Here is the choice. You cannot give every number a name — there are too many. So you pick a step, and name only the numbers it lands on. Everything else is described with the names you have. The step is not a step in the number. It is a step in the exponent. And that matters, because multiplying powers of ten adds their exponents.
A step of one means each name is ten times the one below. A step of two means a hundred times. A step of three means a thousand times. That is the whole design. Choose the step, and the words follow. Take the system used across most of South Asia. A thousand is ten to the three. A lakh is ten to the five. A crore is ten to the seven. An arab is ten to the nine.
Read the exponents: three, five, seven, nine — two at a time. A step of two, so a hundred of one name makes the next. A hundred thousand is a lakh. A hundred lakh is a crore. A hundred crore is an arab. And it keeps going: kharab at eleven, neel at thirteen, padma at fifteen, shankh at seventeen. Maha shankh at nineteen. Nine names, one rule, and it never varies.
Now the other system. A million is ten to the six. A billion is ten to the nine. A trillion is ten to the twelve. Six, nine, twelve — three at a time. A step of three, so a thousand of one name makes the next. A thousand million is a billion. A thousand billion is a trillion. And onward: quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion. Decillion sits at ten to the thirty-three.
Ten names, and again it never varies. Two ladders, then, leaning against the same wall, with rungs at different heights. So where do the rungs line up? Not by looking it up — lay one over the other and see which heights appear in both. Both ladders reach into the stretch from ten to the six up to ten to the nineteen. Inside that stretch, they agree in exactly two places.
The first is ten to the nine: an arab, and a billion. The second is ten to the fifteen: a padma, and a quadrillion. That is it. Two. Everywhere else one ladder has a word and the other has none. Why two, and why those two? One ladder lands on exponents going up in twos. The other lands on exponents going up in threes. They can only meet at a height that both steps reach.
So the meetings happen at the smallest number both two and three divide — which is six. Six is three steps of the first ladder and two steps of the second. That is why the meetings are six apart: nine, then fifteen. And why there is no third meeting in the stretch they share: the next is at twenty-one, above where the shorter ladder stops. Two systems agreeing twice is not a coincidence anybody arranged.
It falls straight out of two and three. Now the part that usually goes unasked. Million, billion, trillion, quadrillion, quintillion, sextillion. Those are not arbitrary sounds. Bi is two. Tri is three. Quad is four. Quint is five. Sext is six. Every one of those names has a counting number at the front. So what is it counting? It is not counting the exponent — a billion is ten to the nine, not ten to the two.
It is not counting the zeros either. It is counting thousands — but carefully. A million is a thousand thousand. One thousand after the first. A billion is a thousand million: two thousands after the first. A trillion, three. So the rule is: take the number in the name, add one, multiply by three. Billion: two, plus one, times three, is nine. Trillion: three, four, twelve. Right again. Run it on all ten names and it is right every time, up to decillion at thirty-three.
A rule that survives ten tests makes a prediction: the name after decillion sits at ten to the thirty-six. Which is exactly where the step of three puts it. These two are not the only ladders that have ever been built. One very old list names only the odd powers, starting at ten to the nine. Ayuta, niyuta, kankara, and on upward through twenty-three names in all. Its last name lands on ten to the fifty-three.
Another list takes twenty-four terms to reach ten to the twenty-three. Twenty-four names ending at twenty-three: so it steps by one, from ten to the nothing. It names every power, one at a time, and stops. A later list does the same to ten to the ninety-six — ninety-seven names. And one goes further still, to ten to the one hundred and forty. The most famous large number of all is on no ladder.
A googol: ten to the hundred. It was named by a child who was asked what to call it, and the name stuck. It is bigger than the number of atoms in the observable universe. Estimates for that run to about ten to the eighty-two. So a googol is eighteen powers of ten above every atom there is. Not eighteen times. Eighteen powers — each one a factor of ten on the last.
But here is what people get wrong. A googol is not the largest number anyone has named — that old list ending at one hundred and forty passed it by forty powers. There is a bigger one, though, and it is built out of the googol. A googolplex: ten to the power of a googol. Say that slowly, because it is a tower three floors high. Ten, raised to ten, raised to a hundred.
The commonest mistake is to hear that as a googol times a googol. It is not. That product is only ten to the two hundred. A googol itself takes a hundred and one digits to write. That product takes two hundred and one digits to write. A googolplex has a googol zeros. There is not enough matter in the universe to write it down. None of this is only decoration.
Counts of living things use these names because nothing else is short enough. Thirty kharab trees: three times ten to the twelve. Eleven neel mosquitoes: one point one times ten to the fourteen. One padma beetles: ten to the fifteen exactly. And these words have ended up on money, too. The largest number ever put on a banknote was ten to the twenty-one — a sextillion, on a note never issued.
A note people did spend reached ten to the fourteen — a hundred trillion — and bought about thirty dollars. Thirteen powers of ten between the number on the paper and what the paper was worth. One last thing. Here is a row from a table of these names. It says trillion. It says ten to the twelve. And beside those, one followed by some zeros. Count them. There are eleven.
A trillion needs twelve, so that written-out numeral is short by exactly one zero. The name was right. The exponent was right. Only the long form was wrong. That is the argument for exponents: a missing zero is invisible, a missing twelve is not. So here is the test, and it is one question long. Given any name for a big number, what step does its system take? Answer that, and it tells you where every other name has to land.
The words are different everywhere you go, and they always will be. The exponent underneath them is the same number in every language on Earth.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- The laws of exponents, and where each one comes fromClass 8 · Ch 2, Power Play
- Scientific notation, and why the standard form is 1 ≤ x < 10Class 8 · Ch 2, Power Play
- Why the nearest power of ten is the only handle on a quantity too big to pictureClass 8 · Ch 2, Power Play
Either side of this one
- Why any number system needs a fixed, ordered sequence of symbolsClass 8 · Ch 3, A Story of Numbers