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
You are accessing this answer with a direct link, so it's being shown above all other answers regardless of its score. You can return to the normal view.
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 $\text{\equiv}$.
0 comment threads
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)
0 comment threads
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.
0 comment threads