# Definitions, Identities, Equations, and Formulas

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Some of the latest posts in Speculations have led me to this attempt at clarification, because I see that some people are very confused when it comes to handling equalities. The point is their meaning. It's subtle, but it's not moot in general; although in some cases it may be.

Not all equalities are the same. There are at least these variants:

Definition (or substituting symbol or replacing symbol, etc.)

Example: fine structure constant,

$\alpha\overset{{\scriptstyle \textrm{def}}}{=}\frac{e^{2}}{4\pi\epsilon_{0}\hbar c}$

Identity (or universal equivalence under known or specified set of previous rules that may be algebraic, geometric, etc.)

Example: constriction between sine and cosine,

$\sin^{2}\theta+\cos^{2}\theta\equiv1$

Equation (proposed equivalence considered in relation to the finding of "solvers" or "solutions")

Example: Pythagorean Theorem assumed true, solve for c2, given c1 and h,

$5^{2}=\left(c_{1}\right)^{2}+3^{2}$

Formula (proposed equivalence under non-universal, somehow hidden, or not necessarily specified assumptions)

Example: Pythagorean Theorem,

$h^{2}\overset{\cdot}{=}\left(c_{1}\right)^{2}+\left(c_{2}\right)^{2}$

Some formulas can become equations once you give values to terms or supply additional information. And yes, sometimes the distinction between what is a formula and what an identity can be blurry, depending on how we look at the defining "valid rules." E.g., the constriction between sine and cosine can be seen as an identity if we take Euler's identities,

$\sin\theta\equiv\frac{e^{i\theta}-e^{-i\theta}}{2i}$

$\cos\theta\equiv\frac{e^{i\theta}+e^{-i\theta}}{2}$

as the point of departure; or a formula, if we adopt Pythagoras' theorem plus geometrical definitions of sine and cosine. It may also depend on assumptions about curvature of space, etc.

Needless to say, most people who use mathematics on a regular basis, don't need to be reminded of these distinctions, because they intuitively know what they're about. The danger is when people start playing with equalities (especially definitions, as I've seen) thinking they have a different value than they really do. Also needless to say, but better said, the symbols for eq., id., form., and def. are not intended for general use, but just to illustrate how confusing all this proves to be to many people.

Edited by joigus

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2 hours ago, joigus said:

Some of the latest posts in Speculations have led me to this attempt at clarification, because I see that some people are very confused when it comes to handling equalities.

Good subject to discuss. +1

However a few points.

I would call this the use of the equals sign "=" not "equalities" because there are differences of usage.

Equalities are only one use of the overworked sign. As you say for most people the context determines which one.

Chemists (and others) use the equals sign as well as arrows to denote a process or the idea of "...A....leads to......B"

Identities are not the same as equalities since

x - 1 = 4 is true for one and only one value of x.

${x^2} - 1 \equiv \left( {x - 1} \right)\left( {x + 1} \right)$

is true for each and every value of x and is an identity.

Strictly we should use the three bar symbol for identities.

Edited by studiot

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11 minutes ago, studiot said:

I would call this the use of the equals sign "=" not "equalities" because there are differences of usage.

No quarrel with this.

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Already between the two of you, fairly well learned and precise, members we have a differing usage of equalities.
Imagine how difficult it is for the rest of us, who haven't been exposed to that kind of rigor/precision since we left University ?
I know, myself, quite often struggle to use the proper terminology; and quite often fail.

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5 minutes ago, MigL said:

Already between the two of you, fairly well learned and precise, members we have a differing usage of equalities.
Imagine how difficult it is for the rest of us, who haven't been exposed to that kind of rigor/precision since we left University ?
I know, myself, quite often struggle to use the proper terminology; and quite often fail.

Actually, @MigL, I think that an excessive attention to precision can be as much crippling as a lack of it. The reason of my post is that I've seen people going around in circles because of their inability to understand that they're using a definition, instead of an equation. That's the opposite end of the spectrum.

TBH, I don't think all the distinctions I've made are all that important. But being able to tell a definition from an equation really is a major mistake, that I don't think you, for example, would ever make.

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Good topic @joigus

I have also seen some confusion due to the common usage of "=" in programming where "x=y" could mean for instance:

-assign to the variable x to point to whatever y points to
-copy the value of y into the variable x
-return the boolean value true if x equals y, otherwise return false

This confuses some individuals* when moving back and forth between math and programming or when switching between programming languages. A typical case is where reals or floating point numbers are involved. In math I thing the following is valid: 1.0/2 = 0.5 which is not necessarily true in computations.

Side note/speculation/question: would there be less confusions if the initial ascii character set have had more math symbols? Both in programming and math?

49 minutes ago, MigL said:

Fortunately we are active on a forum where most discussions are in good faith, and lack of rigor/precision can be resolved in a friendly manner. Apology accepted, in advance.

*) me included, 20+ years of experience does not seem to help.

Edited by Ghideon
grammar

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17 minutes ago, Ghideon said:

Good topic @joigus

I have also seen some confusion due to the common usage of "=" in programming where "x=y" could mean for instance:

-assign to the variable x to point to whatever y points to
-copy the value of y into the variable x
-return the boolean value true if x equals y, otherwise return false

This confuses some individuals* when moving back and forth between math and programming or when switching between programming languages. A typical case is where reals or floating point numbers are involved. In math I thing the following is valid: 1.0/2 = 0.5 which is not necessarily true in computations.

Side note/speculation/question: would there be less confusions if the initial ascii character set have had more math symbols? Both in programming and math?

Fortunately we are active on a forum where most discussions are in good faith, and lack of rigor/precision can be resolved in a friendly manner. Apology accepted, in advance.

*) me included, 20+ years of experience does not seem to help.

Although I've been out of touch with programming for a while, I do remember lots of confusion with languages as "cavalier" as PERL, for example. 😮

Thank you. Very interesting addition to the initial question.

I personally have never found @MigL at fault in rigour. 😲

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From my area of math, the "definitional equality" is usually written as "x := ". For example: "Let f(x) := (x - 1)/(x + 1)".

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7 minutes ago, uncool said:

From my area of math, the "definitional equality" is usually written as "x := ". For example: "Let f(x) := (x - 1)/(x + 1)".

Aaah, yes. I had forgotten that one. +1

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13 minutes ago, uncool said:

From my area of math, the "definitional equality" is usually written as "x := ". For example: "Let f(x) := (x - 1)/(x + 1)".

True! That is also applicable to some programming languages.

10 minutes ago, joigus said:

I personally have never found @MigL at fault in rigour. 😲

Me neither. Which means I lack the rigour required to spot @MigL's possible faults? 🤔

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