# Less than, Equal to, Greater Than Symbols

If A and B are two constant expressions, we write A = B if they are equal, and *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 *being equal*, does not necessarily mean *being the same*.

Also, the statement that involves the symbol "=" may or may not be correct. While *not equal*. But the meaning is just the opposite from "=". While

#### "=" is used to pose a problem

If the expressions A and B are not constant, i.e., if they contain variables, then most often

The terminology varies. I was taught that the statement *equality* or *identity*. If they include variables, *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,

#### "=" 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

#### 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,

Like the symbol of equality, the symbols of comparison, may be used to make a statement or to pose a problem.

In algebra, a statement may include generic variables, like the AM-GM inequality:

By the way, symbol "≤" means "less than or equal to". The

The inequality -x² > x² has no solutions among integers. The inequality

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