I want to know the meaning of ≡ this symbol. I read about it on Wikipedia but didn't actually understand what that means.
I saw this symbol in coordinate geometry where it was given that $A \equiv (1,2)$. But what's wrong with $A = (1, 2)$? Is it wrong? Sometimes, my teacher also represents coordinates as $A(1, 2)$.
Best Answer
As far as my understanding can be trusted, $\equiv$ is often used to draw the reader's attention on the nature of the objects being considered. There is always a notion of equality somewhere behind it, but that is more subtle than a simple equality and highly dependent on the context.
In the case of plane geometry, I interpret $A \equiv (1,2)$ as "$A$ is the point of the plane given by the coordinates $(1,2)$ relatively to the considered coordinate system". There is a bijection between the set of points on the plane and the set of couples of real numbers for a given coordinate system, but a point is not a couple of real numbers.
In arithmetics, we say that $a \equiv b \mod c$ when $a$ and $b$ are congruent. Going further in that direction, it means that $a$ and $b$ represent the same element in the group $\mathbb{Z}/c\mathbb{Z}$, but as element of $\mathbb{Z}$, they might be different.
A third common example is when we write $f \equiv 0$ to say that $f$ is identically $0$. $f$ is not the constant $0$, but rather the constant $0$ function.