Cut a chapati in half. Keep one half. Now cut that in half. Keep one. Do it again. After three cuts you are holding \tfrac{1}{2} \times \tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{8} of the original chapati — and you are holding one of the most instructive pictures in fraction arithmetic. Every "of" in the phrase "half of a half of a half" becomes a multiplication. Every cut makes the strip in your hand half as long as before. Watch the bar shrink and you are watching the rule a \times b = b \times a made visible as area.

Start with one whole

Take a unit bar — a rectangle of length 1, representing the chapati before any cuts. The full bar stands for the whole thing.

A single bar representing one wholeA horizontal rectangle spanning most of the width of the figure, fully shaded in red, labelled as the whole chapati or unit one. A tick mark at the left end shows zero and a tick mark at the right end shows one. 0 1 1 whole
Before any cuts, you are holding the whole chapati — the unit bar, fully shaded.

Every multiplication by \tfrac{1}{2} that follows will halve this shaded region. One halving takes you to the length \tfrac{1}{2}. Two halvings take you to \tfrac{1}{4}. Three to \tfrac{1}{8}. And so on.

The slider — drag how many halvings

Slider controlling the number of halvings applied to the unit barA horizontal bar of fixed outline representing the unit length, with a shaded region on the left whose width halves every time the slider advances. At zero halvings the whole bar is shaded. At one halving, half the bar is shaded. At two halvings, a quarter. At three halvings, an eighth. A readout shows the current fraction in both fraction and decimal form, and a slider below lets the reader drag the number of halvings between zero and eight. so you are holding (1/2)^k of the original 0 1 0 4 8 ↔ drag how many halvings
Drag the slider to pick $k$, the number of halvings. At $k = 1$ the shaded bar is $\tfrac{1}{2}$. At $k = 3$ it is $\tfrac{1}{8}$. At $k = 5$ it is $\tfrac{1}{32}$ and almost invisible. Each halving cuts the visible length in half — so after $k$ halvings, exactly $(1/2)^k$ of the original bar remains shaded.

Three halvings gives you a shaded strip of length \tfrac{1}{8}. Six halvings gives you \tfrac{1}{64} — roughly the width of a pencil line. Eight halvings gives you \tfrac{1}{256}, which is almost invisible on the screen. The bar never quite disappears (you could go on halving forever and never reach zero), but it shrinks geometrically — each step multiplies what is left by \tfrac{1}{2}, so the length decays like powers of two.

Why: each halving operation takes whatever strip is currently shaded and keeps only half of it. Multiplying by \tfrac{1}{2} repeatedly is the same as multiplying by (\tfrac{1}{2})^k once, by the law of exponents. Since (\tfrac{1}{2})^k = \tfrac{1}{2^k}, the shaded length after k halvings is exactly \tfrac{1}{2^k}.

Why multiplying fractions shrinks things

There is a trap here worth naming. If you have only met multiplication for whole numbers — 3 \times 5 = 15, 7 \times 8 = 56 — you picked up the rule of thumb "multiplying makes numbers bigger." With fractions that rule of thumb is wrong. Multiplying by a number less than 1 makes the answer smaller than what you started with.

The slider makes this concrete. You started with 1. After multiplying by \tfrac{1}{2} once, you have \tfrac{1}{2} — smaller. Multiply by \tfrac{1}{2} again and you have \tfrac{1}{4} — smaller still. At no point does the strip ever grow. The reason is simple: "half of anything" means you keep only half and throw the rest away. Each multiplication throws half away. So the result is always smaller.

The full rule is cleaner than the rule of thumb: multiplying by a number greater than 1 grows the result, multiplying by 1 leaves it alone, and multiplying by a number between 0 and 1 shrinks it. The old "multiplication makes things bigger" was really a special case of the first rule, because all the whole numbers you multiplied by (other than 0 and 1) were bigger than 1.

Nested "of" reads as multiplication

The phrase "half of a half of a half" turns into mathematics by reading each of as a multiplication. "Half of x" is \tfrac{1}{2} \times x. So "half of a half" is \tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{4}. And "half of (half of a half)" is \tfrac{1}{2} \times \tfrac{1}{4} = \tfrac{1}{8}.

This reading generalises to any fraction, not just \tfrac{1}{2}.

The order of the multiplications does not matter, because multiplication is commutative. "Half of a third" and "a third of a half" both land on \tfrac{1}{6}. The slider makes this commutativity almost physical — whether you halve twice and then third, or third first and then halve twice, the final shaded region is the same size.

The geometric picture: area in a square

For two fractions, the cleanest picture is not the halving slider but a square cut both ways. Take a unit square. Shade \tfrac{2}{3} of the columns and \tfrac{3}{4} of the rows. The overlap is \tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{1}{2} of the square.

Unit square divided into a grid illustrating two thirds times three quarters equals one halfA unit square divided vertically into three columns and horizontally into four rows, making twelve cells in total. The two leftmost columns are tinted in a light shade representing two thirds. The bottom three rows are tinted in a different light shade representing three quarters. The six cells where the tints overlap are filled with a deeper colour, covering exactly half of the twelve cells and illustrating that two thirds times three quarters equals six twelfths or one half. 2/3 across 3/4 down 6 / 12 = 1 / 2
The square is cut into $3 \times 4 = 12$ cells. Two of the three columns (shaded left-to-right) and three of the four rows (shaded top-to-bottom) overlap in a rectangle of $2 \times 3 = 6$ cells. That rectangle is $\tfrac{6}{12} = \tfrac{1}{2}$ of the square — which is the product $\tfrac{2}{3} \times \tfrac{3}{4}$.

The general rule — numerator times numerator, denominator times denominator — is exactly the rule "overlapping columns times overlapping rows over total cells." The full derivation lives in Fraction Multiplication Is Just Area, and the pattern in that picture is the same pattern hiding inside the halving slider above.

Geometric decay has a life outside fractions

The shrinking-strip picture shows up in a lot of places that have nothing to do with chapatis.

Compound discount. A shop offers a 20\% discount, then a further 20\% off on Diwali. What is the final price as a fraction of the original? 0.8 \times 0.8 = 0.64 — you pay 64\%, not 60\%. The second discount is taken off the already-discounted price, which is smaller, so it removes less in absolute rupee terms.

Half-life in physics. A radioactive sample loses half its atoms every half-life. After k half-lives, only (\tfrac{1}{2})^k of the original atoms remain — the exact same sequence as the halving slider above. Three half-lives leave \tfrac{1}{8} of the sample. Ten half-lives leave \tfrac{1}{1024}, about 0.1\% — essentially nothing.

Binary digit weights. In a binary representation 0.1011, each digit after the point weighs \tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, \tfrac{1}{16}, \ldots — the same geometric halving. This is why computer arithmetic lives inside the geometric series that the halving slider generates.

How many halvings to reach under 1%?

You want the shaded strip to be less than 1\% of the original bar. How many halvings do you need?

You need the smallest k such that (\tfrac{1}{2})^k < 0.01, i.e. 2^k > 100.

Powers of two: 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32, 2^6 = 64, 2^7 = 128.

The first k with 2^k > 100 is k = 7. So after seven halvings, the remaining strip is \tfrac{1}{128} \approx 0.78\% — finally under 1\%.

Why: seven doublings take you past 100, so seven halvings take you past \tfrac{1}{100}. A handy rule of thumb: every ten halvings divides the strip by roughly 1000, because 2^{10} = 1024. So ten halvings reach 0.1\%, twenty halvings reach 0.0001\%, and so on. This is the content of the "binary rule of thumb" 2^{10} \approx 10^3 that computer people quote constantly.

The takeaway

Drag the slider all the way to k = 8. The shaded strip becomes \tfrac{1}{256} of the original. That is the slimmest visible sliver — and yet, if you kept going, you could halve it a thousand more times and it still would not reach zero. Each multiplication by \tfrac{1}{2} takes you closer to zero but never arrives; that is the content of the limit concept you will meet in Limits.

Related: Fractions and Decimals · Fraction Multiplication Is Just Area — (2/3) × (3/4) in One Picture · Multiplying Always Makes Numbers Bigger — So Why Does 1/2 × 1/2 Shrink? · Why Dividing by a Fraction Less Than 1 Grows the Result