What $\equiv$ called?
I saw the text ($\equiv$) few times. When I had searched in internet I got that..
In mathematics, the triple bar is sometimes used as a symbol of identity or an equivalence relation (although not the only one; other common choices include $\sim$ and $\approx$).[7][8] Particularly, in geometry, it may be used either to show that two figures are congruent or that they are identical.[9] In number theory, it has been used beginning with Carl Friedrich Gauss (who first used it with this meaning in 1801) to mean modular congruence: $a\equiv b\pmod N$ if $N$ divides $a − b.$[10][11] It is also used for "identical equality" of functions; one writes $f\equiv g$ for two functions $f, g$ if we have $f(x)=g(x)$ for all $x$ ~ https://en.wikipedia.org/wiki/Triple_bar
I don't have any problem with understanding the "letter". But I was thinking what it actually called. Like as we call $=$ it equal... plus, minus, proportional etc etc. But what it actually called? I don't think triple bar is meaningful cause $\bar{x}$ it's called bar $x.$ But whenever we say triple bar seems like they are trying to represent something just like this but in reality there's no double or triple bar on above of $x$ but triple bar for that symbol isn't looking good to me. Is there anything else to call it?
3 answers
Is congruent to
This satisfies both usages you mentioned and is the most common way I pronounce it/hear it pronounced.
$8 \equiv 3 \pmod{5} \to \text{8 is congruent to 3 mod 5 }$ $\triangle ABC \equiv \triangle DEF \to \text{Triangle ABC is congruent to Triangle DEF}$
Alternatively, "is equivalent to" could work in most cases - in fact, the $\LaTeX$ command for this is $\text{\equiv}$.
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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.
Newer notation: 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 $(83\bmod5) = 3,$ not $\require{cancel} \xcancel{83\bmod(5=3)}$ or anything like that.
Older notation: 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 $(83\equiv98)\bmod5,$ not 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:
Eighty-three is congruent to ninety-eight, modulo five.
The number $5$ is the modulus. 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:
Eighty-three is congruent to ninety-eight, mod five.
I was doing some research on this earlier and found 2 resources that give some guidance:
2 Miscellaneous symbols
...
≡ is identical to or is congruent to
- The Unicode® Standard: https://util.unicode.org/UnicodeJsps/character.jsp?a=≡ (indexed here as well)
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