🪙 Double your money — one dollar, doubled every day
← Curious CalculatorsA dollar doubled every day for a month is $1,073,741,824, and half of that arrives on the last day. That is the whole story of doubling, and nobody believes it the first time. Put in a starting amount and a length — days, weeks, fortnights, months or years — and this page gives the figure exactly to the cent, the pile of $100 notes it would make, where it sits on a ruler of money, and the day each of the famous marks goes past.
The money
Where that sits on a ruler of money
A straight ruler is no use here — the first twenty-nine days of a doubling run look like nothing at all, which is exactly what doubling looks like from the middle of it. So this one is a ruler of ten times per step: a dollar at the left, a thousand, a million, a billion and a trillion going across. The filled part is where the money ends up.
The same run, read every five days
| Point in the run | You have | Mark passed |
|---|
Exact arithmetic, and nothing else. Every figure is worked out by counting the money in cents and doubling it — so the answers are right to the cent however long the run is, which is more than a floating-point calculator can say once the numbers pass seventeen digits. There are no fees, no taxes, no inflation and nobody to double it for you: this is the shape of doubling, not an offer, and a dollar doubled a day is not something that happens to anybody. The pile of notes assumes a $100 note weighs about a gram and is about a tenth of a millimetre thick — the same two figures the Musk net worth page uses on a trillion. Lengths are counted in whole days: a week is 7, a fortnight 14, a month 30 and a year 365, and the run stops at a year because a dollar doubled 365 times has 110 digits in it and would not fit on the page.
Why use this calculator?
Because the answer is unbelievable, and why it is unbelievable is the useful part. Anyone can do the arithmetic of doubling — nobody can feel it. Thirty doublings of a dollar is over a billion dollars, and the shape of the run is the same shape every exponential thing has: nothing happening for three weeks, and then everything happening at once. The same curve runs under a virus, a savings account, a compound fee and a rumour, and the only way to get a feel for it is to watch it happen to a number you put in yourself. That is what this page is for, and the dollar is the cheapest way in.
The chessboard, the paper and the shape of doubling
Two old stories use exactly this arithmetic, and a third fact is the reason it keeps catching people out:
- The chessboard. A traveller asks the emperor for one grain of wheat on the first square of a chessboard, two on the second, four on the third, and so on, doubling to the sixty-fourth square. It sounds modest until about the halfway mark. The whole board comes to 18,446,744,073,709,551,615 grains — and the second half of the board holds about 4.3 billion times as much as the first half, which is the part the emperor did not see coming. Nobody knows who told the story first: the grain changes with the teller, and it is wheat in some versions, rice in others. The arithmetic is the part that is certain.
- The folded paper. A sheet of paper 0.1 mm thick, folded in half, is 0.2 mm; folded again, 0.4 mm. Folded 41 times it would be 219,902 km thick, and folded once more — 42 folds — it would be 439,805 km, which is past the moon. This is the same doubling as the chessboard, and it is impossible to actually do: paper cannot be folded more than about eight times by hand, which is the only reason the story works as a story.
- Half of it always arrives last. Because the money doubles, the last doubling adds exactly what was already there: half of the final total is made on the final day, a quarter of it on the day before, an eighth the day before that. It is true of a dollar, of a thousand dollars and of the sixty-fourth square of a chessboard, and it is why a doubling run feels like nothing at all and then like an avalanche.
- And ten of them is a thousand. Two to the tenth power is 1,024, so every ten doublings multiplies the money by about a thousand — which is why the milestones fall ten doublings apart: a million, a billion and a trillion arrive on days 20, 30 and 40 for a dollar that starts the run.
How does this calculator work?
Doubling once is a multiplication by two, and doubling every day for n days is that same multiplication n times over — which is one power of two:
the money after n days = the starting amount × 2^n
- That is the whole of it. One dollar after thirty days is 2³⁰, which is 1,073,741,824 — the figure on the card. Nothing is added, nothing is invested and no rate is involved: doubling has no parameters to argue about, which is what makes it such a clean way to see an exponential curve.
- The last doubling is always half. The money goes from 2^(n−1) to 2^n, and the difference between those two figures is 2^(n−1) — the amount that was already there. So the last day adds exactly what every day before it added together, and the day before that added half as much again. It is an identity, not a coincidence: it is true on day 2 and on day 200.
- Ten doublings is a thousand times more, because 2¹⁰ is 1,024. That is why the marks fall ten days apart for a dollar: 2²⁰ is 1,048,576, 2³⁰ is 1,073,741,824 and 2⁴⁰ is 1,099,511,627,776 — a million, a billion and a trillion, ten doublings between each.
- Counted in cents, so it stays exact. The money is doubled as a whole number of cents rather than as a decimal, which is why a cent doubled thirty times reads $10,737,418.24 to the cent instead of losing its tail to floating point. The length is converted to whole days before anything is doubled — a month is 30 of them and a year 365 — and the pile of notes is a hundred dollars a note, at about a gram each and about a tenth of a millimetre thick.
Frequently asked questions
Is doubling your money every day realistic?
Nothing like it. The best long runs in real markets have averaged around 10% a year, and at 10% money doubles in about seven years rather than every day — that is the rule of 72 doing its work. A daily doubling is around 2,600 doublings quicker than that, which is why no such opportunity exists for longer than it takes somebody to notice. The compound savings calculator is the same arithmetic at a rate you can actually get, and the coffee vs investing page shows what thirty years of a real rate does to a habit.
Why is half the money made on the last day?
Because doubling adds the amount that is already there. Whatever the money is today, tomorrow it is twice that, so tomorrow's gain is today's total — and today's total is half of tomorrow's. The same argument one day back says a quarter of the total was made the day before, an eighth the day before that. Half the final answer is always made on the final doubling, however long the run is.
So what is a dollar doubled every day for 30 days?
$1,073,741,824 — just over a billion, made from a dollar in a month, with $536,870,912 of it arriving on the thirtieth day alone. It is the answer that makes people check the arithmetic twice, and the arithmetic is right.
What if I start with a cent, or with $100?
It is the same run, scaled. A cent doubled thirty times is $10,737,418.24, which is exactly a hundredth of the dollar answer, and $100 doubled thirty times is $107.37 billion. The starting amount moves the answer, but it does not change the shape of it: the day the money passes a million, a billion and a trillion shifts by however many doublings it takes to cover the starting amount.
Where does the chessboard story come from?
It is an old one, told across Persia, India and China with the grain changing to suit the teller — wheat, rice, or whatever the local staple is — and nobody knows who told it first. What does not change is the arithmetic: sixty-four doublings come to 18,446,744,073,709,551,615 grains, and the second half of the board holds about 4.3 billion times as much grain as the first half. The emperor's mistake was to look at the first few squares.
Why does the page stop at a year, and what are those big number names?
A year of daily doublings is 2³⁶⁵, and that figure has 110 digits — more than would fit on a card, and more than any comparison would help with. The names on the way up are the short scale everybody here uses: a thousand million is a billion, a thousand billion is a trillion, then quadrillion, quintillion, sextillion and so on up to a decillion. Past that the page stops naming them and counts the digits instead, because "octodecillion" is not a word anybody has said out loud and the digit count says more.
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