Less than, Equal to, Greater Than Symbols

If A and B are two constant expressions, we write A = B if they are equal, and A ≠ B, if they are not. For example, for any number or expression N, N = N. 1 = 1, 2.5 = 2.5, x + y² = x + y². On the other hand, 1 ≠ 2.5. One can't go wrong with expressions like N = N because they do not say much. The sign "=" of equality which is pronounced "equal to" has other, more fruitful uses.

"=" is used to make a statement

The symbol of equality "=" is used to make a statement that two differently looking expressions are in fact equal. 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. So, being equal, does not necessarily mean being the same.

Also, the statement that involves the symbol "=" may or may not be correct. While 1 + 1 = 2 is a correct statement, 1 + 2 = 4 is not. The same holds for the symbol "≠", not equal. But the meaning is just the opposite from "=". While 1 + 2 ≠ 4 is a correct statement, 1 + 1 ≠ 2 is not.

"=" is used to pose a problem

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. For example, x + 1 = 4, depending of what x may stand for, may or may not be correct. The request to solve x + 1 = 4 means to find the value (or values) of x, which x + 1 is equal to 4. In this particular case, there is only one value of x which does the job, namely x = 3.

The terminology varies. I was taught that the statement A = B in which A and B is constant, fixed expressions, is called an equality or identity. If they include variables, A = B is called an equation. Nowadays, they use the term "equation" in both cases, the former is being said to be a constant equation.

The reason for the later usage I think is that in algebra a constant expression may contain variable-like symbols to denote generic numbers. For example, (x + y)² = x² + 2xy + y² is a statement that is not supposed to be solved. 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. This usage is similar to the statement of physical laws. For example, in Einstein's law, E = mc², E and m are variables, while c is constant.

"=" is used to define or name an object

In algebra, one may define a function f(x) = x² + 2x³. This is neither a statement, nor a request to solve an equation. This is a convenience definition. 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))), ...

In geometry, as another example, one may introduce point A = (2, 3) and another point B = (-2, 5). The midpoint M = (A + B) /2 = (0, 4) lies on the y-axis.

Symbols "<" and ">" of comparison

Some mathematical objects can be compared, e.g, of two different integers one is greater, the other smaller. Other mathematical objects, complex numbers for one, cannot be compared if the operation of comparison is expected to possess certain properties.

Symbol ">" means "greater than"; symbol "<" means "less than". For example, 2 < 5, 5 > 2. To remember which is which, observe that both symbols have one pointed side where there is just one end, and one split side with 2 ends. The fact that 1 is less than 2 is expressed as 1 < 2, which is the same as 2 > 1, i.e., that 2 is greater than 1. The pointed end with a single endpoint points to the smaller of the two expressions.

Like the symbol of equality, the symbols of comparison, may be used to make a statement or to pose a problem. 2 < 5 is a correct statement. 5 < 2 is incorrect statement. x + 2 < 5 may be correct or not, depending on the value of x. You may be asked to find those values of x for which x + 2 < 5. In which case, by adding -2 to both sides of the inequality we obtain x < 3 which is the solution to x + 2 < 5.

In algebra, a statement may include generic variables, like the AM-GM inequality: (x + y) / 2 ≥ √xy, which is true for all positive x and y.

By the way, symbol "≤" means "less than or equal to". The (x + y) / 2 ≥ √xy, becomes equality for x = y. For example, if x = y = 2, then (x + y) /2 = (2 + 2) /2 = 2.

The inequality -x² > x² has no solutions among integers. The inequality -x² ≥ x² has one solution: x = 0.

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