Sunday Morning Greek Blog

In the English language, words are rarely “inflected” like they are in other languages. The most we do with our regular verbs, for example, is to add an -s in the third person singular, -ed in past tense or past participles, and -ing in gerunds and present participles or present continuous. Occasionally we need to add helping verbs to indicate more nuanced uses, like a form of the “to be” verb for passive voice or a form of “have” for perfective forms of the verb.

In the classical languages like Greek and Latin, such inflections are typically built in to a singular word form. For example, here are the various first-person indicative mood forms of the Latin verb porto with all tense and voice combinations:

Tense	Active Voice	Passive Voice
Present:	porto (I carry)	portor (I am carried)
Future:	portabo (I will carry)	portabor (I will be carried)
Pluperfect:	portaveram (I had carried)	portatus eram (I had been carried)
Imperfect:	portabam (I was carrying)	portabar (I was being carried)
Perfect:	portavi (I have carried)	portatus sum (I have been carried)
Future Perfect:	portavero (I will have carried)	portatus ero (I will have been carried)

Source: Copilot request for first-person Latin verb forms.

Notice that in some cases, one word form in Latin requires up to four English words to translate depending on the ending. Greek is the same way, except inflecting verbs involves not just endings, but could also involve prefixes, infixes, and initial reduplications along with the loss of aspiration for aspirated consonants. The words are intended to be one singular form in most cases, and when we translate them, we treat the original word as a single lexemic unit.

We have a similar “inflection” of mathematical values or “terms” that do not have an extant operational sign (+ – × ÷) or that have such signs enclosed within or under a grouping symbol. These values or terms are (or should be) treated as a unit. Just as we don’t separate the root from the ending of a word in Latin and add the root to the end of the preceding word, neither do we (or should we) separate numbers that are “inflected,” so to speak, to represent a certain method of calculation that should be given priority over signed operations. The latter I am calling “implicit constructions,” because they imply a certain way of calculating based on syntax (position and orientation) that reflects a form of grouping and should be treated as a priority element of any expression.

One of the main functions of this kind of inflection is to indicate formulas for common measurements and values. The circumference of a circle is 2πr; the area of a circle is πr²; velocity is distance over time, v = d/t; etc. The juxtaposition of elements of the formula indicates that you’re working with what should be considered a unified value. This juxtaposition carries over into general mathematics as well. Juxtaposition is a form of grouping, but juxtaposition, like the inflection of the verb, is not a monolithic concept of simply being “side by side” with something else. Implicit constructions in mathematics use juxtaposition combined with orientation of the elements (including grouping symbols when necessary) to show the operational relationship of the individual elements.

At the very basic level, any real number, at least in the Indo-Arabic paradigm[1], is a simple juxtaposition of the powers of the base. A whole number (base 10) is, from right to left, a representation of how many 10ⁿ (n >= 0) in order right to left with implied addition of the place value (i.e., 10ⁿ) multiplied by the number holding the place. If you remember back to basic grade school math, that is how children are taught to understand our number system (ones’ place, tens’ place, hundreds’ place, etc.). In other words, juxtaposition is a foundational element of our number system and should not simply be discarded or replaced when it is used in other ways.

A fraction is a value that is formed by vertical or offset juxtaposition of one number to another with an intervening grouping symbol (vinculum or fraction bar; solidus or slash, respectively) that represents division or ratio but does not necessarily demand that such division be immediately carried out. I give examples of these in this implicit constructions table. Fractions should be considered as a single value first and foremost (because they are grouped) and NOT as a division problem in which the vinculum or solidus is considered equivalent to the obelus (÷) in function and implication. Rational fractions are necessary for precision, especially when the conversion to a decimal involves a nonterminating decimal and you’re working with very large numbers that would not be as precise as needed if we used a two-decimal approximation. For purposes of this paper, I’m working on the assumption that fractions should not be converted to decimal equivalent since this essay is theoretical.

A mixed number is the juxtaposition of a whole number to a fraction, and like a whole number, the side-by-side juxtaposition without any other symbol implies addition. Mixed numbers are considered unique, inseparable values as well when it comes to operations performed on them, but they often must be converted to an improper fraction to work with them more effectively.

Exponents also use juxtaposition to imply their operation. A power is superscripted and juxtaposed to the right of the base, and such superscription implies that the base should be multiplied the number of times indicated by the power. Of course, there are special rules when the power is a fraction, but those aren’t relevant to the discussion as the point is the implication of juxtaposition.

Since division is the opposite of multiplication, the question arises as to what is the opposite of a grouped fraction that implies multiplication? I would call this a “collection” because such a construction implies n sets of a quantity A. By way of example, 2(2 + 2) or 6(1 + 2) are collections if A = (2 + 2) or A = (1 + 2). The 2 (number of sets) is juxtaposed to (2 + 2) or (1 + 2) with the parentheses establishing the boundaries of quantity A.

Here is where many make a critical mistake in interpreting the collection: they fail to recognize the otherwise universal application of juxtaposition in real numbers, fractions, mixed numbers, and exponents that demands those forms be treated as single values. By suggesting one can just willy-nilly replace the juxtapositional relationship with a multiplication symbol and “ungroup” the collection, one violates the sacred bond that juxtaposition has in all other basic number forms. Just as fractions should be treated as a single value, so should collections.

By now you should see how this applies to the viral math expressions that some use to troll the Internet and pounce on the unsuspecting with their flawed view of Order of Operations or PEMDAS. An expression like 8 ÷ 2(2 + 2) should NOT require the undoing of a juxtaposed construction. The juxtaposition demands the “collection” 2(2 + 2) be treated as having a single value and should not be undone by an obelus that does not have grouping (or ungrouping) powers. Otherwise, an expression like 8 ÷ ¾ would be seen as 8 ÷ 3 ÷ 4 rather than inverting the fraction and multiplying (the latter being the official way students are taught to work the problem) if that same principle were applied. The expression 8 ÷ 2(2 + 2) = 8 ÷ 8 = 1, pure and simple. Yet some are so locked into a false concept of Order of Operations that fails to recognize the power of juxtaposition as a form of grouping that they can’t see the forest for the trees.

I have written elsewhere about the other issues that arise with the flawed understanding of juxtaposition. I believe this treatise provides a firmer theoretical basis for the concept of grouping and its proper place in the order of operations than the juvenile charts one finds in many textbooks.

Answering an Objection

One of the objections I often hear about juxtapositional grouping is that it violates the place of the exponent in such an order. That objection, however, is based on a flawed understanding of the role of juxtaposition. One example cited is:

8 ÷ 2(2 + 2)².

Opponents say my theory would interpret the expression as follows:

8 ÷ 2(4)² becomes 8 ÷ 8² (i.e., multiplication first, then exponent), which becomes 8 ÷ 64 or ⅛.

That is not accurate. If you understand that the exponent is also a form of juxtaposed grouping, then there is no issue with performing the exponent operation first and then performing the implicit multiplication, which complies with a standard view of the Order of Operations. Juxtaposed grouping does NOT supersede regular Order of Operations. The problem correctly interpreted then would be

8 ÷ 2(4)² becomes 8 ÷ 2(16) (exponent first, then multiplication), which becomes 8 ÷ 32 or ¼.

To date, no one has ever been able to explain why implicit multiplication is treated differently from other implicit constructions in all my discussions and debates.

The bottom line here is that the juxtaposition itself is grouping, not just the symbols used. A fraction is a combination of the vertical juxtaposition of the numerator with a fraction symbol (vinculum, solidus, slash) grouping the denominator below it. A collection is the horizontal position of a cofactor or coefficient with the other cofactor or coefficient grouped within the parentheses. The concept of dealing with what is in the parentheses first without factoring in juxtaposition of other elements ignores the reality of juxtaposition in every other type of implicit construction and represents an imperfect and immature application of the grouping principle.

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[1] Languages that use letters to represent numbers often do not have a letter to represent the value “0” and are not based solely on a base with powers. In Hebrew and Greek, for example, the respective alphabets represent 1–9, 10–90 (by 10), and 100–900 (by 100) and then repeat for the next three powers of ten using special symbols above the letters. You need 27 symbols/letters for these values instead of 10, and for numbers >=1,000, you need special symbols over (usually) the letters to indicate multiplication by 1,000 for each of the 27 symbols. If there is no value for one of the powers of 10, they simply do not use a letter to represent that.