A percentage is a fraction in disguise

Ask anyone how 3/4 differs from 0.75 and from 75% and you will usually hear that they are "a fraction, a decimal and a percentage" — three different things, each for a different job. That is an illusion produced by appearances. All three notations denote exactly the same rational number and convert into one another without the slightest loss. They differ not in value but in convenience: the fraction bar is the most exact, the decimal point the easiest to compare, and the percentage the easiest to communicate.

The reason we have three notations instead of one is historical. Each arose in a different era under a different practical pressure: common fractions to divide bread and land fairly, decimals to compute fast, percentages to compare things of wildly different size. This piece is the narrative layer above our numbers branch: where these notations came from, what unites them, and where we most often trip over them.

Before the fraction bar

The oldest written parts of a whole are nearly four thousand years old and look nothing like ours. The Babylonians developed a positional system in base 60 and simply extended it rightwards, below one. Sixty has twelve divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60), so a half, a third, a quarter or a sixth all had short, terminating representations — something our decimal system cannot manage. The famous tablet YBC 7289 from the Yale collection preserves an approximation of the square root of two: 1;24,51,10, that is 1 + 24/60 + 51/60² + 10/60³ ≈ 1.414213. The true value is 1.4142136 — six correct decimal places, obtained without a calculator and without algebra.

The Egyptians took another road: they admitted almost exclusively unit fractions, with a one on top, written as a "mouth" sign above a number. Two thirds was the exception and had a symbol of its own. Every other fraction had to be decomposed into a sum of distinct unit fractions — 2/5 was written as 1/3 + 1/15. For splitting loaves among workers this was convenient (everyone got real pieces); for arithmetic it was a nightmare. The Romans, in turn, based their parts on twelfths: the as divided into 12 unciae, which made taxes easy but locked arithmetic into a single grid size.

The modern shape of a fraction — numerator above denominator — came from India. Brahmagupta in the 7th century and Bhaskara in the 12th wrote both numbers one under the other, with nothing in between. The missing fraction bar was added around 1200 by al-Hassar, a mathematician from the Maghreb, and it is what made the notation unambiguous: dividend above the line, divisor below. Fibonacci — Leonardo of Pisa — carried the idea into Europe in Liber Abaci (1202), together with the Hindu-Arabic numerals. He still put the fraction to the left of the whole number, so "two and a half" came out in the reverse of today's order.

Point, comma and the triumph of tens

Decimal fractions appeared in China and in the Islamic world long before Europe — the first works came from al-Uqlidisi (c. 950) and al-Samawal (c. 1172) — but the full theory was built by the Persian astronomer Jamshid al-Kashi. In his treatise on the circumference of the circle (1424) he computed 2π to an accuracy equivalent to sixteen decimal digits, and three years later, in Miftah al-hisab ("The Key to Arithmetic"), he systematised decimal arithmetic.

In Europe the breakthrough arrived in a slim 1585 pamphlet: De Thiende ("The Tenth") by the Flemish engineer Simon Stevin. Its thesis was ruthlessly practical — since operations on decimal fractions reduce to operations on whole numbers, coinage, measures and weights should be decimalised too. Stevin was right about the idea, but his notation was heavy: after each digit he wrote the power of ten in a little circle. The separator itself took decades more to settle — Bartholomaeus Pitiscus used a point in his trigonometric tables in 1595, and John Napier popularised it in his work on logarithms between 1614 and 1617. The split between the Anglophone point and the continental comma hardened back then and survives today — which is why a spreadsheet can still turn 1,5 into fifteen.

Where the % sign came from

Counting "per hundred" is older than the symbol. The emperor Augustus levied the centesima rerum venalium, a one-percent tax on goods sold at auction. In Renaissance Italy, where banking flourished, rates and commissions were quoted per cento, and merchants abbreviated it in their ledgers however they pleased: p 100, p cento, p co. Around 1425 an Italian manuscript shows a shorthand made of a struck-through p and a c with a small loop. Over the 16th century the p evaporated and cento collapsed into two little circles separated by a stroke — 0/0. By the middle of the 17th century the stroke had straightened into a slash, and the % we know went into print. Nobody designed it; it simply wore down into its present shape.

Three notations, one number

Formally the matter is simple. A common fraction a/b says into how many equal parts the whole was cut (the denominator) and how many we take (the numerator). A decimal is just the explicit result of the division a ÷ b. A percentage is a fraction with the denominator nailed down: P% = P/100. Converting takes three moves:

fraction → decimal: divide numerator by denominator, 3/8 = 3 ÷ 8 = 0.375;
decimal → percentage: multiply by 100%, that is, shift the point two places right, 0.375 = 37.5%;
percentage → fraction: put the value over a hundred and reduce, 37.5% = 37.5/100 = 375/1000 = 3/8.

Fraction	Decimal	Percentage	Where you meet it
`1/2`	0.5	50%	half of anything
`1/3`	0.(3)	≈ 33.33%	a three-way split of shares
`2/3`	0.(6)	≈ 66.67%	a qualified majority
`1/4`	0.25	25%	a quarter of the year
`3/4`	0.75	75%	a dominant stake
`1/5`	0.2	20%	the standard discount rate
`1/8`	0.125	12.5%	inches in engineering
`3/8`	0.375	37.5%	precision measurement
`1/10`	0.1	10%	a tithe, a tip
`1/100`	0.01	1%	a penny in a pound

Two rows carry an "≈" next to the percentage, and that is not decorative caution. 1/3 does not equal 33.33%; it equals 33.(3)%. Three such rounded shares add up to 99.99%, not to the whole — which is exactly where those reports whose percentages sum to 100.1% come from.

Why 1/3 never ends

The rule is surprisingly sharp. A fraction a/b in lowest terms has a terminating decimal expansion precisely when the prime factorisation of the denominator contains nothing but twos and fives, that is b = 2^m · 5^n. The reason: the base of our system is 10 = 2 · 5. If the denominator is built only from those factors, it can be scaled up to a power of ten — and the digits after the point run out.

When the denominator holds anything else (3, 7, 11, 13…), the division never leaves a remainder of zero. There are only b − 1 possible remainders modulo b, so after at most b − 1 steps one of them must repeat, and the whole run of digits repeats with it. Hence 1/3 = 0.(3) and 1/7 = 0.(142857) — six digits in the period, because 7 − 1 = 6.

The most interesting part is that the infinity lives in the base, not in the number. In base twelve 1/3 terminates (0.4₁₂) and it is 1/5 that starts repeating. In the Babylonian base 60 both 1/3 (0;20) and 1/5 (0;12) terminate — while 1/7 breaks down just as it does for us. One third is always equally exact; it is ten that cannot write it down.

Percentages that do not come back

Three operations cover 99% of everyday arithmetic: P% of X is X · P/100; if P% of a number is V, the number is V · 100/P; and the change from X₁ to X₂ is (X₂ − X₁)/X₁ · 100%.

Trouble starts when changes are composed, because a percentage always refers to the current base, never the original one. Raise a price of 100 by 20% and you get 120. Cut it by 20% and you get 96 — four percent below where you started. The discount was the same in percentage terms, but it was taken off a larger amount (24 instead of 20). This asymmetry is not a matter of notation: 1.20 · 0.80 = 0.96, and no convention will change that.

Take a real price list. An item costs 250, first goes up by 15%, then down by 20%. After the rise: 250 · 1.15 = 287.50. After the cut: 287.50 · 0.80 = 230. The total change is (230 − 250)/250 = −0.08, that is −8% — not −5%, as subtracting 15 from 20 would suggest.

The same trap hides in VAT. A gross price is 123% of the net price (at a 23% rate), so the net figure comes from division, not from subtracting 23%. For a gross of 123: 123 / 1.23 = 100 net, and 23 in tax. Subtracting 23% would give 123 · 0.77 = 94.71 — a number no accountant will accept. If you want to see these steps laid out one by one, they are in our lesson on percentages.

A percentage point is not a percent

The most common error in news headlines has a single source: a change in a quantity that is itself a percentage can be reported in two ways. Unemployment rises from 4% to 5%. The absolute difference is 5 − 4 = 1 percentage point. The relative difference is (5 − 4)/4 = 0.25, that is 25%. The sentence "unemployment rose by 1%" is simply false — it would mean a rise to 4.04%.

The same goes for a loan. A rate moves from 3.5% to 4.2%: that is a rise of 0.7 percentage points, but the interest cost grows by 0.7/3.5 = 20%. The bank will quote the first number; your budget will feel the second.

Why fractions are so hard

Fractions are the first wall a great many pupils hit, and cognitive psychology has a name for it: the whole number bias. Our cognitive apparatus learns natural numbers first — they have successors, multiplication always makes things bigger, a larger digit means more. A fraction breaks all three intuitions at once, and the brain defends itself by reading a/b not as one value but as two separate numbers.

Hence the three classic errors: adding numerators and denominators separately (1/8 + 1/8 = 2/16 instead of 1/4), judging 1/5 to be larger than 1/4 because five beats four, and being startled that multiplying by a proper fraction shrinks the result. Work by Robert Siegler's group shows the bias does not vanish with age — in adults it returns under time pressure. The remedy is not one more rule but inhibition of the old reflex, and placing the fraction as a single value on the number line. That is exactly why, in our lesson on common fractions, a common denominator appears before any addition does.

The remaining rules then follow. Reducing and expanding work because a/b = (a·k)/(b·k) for k ≠ 0 — the value does not move when both numbers are scaled together. A common denominator is required because 1/3 and 1/4 are pieces of different size, and until they are made uniform there is nothing to add. And the famous "dividing by a fraction is multiplying by its reciprocal" is no trick: if X = (a/b) ÷ (c/d) means X · (c/d) = a/b, multiply both sides by d/c; since (c/d)·(d/c) = 1, what remains is X = (a/b) · (d/c).

What is convention and what is strict

It is worth separating two layers at the end. Almost everything visible is convention: base ten (an accident of anatomy — ten fingers), the fraction bar, comma versus point, the % sign, and the hundred itself as a reference. We could just as well count in per mille or in basis points, and finance does exactly that. What is strict is what notation cannot touch: 1/2, 0.5 and 50% are the same number; reducing does not change a value; 1/3 has no terminating expansion in base ten; and +20% followed by −20% will never bring you home.

So the next time you see a headline about "a rise of 1%", check first whether a percentage point was meant. And if you would rather practise the arithmetic than read about it — exercises with answer entry are waiting in the numbers branch.

A percentage is a fraction in disguise — three ways to write one number