Can You Add Halves Forever and Still Get One?

Step by step

Here is the walk written as numbers. Check them against the widget.

Step	This step	Walked so far	Still to go
1	4 m	4 m	4 m
2	2 m	6 m	2 m
3	1 m	7 m	1 m
4	0.5 m	7.5 m	0.5 m
5	0.25 m	7.75 m	0.25 m
6	0.125 m	7.875 m	0.125 m
7	0.0625 m	7.9375 m	0.0625 m

Two patterns jump out. Each step is the same length as the gap it leaves behind: walk 1 m, and 1 m is left. And the gap halves every time, so after n steps it is 8 m divided by 2 n times: $8 \div 2^n$ meters.

Halves of one whole

To see the walk without the meters, make the whole distance 1, one whole of anything. The steps are then ½, ¼, ⅛, 1⁄16, and so on: each a fraction with 1 on top and a doubled number below. Adding them up:

• After 1 step: ½, and ½ is left.
• After 2 steps: ½ + ¼ = ¾, and ¼ is left.
• After 3 steps: ¾ + ⅛ = ⅞, and ⅛ is left.
• After 10 steps: 1023⁄1024, and 1⁄1024 is left.

The totals are always one piece short of 1: after n steps, the total is 1 − 1/2ⁿ. You can see it in Figure 33.1. The first half fills the left of the square. Each new piece is half of the empty part, so it always fits inside, and what stays empty is a rectangle exactly the size of the last piece.

Two facts that pin the answer down

Nobody can actually do infinitely many additions. So what could “½ + ¼ + ⅛ + … forever” mean? The square gives two facts about the totals:

• a.They never pass 1. Every total is 1 minus the last piece.
• b.They get closer to 1 than any gap you name. Name a gap, a millimeter, an atom, a billionth of a billionth: the gap after n steps is 1/2ⁿ, and halving enough times makes it smaller than yours.

Is there any number other than 1 that the totals could be heading for? Not a number bigger than 1: they never pass 1, so they can’t get close to a number above it. Not a number smaller than 1 either, say 0.9999: the totals pass 0.9999 after 14 steps and keep going. The only number that fits both facts is 1.

So mathematicians agree on this meaning: the sum of infinitely many numbers is the number that the totals settle at, if there is one, which means that the totals come closer to it than any gap you name, and stay that close from then on. They need not ever equal it. A sum that goes on forever is an infinite series; when its totals settle at a number, the series converges, and that number is its sum. So

$$\tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \tfrac{1}{16} + \dots = 1$$

exactly, not “nearly 1”. For your walk, the steps add up to exactly 8 m: if all the steps could be taken, you would be at the door.

Is 0.999… really 1?

The number 0.999…, with nines that never end, is an infinite series too:

$$0.999\ldots = 0.9 + 0.09 + 0.009 + 0.0009 + \dots$$

Its totals are 0.9, 0.99, 0.999… After each nine, the gap to 1 is a tenth of the gap before: 0.1, then 0.01, then 0.001… So, just like the halves, the totals never pass 1 and get closer than any gap you name: 0.999… = 1, exactly. Many students feel that 0.999… must be “the number just below 1”. But there is no such number: between any number below 1 and 1 there is always room for more.

The copy trick

There is a neat way to find the sum without any totals. Call the whole sum S:

$$S = \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \tfrac{1}{16} + \dots$$

Look at Figure 33.1 again. Take away the first half, and what is left, ¼ + ⅛ + 1⁄16 + …, is the same picture again at half the size: every piece is half of a piece of S. So the part after the first half is half of S:

$$S = \tfrac{1}{2} + \tfrac{1}{2}S$$

If S is half of S plus ½, then the other half of S is ½, and S = 1. The same trick works on your walk: the whole walk is 4 m plus half of the whole walk, so half the walk is 4 m and the whole walk is 8 m.

Halves are not the only pieces that settle. Choose a sum, add pieces, and watch where the bar stops.

The quarters, in a picture

Why should quarters add up to a third? Here is a picture, much like the square of halves.

Cut a square into four. The top-right quarter is blue, and with two plain quarters it makes an L of three equal squares. The bottom-left quarter is cut into four the same way, making a smaller L with one blue square of 1⁄16, and so on forever. In every L, exactly one square of the three is blue. The Ls together fill the whole square (all but the corner they close in on), so the blue is a third of the square:

$$\tfrac{1}{4} + \tfrac{1}{16} + \tfrac{1}{64} + \dots = \tfrac{1}{3}$$

Any number of pieces, one rule

In every sum in the widget except the last, each piece is the one before it multiplied by the same fraction: ½, ⅓, ¼, 1⁄10 or ⅔. That fraction is the common ratio, and a sum of this kind is a geometric series, the same kind of sum as the king’s rice, where the ratio is 2. The copy trick works on any of them. Call the first piece a and the ratio r. After the first piece, the rest is the whole sum with every piece multiplied by r:

$$S = a + rS, \quad \text{so} \quad S - rS = a, \quad \text{and} \quad S = \frac{a}{1 - r}$$

Check it on the thirds: a = ⅓ and r = ⅓, so $S = \frac{1/3}{2/3} = \frac{1}{2}$. On 0.9 + 0.09 + …: a = 0.9 and r = 0.1, so $S = \frac{0.9}{0.9} = 1$. And the two thirds: $\frac{2/3}{1/3} = 2$.

What it means: 1 − r is how much each piece shrinks, as a fraction of itself. Pieces with r = 0.99 shrink very little, so they add up to a hundred times the first piece. When r is 1 or more, the pieces don’t shrink, the totals grow past every number, and the formula gives nonsense, as it did for the rice. So it works only when r is less than 1. A negative ratio, pieces that take away and add back in turn, can settle too, as long as it is between −1 and 0 (question d); at −1 or below, the pieces don’t shrink, and there is no sum (question e). In short: the formula gives the sum exactly when r is between −1 and 1.

The last sum in the widget is

$$1 + \tfrac{1}{2} + \tfrac{1}{3} + \tfrac{1}{4} + \tfrac{1}{5} + \dots$$

Its pieces shrink towards nothing: the millionth piece is a millionth. So surely it settles? Press Add 1,000 in the widget a few times. The total does slow down badly. It passes 3 at the 11th piece and 4 at the 31st, but needs 83 pieces to pass 5 and 12,367 to pass 10. After a million pieces it is only about 14.4. It looks as if it is settling. It isn’t.

Oresme’s groups

About the year 1350, the French scholar Nicole Oresme found a way to see why. Put the pieces after the first in groups that end at ½, ¼, ⅛, 1⁄16…:

• ⅓ + ¼ is more than ¼ + ¼, which is ½.
• ⅕ + ⅙ + ⅐ + ⅛: four pieces, each at least ⅛, so more than 4 × ⅛ = ½.
• 1⁄9 to 1⁄16: eight pieces, each at least 1⁄16, so more than 8 × 1⁄16 = ½.

Each group has twice as many pieces as the one before, each piece at least half as big, so every group from the second on is worth more than ½, forever. Add up enough groups and you pass any number at all: 20 groups give more than 10, 200 groups more than 100. So this sum has no sum. It diverges: its totals grow past every number, only very slowly. It is called the harmonic series.

The lesson: pieces that shrink to nothing are not enough for a sum to settle. They must shrink fast enough. Halves shrink by a fixed fraction every time; the harmonic pieces shrink by less and less (from 1⁄999 to 1⁄1000 is hardly a change), and that lets them add up to more than any number.

The same sum also gives a fractal snowflake its area: the snowflake in Can a Shape Have an Edge That Never Ends? grows by pieces each 4⁄9 of the one before, so its area settles at 8⁄5 of its first triangle, while its edge, growing by 4⁄3 each time, has no end at all.

• How Much Rice Is on the Chessboard? Doubling instead of halving: a geometric series that grows without end.
• Can a Shape Have an Edge That Never Ends? A snowflake whose area settles while its edge never does.
• Is There Room at a Full Hotel with Endless Rooms? Infinity again, this time counting rooms instead of adding pieces.
• How Did Madhava Find π Without a Calculator? (Grades 9–12) The Kerala school’s infinite series for π.
• Are Some Infinities Bigger Than Others? (Grades 9–12) Why the fractions can be counted and the decimals can’t.
• What Is the Biggest Number a Computer Can Count To? (Computing, Going Further 6–8) Why a computer’s total of halves becomes exactly 1.
• Grade 10, Sum of a geometric series: the formula for adding the first n pieces of a geometric series.