Could ≡ be used to represent equivalent fractions?

Question

I was thinking about the exact definition of a "triple bar", ≡, which means "identical to". If we are being super formal, wouldn't it be more correct to say that $\frac{1}{2}≡\frac{2}{4}$ instead of $\frac{1}{2}=\frac{2}{4}$, for example, since (if we don't reduce them to decimals) these two numbers are technically the same? Sorry I don't really understand the difference these two operators without going into complicated logic.

Answer 1

If you are being "super formal", then first you need to define what the symbol $\frac{a}{b}$ means. And what $\equiv$ means.

Note that it is not correct to say that $\equiv$ always means "identical to"... that's just one meaning in some contexts. (For example, open any number theory book and you will find something like $2\equiv 7\pmod{5}$, which is true, even though $2$ is not "identical to" $7$).

Now, leaving the precise definition of $\equiv$ to later, let's start with the precise definition of $\frac{a}{b}$. Turns out, there isn't an absolute meaning to it either. There are a few standard ways to give it meaning:

1. You can start by assuming you know what integers are. Then you take the set of ordered pairs $(a,b)$ with $a,b\in\mathbb{Z}$, $b\neq 0$. You define an equivalence relation on this set by $(a,b)\sim (c,d)\iff ad=bc$. Once you prove that this is an equivalence relation, for any integers $a$ and $b$, with $b\neq 0$, you define the symbol $\frac{a}{b}$ to mean "the equivalence class of the ordered pair $(a,b)$". that is, $$\frac{a}{b} = \{(x,y)\in \mathbb{Z}\times(\mathbb{Z}-\{0\})\mid (a,b)\sim (x,y)\}.$$ (And then you define how to identify integers inside this set, addition, multiplication, etc.) With this meaning, then $\frac{1}{2}=\frac{2}{4}$ is literally, "super formally", true. Because the equivalence class of $(1,2)$ is equal to the equivalence class of $(2,4)$.

2. You can define the result of performing a "division" of positive integers, for example geometrically, and after identifying numbers with lengths of line segments, then $\frac{a}{b}$ for positive $a$ and $b$ represents the length obtained by taking a segment of length $a$ and "dividing" by a segment of length $b$. In that interpretation, again $\frac{1}{2}$ is literally, "super formally", equal to $\frac{2}{4}$. You can then extend these "positive rationals" to all rationals by adding negative numbers.

3. Or you could simply define a set of glyphs made up of a horizontal line, an integer above the line, and a nonzero integer below the line. Then you define an equivalence relation on the set of glyphs by saying $$\frac{a}{b} \equiv \frac{c}{d}\iff ad=bc$$ in which case $\equiv$ is appropriate, since "$\frac{a}{b}$" represents a glyph.

4. Or you could define what it means for a "glyph" as in 3 to be in "least terms" (say, $b\gt 0$, $\gcd(a,b)=1$, etc), then define what it means for $\frac{a}{b}$ that is in least terms to be "equal" to the glyph $\frac{c}{d}$, prove that each glyph is "equal" to exactly one glyph in least terms, and then define equality among arbitrary glyphs to mean "they are equal to the same glyph in least terms" (or define this as an equivalence relation, etc.)

In short, if you want to be "super formal", then you need to be formal all the way through, not just at the very end: you need to say exactly what each symbol means. And depending on exactly what each symbol means, $\frac{1}{2}=\frac{2}{4}$ may be literally correct, or it may be an informal gloss, or it may be just incorrect (false as glyphs); and likewise, $\frac{1}{2}\equiv \frac{2}{4}$ may be literally correct, or not.

Answer 2

$\equiv$ means "logically equivalent to" or "congruent" in modular arithmetic (e.g. 26 $\equiv$ 11 (mod 5) since both 26 and 11 have a remainder of 1 after being divided by 5). $\equiv$ is thus used to show two mathematical statements have the same logical meaning or are in the same remainder class, while = is used to show two expressions/terms have the same value. This is why we say "$m\angle A = m\angle B$" (the measure of Angle A equals the measure of Angle B) but $\angle A \cong \angle B$ ("Angle A is congruent to angle B).