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## Notes

Egyptian’s number system is decimal, like most modern ones. However, it did not use repeatable digits. If you write the number 4234 in English, you use the symbol “4” twice, because there are four thousands and four ones, as well as the two hundreds and three tens. But in Egyptian, you have to use four of a glyph which means “thousand”, two of a glyph which means “hundred”, three of a glyph which means “ten”, and four of a glyph which means “one”.

### Basic symbols

The symbols, and the transliteration of the words for each unit, are:

1 (ones) | πΊ | wκ₯ | 1000 (thousands) | πΌ | αΈ«κ£ |

10 (tens) | π | mαΈw | 10,000 (ten-thousands) | π | αΈbκ₯ |

100 (hundreds) | π’ | Ε‘t | 100,000 (hundred-thousands) | π | αΈ₯fn |

Note that the symbol π’ for 100 (Gardiner code V1) is similar to, but not the same as, the shorthand π² for *w* (Gardiner code Z7).

So to write the number 4234 in hieroglyphs, you would write: πΏπ£ππ½ for “four thousands, two hundreds, three tens, four ones”.

If you use Unicode hieroglyphs, note that there are nine versions each of π, π’, πΌ, and π with a range of one to nine of each symbol. For the ones, there are actually eighteen versions each, with the strokes arranged horizontally and vertically; for example, π versus π.

There is a word π¨ *αΈ₯αΈ₯* “one million”, but in Middle Egyptian the word often is used indefinitely for “many”, “countless”.

### Multiplication notation

Infrequently in Middle Egyptian a number symbol would be written above a number of ones marks (or occasionally other symbols) and produce a more multiplication-based notation. This could produce numbers over one million:

π π

ππΌ

7Γ100,000 + 3Γ10,000 = 730,000

π

π’πΊ

101Γ100,000 = 10,100,000

### Zero

The system described above doesn’t use a zero for when a decimal place has zero; there’s simply no glyph of that kind there: πΏπ 4008 (no hundreds and no tens).

The Egyptians appear not to have had a mathematical notion of zero. If they had a zero result in a math problem or in an inventory or the like, they left a blank space or wrote the *nfr* sign π€. The word *nfr* usually means “good, perfect” but in this case, it’s short for π€π
±βπ
ͺπ₯ *nfrw* “deficiency, depletion”.

### Cardinal numbers

The Egyptians rarely wrote the words for numbers out, except for π‘πβπ€ *wκ₯* “one”; they used the hieroglyphs described above instead. But the number words have been reconstructed from Coptic: see table below.

The cardinal numbers are grammatically nouns, but are usually written after the nouns they modify, including any units of measurement: πππππ§πππ’π’ *αΈ₯nqt ds 200* “200 jugs of beer” (lit. “beer, jugs; 200”). Note that since the number itself is written with numeral glyphs and not phonetically, it’s usual in transliteration to do likewise.

### Ordinal numbers

The ordinal numbers show the place of a thing in a sequence: “ninth”, “thirty-first”. These are adjectives since they describe the thing: “which thing? The ninth.”

In Egyptian, the ordinals from “second” to “ninth” are formed by adding π *nw* (masc.) or ππ *nwt* (fem.) to the **root** of the cardinals; see table below. They are usually written with the numeral glyphs and those endings, but can be written out:

π»π or π’βππβπ
±βπ» *snnw* “second”

The ordinal “first” is πΆπͺπ *tpj*, πΆπͺπ *tpt* (or *dpj, dpt*), derived from “on top of”, in turn derived from “head”. The ordinals from “tenth” onward are π *mαΈ₯*, ππ *mαΈ₯t* followed by the cardinal number: ππππΊ *mαΈ₯t-31* “31st (fem.)”

### “Ditto”

The phrase *zp sn* or *zp 2* ππ» “two times, twice” is written as a “ditto” in some texts. (Note that π is Gardiner glyph O50, not Aa1 π which is the uniliteral *αΈ«*; they may easily be confused in some writings.)

π£π€πππ
ππππ» *jb.j n mjwt.j zp 2*

“My heart from my mother, my heart from my mother”

(lit. “My heart of my mother, two times”)

### Fractions

In general, Egyptian fractions could be written by putting πβ *r* above a number: πβπ€ ΒΉββββ. A few basic fractions had special symbols: π Β½ (transl. *gs*), π΄ ΒΌ (also written πβπ½) , π β
(*rwj*), π ΒΎ (*αΈ«mt-rw*).

The Egyptians could only write unit fractions, that is fractions with 1 for a numerator (except for β and ΒΎ). For any other fraction with a higher numerator, it had to be split into multiple fractions: to write Β²Β³ββ β, one must expand it into unit fractions:

Β²Β³ββ
β is more than β
but less than Β½, so we take β
from it;

Β²Β³ββ
β – β
= βΆβΉβββ
β – β΅β°βββ
β = ΒΉβΉβββ
β, which is a bit more than β
, which we take from it;

ΒΉβΉβββ
β – β
= β·βΆββββ – β·β΅ββββ = ΒΉββββ.

So our total is β + β + ΒΉββββ and would be written πβπΌπβππβπ§. A lot of the Rhind Mathematical Papyrus is dedicated to expanding fractions.

### Table of cardinal and ordinal number words

Number | Cardinal (masc.) | Cardinal (fem.) | Ordinal (masc.) | Ordinal (fem.) |
---|---|---|---|---|

1 | π‘πβπ€ wκ₯ | π‘πβππ€ wκ₯ | πΆπͺπ tpj | πΆπͺπ tpt |

2 | sn.wwj^{1} | sn.t | π’βππβπ
±βπ» sn.nw | π’βππβπ
±βππ» sn.nwt |

3 | αΈ«mt.w | αΈ«mt.t | αΈ«mt.nw | αΈ«mt.nwt |

4 | jfd.w | jfd.t | jfd.nw | jfd.nwt |

5 | dj.w | dj.t | dj.nw | dj.nwt |

6 | sjs.w | sjs.t | sjs.nw | sjs.nwt |

7 | sfαΈ«.w | sfαΈ«.t | sfαΈ«.nw | sfαΈ«.nwt |

8 | αΈ«mn.w | αΈ«mn.t | αΈ«mn.nw | αΈ«mn.nwt |

9 | psαΈ.w | psαΈ.t | psαΈ.nw | psαΈ.nwt |

10 | mαΈw | mαΈt | mαΈ₯-10 | mαΈ₯t-10 |

11 | mαΈw-wκ₯ | mαΈw-wκ₯t | (etc.)^{2} | (etc.) |

12, … 19 | mαΈw-sn.wwj, etc. | mαΈw-sn.t, etc. | ||

20 | mαΈwtj | mαΈwtt | ||

30 | mκ₯bκ£ | mκ₯bκ£t | ||

40 | αΈ₯mw | β^{3} | ||

50 | djjw | β | ||

60 | sjsjw | β | ||

70 | sfαΈ«jw | β | ||

80 | αΈ«mnjw | β | ||

90 | psαΈjw | β | ||

100 | β | Ε‘t | ||

200 | β | Ε‘tj | ||

1000 | αΈ«κ£ | β | ||

10,000 | αΈbκ₯ | β | ||

100,000 | αΈ«fn | β | ||

1,000,000 | αΈ₯αΈ₯ | β |

2. For all numbers above 9, the ordinal is

*mαΈ₯*or

*mαΈ₯t*followed by the cardinal.

3. Cardinal tens from 40-90, and higher numbers except 100 and 200, have masculine only. 100 and 200 have feminine only.

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