The Ancient-Greek word, τὰ στοιχει̃α or, when transliterated ‘tà stoicheĩa,’ is a plural form of the 2nd-declension neuter verb, τὸ στοιχει̃ον genitive: του̃ στοιχείου or, when transliterated: ‘tò stoicheĩon,’ genitive: ‘toũ stoicheíou.’
The Ancient-Greek word, ‘tò stoicheĩon,’ can mean ‘an element in a set.’
Figure 1: The elements of this set are alpha, beta, gamma and delta.
The Ancient-Greek word, ‘tò stoicheĩon,’ is formed from the Ancient-Greek masculine noun, ὁ στοι̃χος genitive: του̃ στοίχου or, when transliterated, ‘ho stoĩchos,’ genitive: ‘toũ stoíchou,’ which means ‘steps,’ or ‘a flight of stairs;’ and the Ancient-Greek 2nd-declension neuter nominal suffix, ‘-eĩon,’ genitive: ‘-eíou’ which denotes ‘a means (of),’ ‘an instrument of;’ etc.
Figure 2: a ‘stoĩchos’ or ‘series of steps.’
The term, ‘stoĩchos,’ according to Wiktionary, may be traced back to the indo-european word:
*steigʰ
, which means:
‘climb.’
Hence, etymologically, the Ancient-Greek term, ‘stoicheĩa,’ can be said to mean: ‘the means of climbing up;’ ‘the means of stepping up;’ ‘the means of ascent;’ etc.
This is highly instructive, as, in truth, Elements is a book that is a Jacob’s ladder, of sorts, by which one can ascend, element by element, into the heavens of mathematical knowledge.
Figure 3: With The Elements of Euclid, we advance in our mathematical knowledge element by element. Each element is, conceptually, like a rung, heaving us upwards to Mathematical prowess; to an implicit knowledge of Euclidean Geometry.
Conventional Arithmetic possesses rules for the order of operations. Which operations ought we to evaluate first? In what order ought we to evaluate operations? This is the topic that this chapter wishes to address. ‘Precedence,’ is also sometimes referred to as ‘the order of operations.’
Body:
The Etymological Definition of ‘Precedence:’
Our English noun, ‘precedence,’ is derived from the Latin substantive participle, ‘praecēdentia.’[3] ‘Praecēdentia,’ in Latin, means ‘the abstract concept of which things go before [other things].’
Figure 2: The English arithmetical term, ‘precendence,’ is derived from a compund of the Latin verb ‘cēdēre,’ which means ‘to go.’ Go, or Golang, as a language, is—syntactically—very like the C Programming Language.
Within the context of Arithmetic, ‘precedence,’ etymologically, means ‘the science of determining which operations go before [other operations];’ ‘the science of determining which operations should be evaluated before [other operations].
The Acronym, ‘P.E.M.D.A.S:’
The acronym, ‘P.E.M.D.A.S.,’ stands for:
Parenthesis;
Exponentiation;
Multiplication and Division;
Addition and Subtraction.
The Acronym, ‘P.E.M.D.A.S.,’ can be easily remembered with the Mnemonic phrase:
As we can observe from the above ordered list, some operations share the same level of precedence. For example, the operation, multiplication, and the operation, division, have the same level of precedence. Multiplication and Division share the third level of precedence, in the above list. When we are confronted with an expression or an equation that contains operations at the same level of precedence, seeing that in Anglophone countries, we read from left to right, then we evaluate operations that possess the same level of precedence from left to right. Hence, when two or more operations—within an equation or an expression—share the same level of precedence, then we evaluate them from left to right. Concerning operations at the same level of precedence, we evaluate from beginning at the leftmost operation, and work our way rightwards.
An Example of Precedence:
In the expression:
2 ÷ 1 + 3 × 42 – 5 + ( 3 – 2 )
, we first evaluate the operation in parenthesis, i.e.:
( 3 – 2 )
. When we evaluate:
( 3 – 2 )
, then we obtain the difference:
1
.
This renders the original expression as:
2 ÷ 1 + 3 × 42 – 5 + ( 1 )
or as:
2 ÷ 1 + 3 × 42 – 5 + 1
.
Second, we evaluate the exponentiation operation i.e.:
42
. When we evaluate:
42
, then this obtains for us the power:
16
. This renders our original expression as:
2 ÷ 1 + 3 × 16 – 5 + 1
.
The operations, Multiplication and Division, share the same level of precedence. However, given that the division operation is further to the left, on the page, than the multiplication operation, then we evaluate the division operation before we evaluate the multiplication operation.
Given that the division operation:
2 ÷ 1
is further to the left, on our page than the multiplication operation:
3 × 16
, then we evaluate:
2 ÷ 1
before we evaluate:
3 × 16
.
When we evaluate:
<!–
2\div1
–>
2 ÷ 1
, then we obtain the quotient:
2
. This renders our original expression as:
2 + 3 × 16 – 5 + 1
. Then we proceed to evaluate:
3 × 16
, and this obtains for us the product:
48
. This renders our original expression as:
2 + 48 – 5 + 1
.
The operations; addition, and subtraction; share the same level of precedence. In the above ordered list, they are at the 4th level of precedence. We evaluate these operations as we should find them, beginning at the leftmost, and working our way rightward. Hence, we evaluate:
2 + 48
first. This obtains for us the sum:
50
. This renders our original expression as:
50 – 5 + 1
. We then proceed to evaluate the operation:
50 – 5
, which obtains for us the difference:
45
. This renders our original expression as:
45 + 1
. We then proceed to evaluate the expression:
45 + 1
. This obtains for us the sum:
46
.
This renders our original expression as:
46
. We have thus simplified the expression:
2 ÷ 1 + 3 × 42 – 5 + ( 3 – 2 )
to:
46
. We have observed mathematical precedence οr the order of operations in our simplification of the expression:
<!–
2 \div 1 + 3 \times 42 – 5 + \left ( 3 – 2 )
–>
2 ÷ 1 + 3 × 42 – 5 + ( 3 – 2 )
to:
46
.
Conclusion:
In this chapter, we have endeavoured to gain for ourselves an implicit understanding of precedence as it pertains to basic or conventional arithmetic. Boolean arithmetic, an arithmetic of logic employed in Computer Science, also possesses precedence or an order of operations, which we shall examine in a subsequent chapter. In the next chapter, we shall examine precedence or the order of operations as it specifically applies to the C programming language.
Figure 1: I drew this HTML logo in Scripted SVG. My development skills, especially my ability to develop web images through code, is really beginning to rise to a professional standard.
Figure 2: I mocked up this old website, phouka.com for an e-book that I am putting the finishing touches to, and hope to release upon Amazon, shortly. This website, developed in 2005, employs a pre-html-5 table layout which is now deprecated. Today, the downloading of fonts by a web-accessor is no longer required thanks to @fontface .
Figure 3: I drew this Swift Logo – the Programming Language employed by Apple – in Vector software such as Vector Magic and Inkscape. Apps can be mocked up in Inkscape so as to give the customer a sense of what his/her app’s user experience/ graphical user interface might look like, prior to the app’s development commencing in earnest.
If anyone should require mockups of apps or websites prior to taking this wireframe to a professional website-developer/app-programmer, let me know. Send me a direct message, or something. The advantage of doing this in SVG, is that one can then employ the SVG code and the CSS code generated by Inkscape in the website/app itself.
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The Famous Syllogism[1] in Greek, Latin and English:
Figure 1: I drew this pencil portrait of George Boole (1815-1864) with Pencils and Photoshop. The Mathematical Analysis of Logic was published in 1847, and An Investigation into the Laws of Thought was published in 1854.
Introduction:
Quite early on, in his Mathematical Analysis of Logic, George Boole–whence in programming and computer science we derive the datatype name, ‘Boolean’– introduces this famous syllogism to us, his readers.
Body:
In Ancient Greek:
ὁ Σωκράτης ἐστιν ἄνθρωπος.
πάντης ἄνθρωποι ἐστι θνητοί.
οὖν ὁ Σωκράτης ἐστι θνητός.
When Transliterated:
ho Sōcrátēs estin ánthrōpos.
pántēs ánthrōpoi esti thnētoí.
oũn ho Sōkrátēs esti thnētos.
In Latin:
Sōcratēs est homō.
Omnēs hominēs sunt mortālēs.
Ergō, Sōcratēs est mortālis.
In English:
Socrates is a man.
All men are mortal.
Therefore, Socrates is mortal.
Conclusion:
The Ancient-Greek term, ὁ λόγος or, when transliterated, ‘ho lógos,’[1] means–within the context of logic– ‘statement,’ or ‘argument.’
The Latin 1st-and-2nd-declension adjectival suffix, ‘-ica, -icus, -icum’ means ‘of,’ ‘about,’ ‘concerning,’ ‘pertaining to,’ etc.
Hence, etymologically, ‘logic’ is ‘the study of the truth or falsehood of statements and arguments.’
Conventional arithmetic or Conventional Algebra has quantity for its subject. George Boole developed an algebra, or an arithmetic that had logic as its subject.
Indeed, in his book, The Laws of Thought he terms this ‘arithmetic’ or ‘algebra’ of his ‘a calculus of logic’ by which he meant ‘a system whereby the truth or falsehood of statements/arguments could be analysed.’
Figure 1: I drew this pencil portrait of Socrates (469/470-399 BC) with Pencils. Vēre Sōcratēs est enim mortālis. Truly Socrates is mortal indeed.
Glossary:
calculus (ˈkælkjʊləs) nounplural-luses
a branch of mathematics, developed independently by Newton and Leibniz. Both differential calculus and integral calculus are concerned with the effect on a function of an infinitesimal change in the independent variable as it tends to zero.
any mathematical system of calculation involving the use of symbols
logic an uninterputed formal system. Compare formal language (sense 2)
(plural-li (ˈkælkjʊˌlaɪ) ) pathology a stonelike concretion of minerals and salts found in ducts or hollow organs of the body[C17 from Latin: pebble, stone used in reckoning, from calx small stone, counter]
calcular(ˈkælkjʊlə) adjective relating to calculus
calculous (ˈkælkjʊləs) or calculary(ˈkælkjʊlərɪ) of or suffering from a calculus. Obsolete form: calculose
calculus of variations a branch of calculus concerned with maxima and minima of definite integrals.[1]
It is my contention that the knowledge of Latin and Greek make STEM[1] easier to learn. A huge number of STEM terms are derived from Greek and Latin.
Fig 1: I drew this portrait of George Boole with pencils. George Boole was self-taught and fluent in Latin, Greek and Hebrew by the time that he was 12.
Vel Symbol:
Fig 1: This is the Vel symbol. You may view the Vector at my CodePen Account.
In Formal Logic this symbol represents ‘disjunction.’ The equivalent in Boolean Algebra is ‘Inclusive Or.’ ‘vel’ is Latin for ‘or.’ One sees this quite a bit in liturgical rubrics[2].
The Wedge Symbol
Fig 1: This is the Wedge symbol. You may view the Vector at my CodePen Account.
In Formal Logic this symbol represents “conjunction.” The equivalent in Boolean Algebra is “And.” In Latin, ‘ac’ or ‘atque’ is ‘and.’ Sometimes this symbol is called this. One sees this quite a bit in ecclesiastical Latin.
‘I announce to ye a great joy: we have a Pope!, the most eminent and most revered [forename] lord of the most holy Roman Church, Cardinal [surname], who hath placed upon himself the name [regnal name].’
In the offertory the priest prays:
‘…prō fidēlibus christiānīs vīvīs atque dēfūnctīs…’
‘…for all faithful Christians living and dead…’
In The Young Pope (2016), a Cardinal, disfavoured by Pius XIII/Jude Law, prays this in the frozen wilderness of Alaska, to whence he was banished.
Maths and Comedy 