Fractions, Decimals, and Arithmetic
This page covers the number-language of recipes, prices, statistics, and school maths: fractions (a third, three quarters), decimals — which German punctuates and reads completely differently from English — percentages, and the spoken vocabulary of arithmetic. Two things trip English speakers up more than anything else: German uses a comma where English uses a decimal point, and German fractions are built from the ordinal stem and behave as neuter nouns. Once those two facts click, the rest is straightforward.
Fractions: Ordinal Stem + -tel, Neuter Noun
A German fraction is made by taking the ordinal number (third, fourth, fifth...) and adding -tel (for 2–19) or -stel (for 20 and up). The result is a neuter noun, written with a capital letter.
- drei (three) → ordinal dritt- → das Drittel (a third)
- vier (four) → ordinal viert- → das Viertel (a quarter)
- fünf (five) → ordinal fünft- → das Fünftel (a fifth)
- zwanzig (twenty) → das Zwanzigstel (a twentieth)
- hundert → das Hundertstel (a hundredth)
This is why you should learn ordinals first — fractions are a derived form of them. The link to Viertel (quarter) is the same word you met telling time (Viertel nach drei).
| Number | German | Notes |
|---|---|---|
| 1/2 | ein halb / die Hälfte | irregular — see below |
| 1/3 | ein Drittel | das Drittel |
| 2/3 | zwei Drittel | numerator stays cardinal |
| 1/4 | ein Viertel | das Viertel |
| 3/4 | drei Viertel | das Viertel (no plural -s here) |
| 1/5 | ein Fünftel | das Fünftel |
| 7/10 | sieben Zehntel | das Zehntel |
| 1/100 | ein Hundertstel | -stel from 20 up |
The numerator is just the cardinal number placed in front: zwei Drittel (two thirds), drei Viertel (three quarters), fünf Achtel (five eighths). Unlike English, the fraction noun does not take a plural ending here — it is drei Viertel, never drei Viertels.
Drei Viertel der Befragten waren mit dem Ergebnis zufrieden.
Three quarters of those surveyed were satisfied with the result.
Gib bitte ein Drittel des Mehls in die Schüssel.
Add a third of the flour to the bowl, please.
Nur ein Fünftel der Bewerber wurde eingeladen.
Only a fifth of the applicants were invited.
The Special Case: halb, die Hälfte, anderthalb
The fraction "one half" does not follow the -tel pattern. German has three distinct words depending on how you use it:
- halb — an adjective, declined like any adjective: ein halbes Brot (half a loaf), eine halbe Stunde (half an hour), zwei halbe Liter.
- die Hälfte — a noun meaning "the half (of something)": die Hälfte der Klasse (half the class). Use this when "half" is the thing you're talking about.
- anderthalb / eineinhalb — both mean 1½ and are fixed, indeclinable words. anderthalb is slightly more old-fashioned; eineinhalb is transparent ("one and a half"). Higher mixed numbers follow the same pattern: zweieinhalb (2½), dreieinhalb (3½).
Wir warten schon eine halbe Stunde auf den Bus.
We've already been waiting half an hour for the bus.
Die Hälfte des Kuchens ist schon weg.
Half the cake is already gone.
Der Film dauert anderthalb Stunden.
The film lasts an hour and a half.
Decimals: the Comma, Spoken Komma
Here is the change English speakers must internalise: German uses a comma as the decimal separator, and a point (or space) as the thousands separator — the mirror image of English.
- English 3.5 = German 3,5
- English 1,000.75 = German 1.000,75
When read aloud, the comma is spoken as the word Komma, and — crucially — the digits after the comma are read one by one, not as a whole number.
| Written | Spoken | English equivalent |
|---|---|---|
| 3,5 | drei Komma fünf | three point five |
| 3,14 | drei Komma eins vier | three point one four |
| 0,75 | null Komma sieben fünf | zero point seven five |
| 12,08 | zwölf Komma null acht | twelve point oh eight |
Die Kreiszahl Pi ist ungefähr drei Komma eins vier.
The number pi is approximately 3.14.
Das Paket wiegt zwei Komma fünf Kilo.
The parcel weighs 2.5 kilos.
Percentages: Prozent
The word is das Prozent, and after a number it stays singular (like other measure nouns): fünf Prozent, hundert Prozent. The symbol is %, read Prozent.
Die Mehrwertsteuer beträgt in Deutschland neunzehn Prozent.
VAT in Germany is nineteen percent.
Fast fünfzig Prozent der Befragten waren unentschlossen.
Almost fifty percent of those surveyed were undecided.
To express a multiple, German uses -fach or doppelt/dreifach: doppelt so viel (twice as much), die dreifache Menge (three times the amount).
Arithmetic: Saying Sums Out Loud
The spoken vocabulary of basic maths:
| Operation | German word | Symbol |
|---|---|---|
| addition | plus / und | + |
| subtraction | minus / weniger | − |
| multiplication | mal | × |
| division | (geteilt) durch | ÷ |
| equals | ist (gleich) | = |
| power | hoch | xy |
The verb for "equals" is simply ist (is), optionally reinforced as ist gleich (literally "is equal").
Drei plus vier ist sieben.
Three plus four is seven (3 + 4 = 7).
Acht mal sieben ist sechsundfünfzig.
Eight times seven is fifty-six (8 × 7 = 56).
Zwanzig geteilt durch fünf ist gleich vier.
Twenty divided by five equals four (20 ÷ 5 = 4).
For powers, hoch sits between base and exponent: zwei hoch drei = 2³ = acht. The square root is die Quadratwurzel (or just Wurzel aus): die Wurzel aus neun ist drei (the square root of nine is three). (academic) registers prefer ist gleich and the precise terms; casual speech uses ist, und, mal.
Zwei hoch drei ist acht.
Two to the power of three is eight (2³ = 8).
Common Mistakes
❌ drei Punkt fünf
Incorrect — German decimals use Komma, never Punkt.
✅ drei Komma fünf (3,5)
Three point five — read with Komma.
❌ drei Komma vierzehn (für 3,14)
Incorrect — digits after the comma are read individually.
✅ drei Komma eins vier (3,14)
The digits after the comma are read one by one.
❌ drei viertels / zwei drittels
Incorrect — the fraction noun takes no plural -s.
✅ drei Viertel / zwei Drittel
Three quarters / two thirds — capitalized neuter nouns, no plural ending.
❌ fünf Prozente der Leute
Incorrect — Prozent stays singular after a number.
✅ fünf Prozent der Leute
Five percent of people.
❌ ein und ein halb Stunden
Incorrect — 1½ is a single fixed word.
✅ anderthalb Stunden / eineinhalb Stunden
An hour and a half.
Key Takeaways
- Fractions = ordinal stem + -tel (or -stel from 20 up), capitalized neuter nouns: das Drittel, das Viertel; the numerator is a plain cardinal: zwei Drittel. No plural -s.
- "Half" has three forms: adjective halb, noun die Hälfte, and the fixed anderthalb / eineinhalb for 1½.
- Decimals use a comma (3,5), spoken Komma, with the digits after it read individually.
- Prozent stays singular after a number.
- Arithmetic: plus, minus, mal, geteilt durch, ist (gleich), hoch.
Now practice German
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Start learning German→Related Topics
- Ordinal NumbersA2 — Forming German ordinals with -t (1-19) and -st (from 20), the irregulars erste, dritte, siebte and achte, why ordinals take adjective endings (am zweiten Mai), and the period-as-ordinal-marker (1. = erste).
- Hundreds, Thousands, MillionsA2 — Building large German numbers as single words up to a million, the reversed decimal comma and thousands dot (1.000,5), and the high-stakes false friend Milliarde = billion, Billion = trillion.
- Quantities, Measurements, and CountingA2 — Why German says zwei Glas Bier (not Gläser) and eine Tasse Kaffee (no 'of') — the singular-unit rule, feminine exceptions, and ein paar vs ein Paar.
- Telling TimeA2 — How to tell time in German, including the trap that makes English speakers miss appointments: halb drei means 2:30, not 3:30.
- Cardinal Numbers 0-20A1 — The German numbers null to zwanzig, including the irregular teens elf and zwölf, the dropped letters in sechzehn and siebzehn, and why the count eins becomes ein before a noun.