How Long Is a Crore Seconds?

Counting up from a minute

Start with things you can feel. A minute is 60 seconds. An hour is 60 minutes, so 3,600 seconds. A day is 24 hours, so 86,400 seconds. A year of 365¼ days (the quarter for the leap years) is 31,557,600 seconds, a little more than 3 crore.

Now climb up the number names, one at a time. Work out each line before you open the answer.

Look at how the times jump. Going from a lakh to a crore multiplies the seconds by 100: from a day to almost four months. Going from a crore to an arab multiplies by 100 again: from four months to more than thirty years. The names sound like small steps, one after the other, but each step in Nepal’s list is 100 times the one before.

Counting out loud

Bibek’s plan has a second problem. Saying “one” takes less than a second, but try saying 99,99,999, unansaya lakh unansaya hajar nau saya unansaya, in one second. Most of the numbers below a crore have seven digits, and saying them takes several seconds each.

Here is a quick test of your feeling for big numbers. A line runs from 0 at one end to 1 arab at the other. Where on it does 1 crore belong? Don’t calculate: drag the marker where it feels right, then check.

If your mark was far to the right, you are in good company. In 2013 three researchers in the United States, David Landy, Noah Silbert and Aleah Goldin, asked adults to put large numbers on a line like this one. About half placed them in proportion, as on a ruler. Nearly all the others put 1 million about halfway between 1 thousand and 1 billion, as if the words thousand, million, billion, trillion were evenly spaced steps, like 1, 2, 3, 4. They are not. On a line from 0 to a billion, a million sits a thousandth of the way along: on a line one meter long, one millimeter from the zero.

Same places, two ways to group

Every number we write is built from place value: each place is worth 10 times the place to its right. The ones, the tens, the hundreds, the thousands, the ten thousands, and so on, every place ten times the last, for ever. Nothing about the places changes from one country to another.

What changes is how we group the places to read them. Nepal’s way, which school books call the national system, puts the first comma after three digits and then one after every two: 1,00,00,000. The international system, used in most of the world, puts a comma after every three digits: 10,000,000. The groups are called periods, and each period has a name.

Because one system groups by twos and the other by threes, their names land on different places. They line up only now and then:

Nepal’s name	Digits	International name
1 lakh	1,00,000	100 thousand
10 lakh	10,00,000	1 million
1 crore	1,00,00,000	10 million
10 crore	10,00,00,000	100 million
1 arab	1,00,00,00,000	1 billion
1 kharab	1,00,00,00,00,000	100 billion
10 kharab	10,00,00,00,00,000	1 trillion

So a million is ten lakh, and a crore is ten million. An arab and a billion are the same number. A trillion is ten kharab.

Counting the zeros: powers of ten

When numbers get this big, the commas and names stop helping. Count the zeros instead. A crore is 1 followed by 7 zeros: ten multiplied by itself 7 times, which you write as 10⁷ (Grade 8’s indices). An arab is 10⁹. A kharab is 10¹¹. Each extra zero is one more multiplication by ten.

A number that isn’t a plain 1 followed by zeros is written as a number from 1 up to (but not including) 10, times a power of ten. This is scientific notation, which Grade 8 introduces. Nepal’s people at the 2021 census, 2,91,64,578, become

$$29{,}164{,}578 = 2.9164578 \times 10^{7} \approx 2.9 \times 10^{7}.$$

A calculator or a spreadsheet often shows such a number as 2.9164578E7 or 2.9164578E+07: the E means “times ten to the power”, so E7 is × 10⁷. A tiny number shows as, say, 4E-04, that is, 4 × 10⁻⁴.

The power of ten tells you the size at a glance: 10⁷ means “tens of millions”, “crores”. The first part, 2.9, says how many. The power of ten is called the number’s order of magnitude. Two numbers with the same order of magnitude are roughly alike in size; one order of magnitude more means about ten times as big.

How many times bigger?

The real question about big numbers is usually not “how big?” but “how many times bigger?” Powers of ten make that easy. An arab divided by a crore is

$$\frac{10^{9}}{10^{7}} = \frac{10 \times 10 \times \cdots \times 10 \ (9 \text{ tens})}{10 \times 10 \times \cdots \times 10 \ (7 \text{ tens})} = 10^{2} = 100.$$

Seven of the tens on top cancel with the seven below, and two are left. That is why you subtract the powers when you divide, and add them when you multiply: 10³ × 10⁴ = 10⁷, because three tens and four tens make seven tens.

Choosing the right unit

A number of seconds that big isn’t much use until you turn it into a unit you can picture. Divide by the seconds in one of that unit, and the powers of ten tell you what’s sensible.

Now take the seconds backwards. Start from today, and step back 10³ seconds, then 10⁴, then 10⁵, each step ten times as far as the one before. Where do you land?

Two things stand out. First, a few steps cover everything. Nine steps from a thousand seconds take you from a quarter of an hour ago to the Ice Age; five more pass the dinosaurs and reach a time when the only life on Earth was microbes. Each step is the same small move on the slider, but it multiplies the time by ten.

Second, from 10⁸ seconds on, the digits 3, 1, 7 come back at every step: 3.17 years, 31.7 years, 317 years, 3,170 years, and so on. That isn’t a coincidence. A year is about 3.16 × 10⁷ seconds, so 10⁸ seconds is 10⁸ ÷ (3.16 × 10⁷) ≈ 3.17 years, and every ten times more seconds is ten times more years with the same digits. Multiplying by 10 only moves the decimal point.

A stack of paper

A sheet of ordinary copy paper is about 0.1 mm thick. Count out sheets and stack them, one on top of another.

When will you be an arab seconds old?

All of this works only if everyone means the same number by the same word. They don’t always.

A billion used to be bigger. In Britain, a billion traditionally meant a million million, 10¹². In 1974 the Prime Minister told Parliament that “billion” was now used internationally to mean a thousand million, and that British ministers would use it that way too. Many languages still use the old meaning: in French a billion is 10¹², and 10⁹ is a milliard; in German 10⁹ is a Milliarde. A French newspaper’s “un billion” is a thousand times an American’s “one billion”.

Old names changed their size too. The Shukla Yajurveda, one of the Vedas, lists names for the powers of ten from eka (1) up to parārdha (10¹²). In that list arbuda is 10⁷, the size of today’s crore, while today’s Nepali arab is 10⁹. Later texts don’t agree with each other on the names above 10⁸, and even today, as Topic 1 found, lists of Nepali names above kharab disagree.

• How Much Rice Is on the Chessboard? Numbers that grow by doubling, and how fast they run past every name on this page.
• How Much Rice Does Nepal Eat in a Day? Using powers of ten to estimate big numbers you can’t count.
• Is the Answer Roughly Right? Rounding to check a calculation in your head.
• Why Is a Magnitude 8 Earthquake Not Just a Bit Bigger Than a 7? (Grades 9–12) Scales where each step is ten times the last, like the slider in this topic.
• What Is the Biggest Number Anyone Has Used? (Grades 9–12) Numbers that make a tallakshana look tiny.
• What Is the Biggest Number a Computer Can Count To? (Computing, Going Further) Why a computer’s numbers run out, and what happens when they do.