In geometry, as another example, one may introduce point A = (2, 3) and another point B = (-2, 5). After it is given, we may talk of the powers of function f, its derivative f', or of its iterates f(f(x)), f(f(f(x))). This is neither a statement, nor a request to solve an equation. In algebra, one may define a function f(x) = x² + 2x³. For example, in Einstein's law, E = mc², E and m are variables, while c is constant. This usage is similar to the statement of physical laws. It simply says that the two expressions, (x + y)² on the left, and x² + 2xy + y² on the right are equal regardless of specific values of x and y. For example, (x + y)² = x² + 2xy + y² is a statement that is not supposed to be solved. The reason for the later usage I think is that in algebra a constant expression may contain variable-like symbols to denote generic numbers. Nowadays, they use the term "equation" in both cases, the former is being said to be a constant equation. If they include variables, A = B is called an equation. I was taught that the statement A = B in which A and B is constant, fixed expressions, is called an equality or identity. In this particular case, there is only one value of x which does the job, namely x = 3. The request to solve x + 1 = 4 means to find the value (or values) of x, which x + 1 is equal to 4. For example, x + 1 = 4, depending of what x may stand for, may or may not be correct. If the expressions A and B are not constant, i.e., if they contain variables, then most often A = B means a request to find the values of the variables, for which A becomes equal to B. While 1 + 2 ≠ 4 is a correct statement, 1 + 1 ≠ 2 is not. But the meaning is just the opposite from "=". The same holds for the symbol "≠", not equal. While 1 + 1 = 2 is a correct statement, 1 + 2 = 4 is not. So, being equal, does not necessarily mean being the same.Īlso, the statement that involves the symbol "=" may or may not be correct. For example, 1 + 1 does not look like 2 but the definitions of the symbols 1, 2, +, and the rules of arithmetic tell us that 1 + 1 = 2. The symbol of equality "=" is used to make a statement that two differently looking expressions are in fact equal. The sign "=" of equality which is pronounced "equal to" has other, more fruitful uses. One can't go wrong with expressions like N = N because they do not say much. For example, for any number or expression N, N = N. If A and B are two constant expressions, we write A = B if they are equal, and A ≠ B, if they are not. Less than, Equal to, Greater Than Symbols
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