Post History
#2: Post edited
by
Michael Hardy
·
2026-06-16T18:22:01Z (3 months ago)
- It depends upon the context.
- I am mainly posting to address a kind of confusion that often arises. There are two conventions of terminology and notation in modular arithmetic, and one should be clear about which is which.
- <b>Newer notation:</b> I suspect this one was introduced around the middle of the 20th century. It is that $a\bmod b$ means the remainder when $a$ is divided by $b.$ Thus $83\bmod5=3.$ The correct parsing is $(83\bmod5) = 3,$ <b>not</b> $\require{cancel} \xcancel{83\bmod(5=3)}$ or anything like that.
<b>Older notation:</b> The older convention was introduced in the book _Disquisitiones Arithmeticae_, by Carl F. Gauss (rhymes with "house" as pronounced by most if not all English-speaking people), published in 1801. The older convention is the most prevalent one among mathematicians and the newer perhaps among computing experts. According to the older convention one may write $83\equiv98\bmod5$ and it is parsed as $(83\equiv98)\bmod5,$ <b>not</b> as $\xcancel{83\equiv(98\bmod5)}.$ It means that $83$ and $98$ both leave the same remainder when divided by $5.$ In words, one says:- <b>Eighty-three is congruent to ninety-eight, modulo five.</b>
- The number $5$ is the <b>modulus</b>. The word "modulo" is a different case of that Latin word (modulus is nominative and modulo is ablative or dative or something; ask your Latin teacher). Often in speaking it is abbreviated just as in the mathematical notation, thus:
- <b>Eighty-three is congruent to ninety-eight, mod five.</b>
- It depends upon the context.
- I am mainly posting to address a kind of confusion that often arises. There are two conventions of terminology and notation in modular arithmetic, and one should be clear about which is which.
- <b>Newer notation:</b> I suspect this one was introduced around the middle of the 20th century. It is that $a\bmod b$ means the remainder when $a$ is divided by $b.$ Thus $83\bmod5=3.$ The correct parsing is $(83\bmod5) = 3,$ <b>not</b> $\require{cancel} \xcancel{83\bmod(5=3)}$ or anything like that.
- <b>Older notation:</b> The older convention was introduced in the book _Disquisitiones Arithmeticae_, by Carl F. Gauss (rhymes with "house" as pronounced by most if not all English-speaking people), published in 1801. The older convention is the most prevalent one among mathematicians and the newer perhaps among computing experts. According to the older convention one may write $83\equiv98\bmod5$ and it is parsed as $(83\equiv98)\bmod5,$ <b>not</b> as $\xcancel{83\equiv(98\bmod5)}.$ It means that $83$ and $98$ both leave the same remainder when divided by $5.$ Or, equivalently, it means $83-98$ is divisible by $5.$ In words, one says:
- <b>Eighty-three is congruent to ninety-eight, modulo five.</b>
- The number $5$ is the <b>modulus</b>. The word "modulo" is a different case of that Latin word (modulus is nominative and modulo is ablative or dative or something; ask your Latin teacher). Often in speaking it is abbreviated just as in the mathematical notation, thus:
- <b>Eighty-three is congruent to ninety-eight, mod five.</b>
#1: Initial revision
by
Michael Hardy
·
2026-06-16T18:20:39Z (3 months ago)
It depends upon the context.
I am mainly posting to address a kind of confusion that often arises. There are two conventions of terminology and notation in modular arithmetic, and one should be clear about which is which.
<b>Newer notation:</b> I suspect this one was introduced around the middle of the 20th century. It is that $a\bmod b$ means the remainder when $a$ is divided by $b.$ Thus $83\bmod5=3.$ The correct parsing is $(83\bmod5) = 3,$ <b>not</b> $\require{cancel} \xcancel{83\bmod(5=3)}$ or anything like that.
<b>Older notation:</b> The older convention was introduced in the book _Disquisitiones Arithmeticae_, by Carl F. Gauss (rhymes with "house" as pronounced by most if not all English-speaking people), published in 1801. The older convention is the most prevalent one among mathematicians and the newer perhaps among computing experts. According to the older convention one may write $83\equiv98\bmod5$ and it is parsed as $(83\equiv98)\bmod5,$ <b>not</b> as $\xcancel{83\equiv(98\bmod5)}.$ It means that $83$ and $98$ both leave the same remainder when divided by $5.$ In words, one says:
<b>Eighty-three is congruent to ninety-eight, modulo five.</b>
The number $5$ is the <b>modulus</b>. The word "modulo" is a different case of that Latin word (modulus is nominative and modulo is ablative or dative or something; ask your Latin teacher). Often in speaking it is abbreviated just as in the mathematical notation, thus:
<b>Eighty-three is congruent to ninety-eight, mod five.</b>