Addition of Equations

Usually an equation is an algebraic shorthand for a request to determine values of a variable (unknown) for which two functions become equal. This is expressed as

f(x) = g(x)

For example x² + 3 = 2x. A similar notation, i.e. the equality symbol "=", is also used to convey the idea that two quantities are equal unconditionally, i.e. for all values of the variable, for example, (x-1)² = x² - 2x + 1. Such an expression is usually called an identity. In other cases, where the fact that the "left" and "right" side of the identity may contain a variable is not important, the identity is called equality. For example, if f(x) = (x-1)² and g(x) = x² - 2x + 1, then f = g is such an equality. 2 + 3 = 5 is another example.

In all three cases, two quantities that are claimed to be equal are written side by side with the equality symbol "=" in between. Let we have two such assertions

A = B and C = D

We can construct a third assertion by equating the sums of the left and right sides: A + C = B + D. This is how we add equations, identities, and equalities. We may consider a space of equations with 0 = 0 as the zero element, and -A = -B as the inverse of A = B but I must confess of being unaware of any useful results that follow from such a formalization.

The equality symbol plays a very special role in the whole of science and not only in mathematics. The notion of equality is probably more universal and basic than any other in the human thesaurus. Geometric shapes are equal (congruent in the modern terminology) when they overlap each other. Topologically, shapes are equal when they can be continuously transformed into one another. Functions are equal when they take on the same values for every point they are defined on. Sets are equal when they contain exactly the same elements. In every example the criteria of equality is different for different kinds of objects. However, the basic properties of equality do not depend on a particular family of objects being equated.

Euclid's Elements, an absolutely unique book written more than 2300 years ago and that since underwent more than 2000 editions, includes the notion of equality and formulates its properties among the Common Notions. As a point of reference, the Elements, axiomatic bible of Mathematics, consists of

Definitions - attempts to define the objects of mathematical study
Postulates - "self-evident" truths about mathematical objects
Theorems - facts about mathematical objects that logically followed from axioms
Common Notions - self-evident truths not specific to mathematics

Here are the first three of the Elements' Common Notions:

Things which are equal to the same thing are also equal to one another.
If equals be added to equals, the wholes are equal.
If equals be subtracted from equals, the remainders are equal.

Euclid was unaware of algebra but in the modern, algebraic notations, addition of equations expresses nothing more than the same simple truths that have been recognized and annunciated thousands of years ago and taught ever since. It would probably be considered a flippancy to remark that, being germane to human thought, these truths have been used by the unaware humankind long before the first edition of the Elements.

References

W. Dunham, Journey through Genius, Penguin Books, 1991

What Can Be Added?

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