Why Does Doubling Get Out of Hand?

Step 1 of 5

Each step adds more than the last.

Adding always adds the same amount. Doubling adds however much you already have, so every step is bigger than the one before.

Step 2 of 5

The early steps look harmless.

One, two, four, eight. For the first several doublings the numbers stay small enough that nothing seems to be happening.

Step 3 of 5

Then one step overtakes everything before it.

Every doubling is bigger than the whole history that came before. The last step of any doubling is more than all the others put together.

Step 4 of 5

Rice on a chessboard is the old demonstration.

One grain on the first square, doubling each time. By the last square the pile is more rice than the world has ever grown.

Step 5 of 5

Ten doublings is about a thousand times.

It is a useful handhold: every ten doublings multiplies by roughly a thousand. Twenty doublings is a million, thirty is a billion.

This is why anything that spreads by doubling — a rumour, a virus, a chain letter — is either negligible or overwhelming and rarely in between.