Numbers

• Basic symbols
• Multiplication notation
• Zero
• Cardinal numbers
• Ordinal numbers
• “Ditto”
• Fractions

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:

Note that the symbol 𓍢 for 100 (Gardiner code V1) is similar to, but not the same as, the shorthand 𓏲 for w (Gardiner code Z7).

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

If you use Unicode hieroglyphs, note that there are nine versions each of 𓎆, 𓍢, 𓆼, and 𓂭 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, 𓐁 versus 𓐋.

There is a word 𓁨 ḥḥ “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:

 𓆐 𓂭
𓐀𓏼
7×100,000 + 3×10,000 = 730,000

𓆐
𓍢𓏺
101×100,000 = 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: 𓆿𓐁 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 𓄤. The word nfr usually means “good, perfect” but in this case, it’s short for 𓄤𓅱‌𓅪𓏥 nfrw “deficiency, depletion”.

Cardinal numbers

The Egyptians rarely wrote the words for numbers out, except for 𓌡𓂝‌𓏤 wꜥ “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: 𓎛𓈎𓏏𓏋𓂧𓊃𓏊𓍢𓍢 ḥnqt 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 𓏌 nw (masc.) or 𓏌𓏏 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:

𓏻𓏌 or 𓌢‌𓈖𓏌‌𓅱‌𓏻 snnw “second”

The ordinal “first” is 𓁶𓊪𓏭 tpj, 𓁶𓊪𓏏 tpt (or dpj, dpt), derived from “on top of”, in turn derived from “head”. The ordinals from “tenth” onward are 𓎔 mḥ, 𓎔𓏏 mḥt followed by the cardinal number: 𓎔𓏏𓎐𓏺 mḥt-31 “31st (fem.)”

“Ditto”

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

𓄣𓏤𓀀𓈖𓅐𓏏𓀀𓊗𓏻
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 𓂋‌ r above a number: 𓂋‌𓍤 ¹⁄₃₀₀. A few basic fractions had special symbols: 𓐞 ½ (transl. gs), 𓏴 ¼ (also written 𓂋‌𓏽) , 𓂌 ⅔ (rwj), 𓂍 ¾ (ḫmt-rw).

The Egyptians could only write unit fractions, that is fractions with 1 for a numerator (except for ⅔ and ¾). For any other fraction with a higher numerator, it had to be split into multiple fractions: to write ²³⁄₅₀, one must expand it into unit fractions:

²³⁄₅₀ is more than ⅓ but less than ½, so we take ⅓ from it;
²³⁄₅₀ – ⅓ = ⁶⁹⁄₁₅₀ – ⁵⁰⁄₁₅₀ = ¹⁹⁄₁₅₀, which is a bit more than ⅛, which we take from it;
¹⁹⁄₁₅₀ – ⅛ = ⁷⁶⁄₆₀₀ – ⁷⁵⁄₆₀₀ = ¹⁄₆₀₀.

So our total is ⅓ + ⅛ + ¹⁄₆₀₀ and would be written 𓂋‌𓏼𓂋‌𓐁𓂋‌𓍧. A lot of the Rhind Mathematical Papyrus is dedicated to expanding fractions.