Less than or equal sign

If I know for two numbers a and b that

$${a < b }$$

Then is it correct to say that

$$ a \leq b $$

I know that the second statement is true as long as the first one is. It seems OK as it is true but from the other side it seems kinda weird (to me), to say that, if you know, that a is strictly less that b.It's like you lose some extra information.

You have a sound understanding of the matter:

It is true that $a\lt b \implies a\leq b$, since $$a \leq b \iff (a\lt b\;\text{or}\;a = b)$$

If you need to prove $a \leq b$, then it suffices to prove either $a < b$ or $a = b$.

On the other hand, if you need (or want) to prove $a$ is strictly less than $b$, then it is necessary to prove $a < b$.

There can be, indeed, some loss of information, just as we lose information when, while knowing $x = 3$, we assert only the (true) claim that $x \leq 3$.

But there are examples where we can make a claim for a wider range of values by using the inclusive $\leq$.

I don't know if this example will be of any help, but if you are trying to prove, by induction, that $n^2 \leq 2^n$ for $n>3$, in your inductive proof, we first have the base case $n = 4$, where the inequality holds precisely because equality holds: $4^2 = 2^4$, hence $4^2 \leq 2^4$ is true.

But in the inductive step, you might use the following chain: $$(n+1)^2 = n^2 + 2n + 1 \leq 2^n + \underbrace{2n + 1}_{\large <\; 2^n,\,n\geq 3} \lt 2^n + 2^n = 2\cdot 2^n = 2^{n+1}$$

So in the inductive step, even though we find we have a strict inequality involved, the point of the induction is to prove $n^2 \leq 2^n$, for $n \gt 3$ and because the base case is true (because equality holds), we can actually say more by affirming the NON-strict inequality (since the range of $n$ for which the proposition is true is greater with $\leq$ than with $\lt$.)

In the end, the amount of information conveyed by the choice of using $\lt$ vs. $\leq$ depends, as do many choices, on context.

It is correct to say that if $a < b$, then $a \le b$. And yes, I completely agree with you: if explicitly being shown $a < b$, there is a certain "loss" of information when we write $a \le b$ instead of being concrete.

"$\le$" is, however, very useful in real-life, practical mathematical problems such as optimization when you want to say that one quantity does not exceed another.