Prozentrechnung

The word is Latin by way of Italian: per centum, "by the hundred". Rome was computing in hundredths long before anyone wrote a percent sign. Augustus taxed goods sold at auction at one hundredth of the price — the centesima rerum venalium — and Tacitus records the rate being cut to a two-hundredth once revenue from newly annexed territory came in. Roman lending used the same denominator: interest was quoted as centesimae, hundredths of the principal per month, so what a modern contract would call 12% a year was one hundredth a month.

Hundredths became a calculating habit in the commercial arithmetic of late medieval Italy. The merchant handbooks of the fourteenth and fifteenth centuries worked profit, loss, interest and tax "per cento", and the Hindu–Arabic numerals that spread after Liber Abaci made that kind of computation practical for people who were not specialists. Florian Cajori, in his History of Mathematical Notations (1928), traced the symbol itself through those manuscripts: an Italian hand of about 1425 abbreviates per cento as "per" followed by a small circle above a stroke; the "per" gradually drops away; and by the seventeenth century the surviving mark has drifted into something close to the modern %, with a second circle below the bar. The frequently repeated claim that the sign is simply a stylized 0/0 is folk etymology — the manuscript trail points to an abbreviation, not a fraction.

Finance kept subdividing. One tenth of a percent is a per mille, written ‰, still used for property tax mill rates and, in much of Europe, for blood alcohol limits. One hundredth of a percent is a basis point, which exists precisely because "a quarter point" is ambiguous: it could mean a quarter of a percentage point or a quarter of a percent of the current rate. That ambiguity is the oldest trap in the subject. A rise from 4% to 5% is one percentage point and a 25% relative increase, and the two are still routinely confused in news coverage.

Wichtige Eigenschaften

• x% means x/100, so converting a percentage to a decimal is division by 100 and never anything else.
• Percentages of percentages multiply: 10% of 20% is 2%, not 30%.
• Successive changes compose multiplicatively, so +10% followed by −10% leaves 0.99 of the original — a 1% net loss, not a return to the start.
• A move from p% to q% is (q − p) percentage points; calling it (q − p)% means something different and usually wrong.
• An increase of p% is undone by a decrease of 100p/(100 + p)%, so a 25% rise is reversed by a 20% fall.
• If A is x% of B then B is y% of A with xy = 10,000, for any non-zero A and B.
• One basis point is 0.01%, so 100 basis points make 1% and 10,000 make the whole quantity.
• One per mille (‰) is 0.1%, which makes 1% equal to 10‰.

Wo sie auftauchen

• Consumer credit must be advertised with an annual percentage rate by law in the United States under the Truth in Lending Act and across the EU under the Consumer Credit Directive, so that offers with different fee structures can be compared on one number.
• Central banks and bond markets move rates in basis points — a "25 basis point cut" is 0.25 percentage points — specifically to avoid the percent versus percentage-point ambiguity.
• Drink-driving limits are written as a percentage of blood alcohol by volume: 0.08% in most US states, and 0.05% in Utah.
• Value-added tax and sales tax are quoted as percentages added to a net price, which is why reverse percentages matter — removing 20% VAT from a gross price of 120 gives 100, not 96.
• Election coverage reports both poll shares and margins of error in percentage points, and a lead inside the margin of error is a statement about sampling, not a prediction.
• Battery and download indicators show percentages of an estimated total rather than a measured one, which is a display convention rather than a precise measurement.