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A two part question.

1

True or False: when working with an equation or inequality, everything that you do is either:

  • a substitution, or
  • an operation performed on each side

Note that algebraic or numerical simplifications are substitutions - $2+2=4$, so we're free to substitute $4$ where $2+2$ is present in an equation.

2

Is there a generic name for 'performing an operation on both sides' of an equation/inequality?

Transformation? Equivalence derivation? I think that I've heard 'transpose' for the case of adding/subtracting something to both sides in order to eliminate it from one side, but I don't believe that this is used for, say, doubling or squaring both sides.

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  • $\begingroup$ Does factoring count as a simplification? $\endgroup$ – JB King Dec 11 '15 at 15:49
  • $\begingroup$ @JBKing Whether or not it's a simplification, a factorization would be an example of a substitution. $\endgroup$ – ColinK Dec 11 '15 at 15:53
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    $\begingroup$ I usually call it "balancing the equation" since you are are doing the same operation on both sides. $\endgroup$ – John Odom Dec 11 '15 at 16:33
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    $\begingroup$ @JohnOdom That's an interesting take, but a nitpick might be that the equation was already balanced. Rebalancing? $\endgroup$ – ColinK Dec 11 '15 at 16:38
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    $\begingroup$ I think (2) is formalized by the notion of congruence: books.google.ie/… $\endgroup$ – bolbteppa Dec 12 '15 at 6:07

12 Answers 12

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Is there a generic name for 'performing an operation on both sides' of an equation/inequality?

Yes. The name is algebra (maybe "doing algebra" is more appropriate and/or sounds better).

Here there are three sources:

  • The word "algebra" comes from a book written in 830 by the astronomer Mohammed ibn Musa al-Khowarizmi (c. 825), titled Al-jabr w'al muqâbala. The word al-jabr meant "restoring," in this context, restoring the balance in an equation by placing on one side of an equation a term that has been removed from the other; thus if $-7$ is removed from $x^2 - 7 = 3$, the balance is restored by writing $x^2 = 7 + 3$. Al' muqâbala meant "simplification," as by combining $3x$ and $4x$ into $7x$ or by subtracting equal terms from both sides of an equation. [...] When al-Khowarizmi's book was first translated into Latin in the twelfth century, the title was rendered as Ludus algebrae et almucgrabalaeque, though other titles were also used. The name of the subject was finally shortened to algebra. (Kline's book)
  • He [al-Khwarizmi] did this in his book al-jabr w al-muqabalah. “Al-jabr” (from which stems our word “algebra”) denotes the moving of a negative term of an equation to the other side so as to make it positive, and “al-muqabalah” refers to cancelling equal (positive) terms on both sides of an equation. These are, of course, basic procedures for solving polynomial equations. Al-Khwarizmi (from whose name the term “algorithm” is derived) applied them to the solution of quadratic equations. (Kleiner's book)
  • The mathematical sense [of the word algebra] comes from the title of a 9th-century Arabic book ilm al-jabr wa'l-mukabala, ‘the science of restoring what is missing and equating like with like’, written by the mathematician al-Kwarizmi. (Oxford Dictionaries)

EDIT

@BillDubuque pointed out that a correct answer for the question should give a name for the following rule: equalities are preserved by "performing an operation on both sides" (see the comments in this post).

In the Patrick Suppes terminology, the name of this rule can be Consequence of the Rule Governing Identities.

Rule Governing Identities (RGI): If $S$ is an open formula, from $S$ and $t_1=t_2$, or from $S$ and $t_2=t_1$ we may derive $T$, provided that $T$ results from $S$ by replacing one or more occurences of $t_1$ in $S$ by $t_2$. Moreover, the identity $t=t$ is derivable from the empty set of premises. (Suppes book)

Remark 1: Given an operation $f$, let $S$ be the formula $f(z)=f(z)$. Then, by the RGI, $$x=y\quad\Longrightarrow\quad f(x)=f(y).$$ In words, "equalities are preserved by performing an operation on both sides".

Remark 2: The RGI also justifies general substitutions in equalities. For example, $x+y=2$ and $x=y+3$ implies $(y+3)+y=2$ (here, $S$ is the formula $x+y=2$). To understand the word "general" see the comments in the ASKASK's answer.

Remark 3: Other names (probably, the usual ones) are Replacement, Substitution Property and Substitution by equality (see other answers and comments).

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    $\begingroup$ @ColinK This ancient, restricted meaning of "algebra" is not the correct modern name that you seek. $\endgroup$ – Bill Dubuque Dec 11 '15 at 18:35
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    $\begingroup$ @BillDubuque According this source, my answer provides a correct modern name for the procedure. Of course, the word "algebra" has other meaning (as the word "analysis" that can be a "examination" or a "branch of mathematics" or something else). $\endgroup$ – Pedro Dec 11 '15 at 19:10
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    $\begingroup$ @Colink The highly overloaded and imprecise phrase "doing algebra" can mean so many different things that it makes no sense to attempt to use it to name this very specific property. This answer is highly misleading. $\endgroup$ – Bill Dubuque Dec 11 '15 at 19:23
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    $\begingroup$ @ColinK Is taking the numerator of a fraction "algebra"? If so then doing that to $\frac{1}2 = \frac{2}4$ yields $ 1 = 2$. If that is not "algebra" then what precisely is "algebra" and why does it enable said property? $\endgroup$ – Bill Dubuque Dec 11 '15 at 19:47
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    $\begingroup$ @BillDubuque I would say context is very important for all kinds of definitions. In the context of distinguishing this kind of math from what is commonly called "arithmetic", I submit there is no better word than "algebra". While any undergraduate math major quickly learns that the word "algebra" formally means something else (or maybe something larger), for 98% of the world (and approximately 100% of high school students), using the word "algebra" in this way is a good way to clearly communicate what one is talking about. Outside of the post-secondary school context, this is a good answer. $\endgroup$ – Todd Wilcox Dec 11 '15 at 20:01
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"Performing the same operation on both sides of an equality" is called the substitution property.

"Performing the same operation on both sides of an inequality" is not always possible. For instance, adding a constant is ok but multiply by one is not if the constant is negative.

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    $\begingroup$ Well, if you know the constant is negative then it just reverses the direction of the inequality. $\endgroup$ – Random832 Dec 11 '15 at 19:10
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    $\begingroup$ The linked article should do a better job of stressing the key point that this property holds because we are substituting/replacing an argument by an equal value in a mathematical (set-theoretic) function. For more general "functions" it may fail, e.g. in computer science, for programming functions - see pure function, deterministic algorithm, referential transparency. $\endgroup$ – Bill Dubuque Dec 13 '15 at 19:14
  • $\begingroup$ Let try that comment again! Its pretty tragic that this is called the substitution property; surely the substitution property should be $A = B \rightarrow A(x:= J) = B(x:=J)$, where $A(x:=J)$ denotes the result of replacing every occurrence of $x$ in $A$ with $J$. $\endgroup$ – goblin Jun 28 '16 at 7:17
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For Q1, I would say that yes, roughly everything is either a substitution or an operation performed on both sides. In fact, I would argue that "an operation performed on both sides" is actually just another form of substitution.

For Q2, whenever you do something to both sides, you are just applying some function to both sides. Hence, you are using the property: $$ x=y \implies f(x)=f(y)$$

Which must hold true for all functions (by the definition of a function). For example, if you have $x^2=5$, and you squareroot both sides, you are just evaluating the function $f(x)=\sqrt{x}$ at the values $x^2$ and $5$, which must both be equal. Thus, you are simply just substituting the two values into the function and looking at the new equality you have.

EDIT just to clarify: The example of applying the squareroot function to $x^2=5$ is only valid when you note that $\sqrt{x^2}=|x|$, so the equality you are left with is $|x|=\sqrt{5}$

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    $\begingroup$ The principle you state ifor Q2 is a good one, but works only when you're careful. The square root function isn't a good example. $x^2 = 5$ has two solutions, but $x = \sqrt{5}$ has just one. $\endgroup$ – Ethan Bolker Dec 11 '15 at 16:00
  • $\begingroup$ Could you elaborate on 'an operation performed on both sides' being a substitution? I don't see it, at least as far as my meaning of 'substitution' goes (replacing an expression with another which is equal to it). $\endgroup$ – ColinK Dec 11 '15 at 16:01
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    $\begingroup$ Take my squareroot function example. You are given that $x^2=5$, and also that $\sqrt{x^2}=\sqrt{x^2}$. In the second equality, substitute the second $x^2$ with $5$ to get the new equality. By "doing the same operation to both sides", you are just substituting your original equation into the equation $f(x)=f(x)$ $\endgroup$ – ASKASK Dec 11 '15 at 16:04
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    $\begingroup$ @EthanBolker When you take the square root of $x^2$, the result is not $x$, but rather $|x|$. So the square root function is a great example, but not if it is used incorrectly. $\endgroup$ – mweiss Dec 11 '15 at 17:51
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    $\begingroup$ @ColinK This property of mathematical functions has more widely-used names in more general contexts where it may fail for various reasons, e.g. in computer science, for programming functions, e.g. see pure function, deterministic algorithm, referential transparency. See also my comments here. $\endgroup$ – Bill Dubuque Dec 13 '15 at 17:50
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In education the term balancing is used for the process of "doing the same thing to both sides". One may refer to the "balance method" of solving equations, as contrasted with function machine methods.

There is a weakness in this terminology, if the equation was already in "balance" then it is arguable that you are not balancing it, rather you are maintaining the balance that already existed. And if the equation is not balanced (ie it expresses a falsehood) then applying a function to both sides may or may not bring it into balance. Nevertheless this terminology is recognised and used (Example link)

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The key property here is well-defined. A function is well-defined if its value is independent of the name given to its input. For example, $f(x)=x^2$ is well-defined, so $f(3)=f(2+1)$, because $3=2+1$. This takes us from $3=2+1$ to $3^2=(2+1)^2$ (i.e. squaring both sides).

For another example, $g(x)=x+3$ is well-defined, so $g(5)=g(4+1)$, which takes us from $5=4+1$ to $5+3=4+1+3$ (i.e. adding $3$ to both sides).

Generally any function we work with must be well-defined, it is a "mandatory" property of functions. When defining a new function, we need to verify that it is indeed well-defined. For example, consider a function from equivalence classes modulo $n$. It needs to give the same result regardless of which element of the equivalence class you choose.

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An equation is a logical statement, which equates two mathematical formuae. If that statement is true, then we can transform it into another equation which is also true. This is called derivation. Derivation can involve doing similar algebra on both sides, like adding the same term to both sides, or substituting a value for a variable across the entire equation, or other substitution, or doing some independent rearrangements on either or both sides which preserve their meaning.

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$1.\, \rm False$

Consider the statement:

$$ 0=1 $$ We clearly have

$$0=1\rightarrow \text{Apple}=\text{Orange}$$

Which is true and it's not a substitution or some kind of operation like that.

$\rm 2.\, $I'd say it's just applying a function to both sides.

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  • $\begingroup$ I don't understand your first part. How are you jumping from $0=1$ to Apple$=$Orange? If you are claiming that it is vacuously true, then you are definitely not manipulating the equation. $\endgroup$ – ASKASK Dec 11 '15 at 16:06
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    $\begingroup$ Consider the map $f:\{0,1\}\to \{\text{Apple,Orange}\}, 0\mapsto \text{Apple}, 1\mapsto \text{Orange}$. $\endgroup$ – YoTengoUnLCD Dec 11 '15 at 16:08
  • $\begingroup$ Then I would argue, as in my post, that you are substituting an equality into that function. $\endgroup$ – ASKASK Dec 11 '15 at 16:08
  • $\begingroup$ @ASKASK Oops, I misread your question. OP asks 'when working with equalities/inequalities, all we do is...' I understood this as 'What can we do to draw conclusions from a given equality'? And as I showed, if you have an equality like $0=1$ you don't need to do anything at all! You can infer everything from it, that was my point. $\endgroup$ – YoTengoUnLCD Dec 11 '15 at 16:13
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I commonly see these titled separately as the "Addition property of equality", "Subtraction property of equality", "Multiplication property of equality", and "Division property of equality". In the past I've searched for, and really wanted to find, a single unifying term for them but never found any in common usage.

Example at hotmath.com.

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In answer to number one, no, you do not always have to do the same on each side of an equation/inequality.

I remember doing a problem some time ago with an "epsilon >0" or some such inequality, and it is helpful to be able to add or subtract any (positive) number in some cases. For example, assuming real numbers:

a > 1

b^2 + a = c

You could just as well say a + 8 > 1 (though it is not equivalent)

You can find that b^2 = c - a and deduce:

a + (b^2) > 1

a + (c - a) > 1

c > 1

Unfortunately I can't remember what application this was useful for - perhaps probability or distribution.

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I agree with Pedro, Algebra is the name, the Arabic word "Al-Jabr" was not translated to English and used almost as it is (Aljebra). Al-jabr in Arabic can be defined as the action of fixing something broken( putting things together) So if you add something to one side of equation only, "you broke it", therefore you need to fix it "AlJabr" by adding the same quantity to the other side.

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To do the same thing on the both sides of an equality is a special case of substitution. Substitution is valid in any mathematical relation and the balance rule is a consequence of substitution: if $a=b$, then substitute $a$ with $b$ to the right in $a+c=a+c$ and get $a+c=b+c$.

Substitution is more general and abstract, the balance principle is less general and more straight forward and pedagogical. But due to Kurt Gödel there are only two possibilities for a mathematician to make a conclusion: substitution or modus ponens:

"The class of provable formulas is defined to be the smallest class of formulas that contains the axioms and is closed under the relation ’immediate consequence’, i.e. formula c of a and b is defined as an immediate consequence in terms of modus ponens or substitution." Jean van Heijenoort, 1976, ”From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931”, p. 601, Harvard University Press.

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From a categorial perspective, this might be called the functoriality of functions.

In particular, observe the similarities:

  1. If $f : X \rightarrow Y$ is a function between sets and $x,x'$ are elements of $X$, then we get an implication like so: $$x = x' \rightarrow f(x) = f(x')$$

  2. If $F : \mathbf{C} \rightarrow \mathbf{D}$ is a functor between categories and $X,X'$ are objects of $\mathbf{C}$, then we get a function like so: $$\mathrm{Hom}(X,X') \rightarrow \mathrm{Hom}(FX,FX')$$

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