The Logical Permutations of an And Gate Realised in Javascript

 

Figure 1: The JavaScript Logo. I drew this with pencils.

Figure 2:  The schematic symbol for an And Gate.

The logical permutations of an And Gate are as follows:

0 ∧ 0 = 0

0 ∧ 1 = 0

1 ∧ 0 = 0

1 ∧ 1 = 0

The above can be read, in English, as follows:

Zero conjunction zero equals zero.

Zero conjunction one equals zero.

One conjunction zero equals zero.

One conjunction one equals one.

Alternatively:

Zero AND zero equals zero.

Zero AND one equals zero.

One AND zero equals zero.

One AND one equals one.

We can realise the above Boolean Logic in Digital circuitry, as follows:

Figure 3:  An AND Gate.  Notice how, this time, the switches, x and y are in series.  In the Inclusive-Or gate, the switches, x and y were in parallel.  The above-depicted And Gate represents the Boolean Equation, 0 ∧ 0 = 0

Figure 4:  This And gate represents the Boolean Equation, 0 ∧ 1 = 0

Figure 5:  This And gate represents the Boolean Equation, 1 ∧ 0 = 0

 

Figure 6:  This And gate represents the Boolean Equation, 1 ∧ 1= 1

It is possible to translate Boolean Equations into Conventional-Arithmetic Equations, and to obtain the same logical result:

x ∧ y = x ( y)

Because:

x conjunction y

in Boolean Arithmetic equates to:

x multiplied by y 

in conventional arithmetic, the logical output of an And Gate is termed:

the logical product

.

0 ∧ 0 = 0 (0)  = 0

0 ∧ 1 = 0 (1)  = 0

1 ∧ 0 = 1 (0)  = 0

1 ∧ 1 = 1(1)  = 1

It is possible to use the above logical translations so as to create buttons in JavaScript that calculate the logical permutations of an And gate.

Figure 7:  The HTML-5 file that we need to program a web-app that calculates the truth-table of an And gate.

Figure 8:    In the above-depicted JavaScript file, we declare four functions.  Each function describes a logical permutation of the And Gate.

In the web-app that we have just programmed,  we have created 4 buttons.  Each button calculates a logical permutation of the And gate and displays the result in a dialog box.

Figure 9:  A Screenshot of the web-app that we have just programmed in JavaScript.

You can test the above-described web-app for yourself by clicking on the following link:

 And Gate Web-application

 

The Mathematical Distinction that Exists between Precision and Accuracy.

Click the below link so as to view a Microsoft-Word version of this blog post.

the_mathematical_distinction_that_exists_between_precision_and_accuracy.docx

Click the below link so as to view a pdf version of this blog post.

the_mathematical_distinction_that_exists_between_precision_and_accuracy.pdf

 

Introduction:

In everyday common parlance, the terms ‘precision’ and ‘accuracy’ are synonyms. However, this is not so in Mathematical jargon. In mathematical jargon, there exists an important difference in meaning between the terms ‘precision’ and ‘accuracy.’ It is the purpose of this chapter to explicate this difference in meaning.

Body:

If one should derive:

π

to eleven decimal places by dividing 22 by 7:

22/7

, then the result of this calculation:

3.14285714286

will be more precise than:

π

derived properly – i.e. through differentiation – to eight decimal places:

3.14159265

, but the latter answer will be more accurate than the former.

In mathematics, the term, ‘precision,’ denotes how many decimal places the result of a calculation can run to[1], whereas the term, ‘accuracy,’ denotes how close to the correct answer the result of a calculation is.

In mathematics, the term, ‘precision’ denotes how many significant digits that a mathematician should use to render a quantity, whereas the term ‘accuracy’ denotes how close to the correct quantity the quantity rendered in digits is.

An Etymological Definition:

The English term, ‘precision’ comes from the Latin 3rd-declension feminine noun, ‘praecīsiō, praecīsiōnis,’ which means a ‘cutting off[2] .’ After how many significant digits ought we to cut the answer off?

The verb, ‘caedere,’ can also mean ‘to kill[3] .’ After how many significant digits do we kill off the trail?

After our dividing 22 by 7:

22/7

, we cut the answer off after 11 significant digits following the decimal point:

3.14285714286

.
We use a total of 12 significant digits in our rendering of:

π

.

After arriving at:

π

, properly – i.e. through differentiation – we cut the answer off after 8 significant digits following the decimal point:

3.14159265

.

We use a total of 9 significant digits in our rendering of:

π

.

In the first case, we employ 12 significant digits in our rendering of:

π

, and, in the latter case, we employ 9 significant digits in our rendering of:

π

. Therefore, again, to restate, the former rendering of:

π

is more precise than the latter, however, the latter rendering of:

π

is more accurate than the former.

Precision in Python:

It is possible to modify the precision of π, in Python, by applying formatting[4]

Figure 1:  We have, here, created a program that prints out π with varying degrees of precision; from the most precise: with 6 significant figures trailing the decimal point; to the least precise: with 0 significant figures trailing the decimal point.

Figure 2:  What the previous Python script outputs.

Epilogue: What is π?

Figure 3:  The iconic symbol.  π is the sixteenth letter of the Ancient-Greek Alphabet.  This is its minuscule form.  It is conjectured that the Ancient-Greek letter, π, was employed so as to represent the length of a circle’s circumference, as the letter, ‘pi’ can be taken as a contraction of the term, ‘perimeter.’

π comes into play with the radians of Trigonometry, a crucial component of Computing Mathematics, so we may as well digress a little and ask ourselves what π actually is.

 

π symbolises and denotes the ratio that exists between the length of a circle’s diameter and the length of a circle’s circumference.

 

Figure 4:  π is equal to the ratio that exists between the length of a circle’s diameter, and the length of a circle’s circumference.

The ratio:

Diameter : Circumference

can be expressed as the ratio:

1 : 3.14159…

Glossary:

accuracy

• noun.

(plural. accuracies)

     [mass noun] the quality or state of being correct or precise:  we have confidence in the accuracy of the statistics.

 

• TECHNICAL the degree to which the result of a measurement, calculation, or specification conforms to the correct value or standard: the accuracy of radiocarbon dating | [count noun] accuracies of 50-70 per cent. Compare with PRECISION. [1]

<ETYMOLOGY>  From the Latin 3^rd-declension Latin noun, ‘accurātiō, accurātiōnis,’ which means ‘accuracy,’ ‘carefulness.’  From the Latin preposition, ‘ad,’ which means ‘towards;’ and the Latin 1^st-declension feminine noun, ‘cūra, cūrae,’ which means ‘care;’ and the Latin 3^rd-declension nominal suffix, ‘-ātiō’ which denotes a state of being.  The etymological sense of the English term, ‘accuracy,’ therefore, is ‘the state of being oriented towards carefulness.’

 

 

accurate

• adjective.

1. (especially of information, measurements, or predictions) correct in all details; exact: accurate information about the illness is essential.
   • (of an instrument or method) capable of giving accurate information: an accurate thermometer.
   • providing a faithful representation of someone or something: the portrait is an accurate likeness of Mozart.
2. (with reference to a weapon, missile, or shot) capable of or successful in reachingthe intended target: reliable, accurate rifles | a player who can deliver long accurate passes to the wingers.

<DERIVATIVES> late 16^th century: from Latin accuratus ‘done with care’, past participle of accurare, from ad- ‘towards’ + cura ‘care’.[2]

 

<ETYMOLOGY>  From the Latin 1^st-conjugation verb, ‘accūrō, accūrāre, accūrāvī, accūrātum,’ or ‘adcūrō, adcūrāre, adcūrāvī, adcūrātum,’ which means ‘to give close attention to,’ ‘to be careful.’[3]

From the Latin preposition, ‘ad,’ which means ‘towards,’ and the Latin 1^st-declension feminine noun, ‘cūra, cūrae,’ which means ‘care.’

precise

• adjective.

marked by exactness and accuracy of expression or detail: precise directions | I want as precise a time of death as I can get.

• (of a person) exact, accurate, and careful about details: the director was precise with his camera positions.
• [attributive] used to emphasize that one is referring to an exact and particular thing: at that precise moment the car stopped.

<DERIVATIVES> preciseness noun.

<ORIGIN> late Middle English: from Old French precis, from Latin praecis- ‘cut short’, from the verb praecidere, from prae ‘in advance’ + caedere ‘to cut’[4].

 

<ETYMOLOGY> From the Latin 1^st-and-2^nd-declension adjective, ‘praecīsa, praecīsus, praecīsum,’ which means ‘broken off,’ ‘abrupt.’  From the Latin verb, ‘praecīdō, praecīdere, praecīdī, praecīsus,’ which means ‘to cut off in front,’ ‘cut off.’[5]

From the Latin preposition, ‘prae,’ which means ‘before,’ ‘beforehand;’ and the Latin 3^rd-conjugation verb, ‘caedō, caedere, cecīdī, caesum,’ which means ‘to cut.’  The etymological sense of ‘precise’ – like ‘concise’ – is ‘cut off.’

 

 

precisely

in exact terms; without vagueness: the guidelines are precisely defined.

• exactly (used to emphasize the complete accuracy or truth of a statement): at 2.00 precisely, the phone rang | kids will love it precisely because it will irritate their parents.
• used as a reply to confirm or agree with a previous statements: ‘You mean it was a conspiracy?’ ‘Precisely.’ .[6]

 

precisian

chiefly ARCHAIC a person who is rigidly precise or punctilious, especially as regards religious rules.

<DERIVATIVES> precisianism noun.[7]

precision

• [mass noun]

the quality, condition, or fact of being exact and accurate: the deal was planned and executed with military precision.

• [as modifier] marked by or adapted for accuracy and exactness: a precision instrument.
• TECHNICAL refinement in a measurement, calculation, or specification, especially as represented by the number of digits given: a technique which examines and identifies each character with the highest level of precision | [count noun] a precision of six decimal figures.

Compare with ACCURACY.

<ORIGIN> mid 18^th century: from French precision or Latin praecīsiō(n-), from praecīdere ‘cut off’ (see PRECISE).[8]

<ETYMOLOGY>  From the Latn 3^rd-declension feminine noun, ‘praecīsĭo, praecīsĭōnis,’ which means ‘a cutting off,’ ‘the piece cut off.’  In rhetoric, ‘praecīsĭo,’ means ‘a breaking off abruptly.’  In Late Latin, ‘praecīsĭo,’ can mean ‘an overreaching.’ [9]

 

radian

[GEOMETRY] a unit of measurement of angles equal to about 57.3°, equivalent to the angle subtended at the centre of a circle by an arc equal in length to the radius.[10]

<ETYMOLOGY>  From the Scientific Latin 1^st-and-2^nd-declension adjective, ‘radiāna, radiānus, radiānum’ which means ‘of the radius,’ ‘concerning the radius,’ ‘denoting the radius.’  From the Latin 2^nd-declension masculine noun, ‘radius, radiī’ which means ‘a geometer’s rod[11] [for measuring the distance between the centre of a circle and the circumference],’ and the Latin 1^st-and-2^nd-declension adjectival suffix,      ‘-iāna, -iānus, -iānum,’ which means ‘of,’ ‘concerning,’ ‘denoting.’  The etymological sense of the English noun ‘radian’ is ‘that unit of measurement of angles that concerns radiuses.’

 

 

[1]  Oxford University Press.  Oxford Dictionary of English (Electronic Edition). Oxford. 2010.  Loc 4388.

 

[2]  Oxford University Press.  Oxford Dictionary of English (Electronic Edition). Oxford. 2010.  Loc 4388.

[3] Perlingua Language Tools.  www.perlingua.com   Version 2.1 (Kindle Edition.)  2013.  Latin English Lexicon.  Thomas Mc Carthy.  Loc. 1075.

[4] Oxford University Press.  Oxford Dictionary of English (Electronic Edition). Oxford. 2010.  Loc. 554162.

[5] Perlingua Language Tools.  www.perlingua.com   Version 2.1 (Kindle Edition.)  2013.  Latin English Lexicon.  Thomas Mc Carthy.  Loc. 76264.

 

[6] ibid.  Loc. 554172

[7] ibid.  Loc. 554179

[8]  ibid.  Loc. 554191

[9]  Perlingua Language Tools.  www.perlingua.com   Version 2.1 (Kindle Edition.)  2013.  Latin English Lexicon.  Thomas Mc Carthy.  Loc. 76264.

[10]  Oxford University Press.  Oxford Dictionary of English (Electronic Edition). Oxford. 2010.  Loc. 578311.

[11]  Perlingua Language Tools.  www.perlingua.com   Version 2.1 (Kindle Edition.)  2013.  Latin English Lexicon.  Thomas Mc Carthy.  Loc. 87154.

 

---

[1]

See the Oxford Dictionary of English’s TECHNICAL definition of ‘precision’ in the glossary below.

[2]

From the Latin preposition ‘prae,’ which means ‘before,’ and the Latin 3rd-conjugation verb, ‘caedō, caedere, cecīdī, caesum,’ which means ‘to cut.’

[3]

For instance, a ‘suicide’ is a man or woman who kills himself/herself. From the Latin 1st-and-2nd-declension possessive adjective, ‘sua, suus, suum,’ which means ‘his, her, its;’ and the Latin 3rd-conjugation verb, ‘caedō, caedere, cecīdī, caesum,’ which means ‘to kill.’ The etymological sense of ‘suicide,’ therefore is a man or woman who kills himself/herself.

[4]

See the blogpost: Formatting Numbers in Python.

Must Do is a Great Master.

The title of this piece comes from a Monaghan-ism that has a parallel in non-localised English.

As part of our Networking-and-Systems Apprenticeship, we have to draw Network Topologies.  I am watching a Microsoft MVA video series on how to do this:

Video 1:  A Microsoft Video Playlist on Networking Fundamentals.

The chaps at Microsoft recommend some of their proprietary software for the task: Microsoft Visio.

I looked up Microsoft Visio on their online store, and, to my horror, it cost in excess of €700.  It amazes me that triple-A videogames can employ hundreds of developers that can work years of man-hours between them, and, yet, they can retail their software at €60-€70.  How come non-gaming software can sometimes retail at the €1000+ mark?

So, I am not spending money – that I do not have – on this.  Eventually, I would like to set up a development company using the Microsoft Bizspark program, and, I think, Microsoft will give me all of their software for free, upon the proviso that I develop for their platform.

So I looked for a coding solution.

I took a low-quality raster screenshot of the server icon featured in their video:

Figure 1: A low quality raster-screenshot of a server icon.

I then began to manually describe this Server Icon in SVG[1] code:

Figure 2:  A manually programmed svg file that I produced.

Below is a capture of the high-quality vector image that the code produced:

Figure 3:  A capture of what the simple svg file, above, produces.

Sometimes, lacking the proper resources necessary to – ordinarily – complete a task, forces one to seek more creative solutions.

As in the adage:

Necessity is the Mother of Invention

Indeed, it is necessity – and not plenty – that it the mother of invention and innovation.

---

[1] Scalable Vector Graphics.

The Epistemology of Algorithms.

Below is a Word-Document Version of this blog post:

final_epistomology_recovered_algorithms

Below is a pdf Version of this blog post:

final_epistomology_recovered_algorithms

---

Figure 1:  The Ancient-Greek word ‘EPISTEME’ which means ‘knowledge’ wrought into the Blacksmith’s monument outside Monaghan Courthouse. Monaghan is such a cultured little burg!

In science and philosophy, Epistemology is the study of knowledge, and what constitutes knowledge.

 

In Computer programming, we have to think epistemologically about knowledge; what constitutes knowledge; and what forms it may take.

 

In Computer programming, knowledge is deemed to have two forms:

1. Declarative,
2. Imperative.

The term, ‘imperative,’ denotes a command. ‘Imperative Knowledge’ is knowledge that instructs one on how to do something by giving him/her a set of commands.

 

If I told you that porridge consisted of oats and heated water, then this would be ‘declarative knowledge.’

 

If, instead, I gave you a set of instructions on how to make porridge such as:

RECIPE:

 

INGREDIENTS:

 

• Oats
• Water

 

METHOD:

 

Measure out 80 grams of oats. Measure out 160 millilitres of water. Combine the oats and water in a pot. Heat the pot over a hob until it reaches boiling point, stirring all the while. Keep the oats-and-water mixture at boiling point for three minutes. Take the porridge off the hob. Serve.

 

then the above would be an example of imperative knowledge.

 

Above, we see the two types of knowledge in action: declarative and imperative. The declarative form of knowledge tells you what porridge is. The imperative form of knowledge consists of a series of instructions that enables you to make porridge.

 

As you can see, the verbs that I use in telling you how to make porridge are, grammatically, in the imperative mood:

 

‘measure…’

‘combine…’

‘heat…’

‘keep…’

‘take…’

‘serve…’

 

In grammar, the imperative mood denotes a verb in its command form.

 

A series of commands that enables one to prepare a foodstuff is termed a ‘recipe.’  Very often, Computer Scientists will refer to algorithms as ‘recipes.’

In cookery, a recipe is a series of commands that allows one to prepare a foodstuff.

In computer science, an algorithm – or recipe – is a series of commands that allows one to solve a computational problem.

In computer science, an algorithm – or recipe – is a series of commands that allows one to accomplish a task computationally.

 

 

Glossary:

declarative

• adjective.

of the nature of or making a declaration: declarative statements.

[GRAMMAR] (of a sentence or phrase) taking the form of a simple statement.

[COMPUTING] denoting high-level programming languages which can be used to solve problems without requiring the programmer to specify an exact procedure to be followed.

noun.

a statement in the form of a declaration.

[GRAMMAR] a declarative sentence or phrase.

<DERIVATIVES> declaratively adverb. [1]

<ETYMOLOGY> from the Latin 1st-and-2nd-declension adjective, ‘dēclārātīva, dēclārātīvus, dēclārātīvum,’ which means ‘pertaining to the making quite clear.’ From the Latin 1st-conjugation verb, ‘dēclārō, dēclārāre, dēclārāvī, dēclārātum,’ which means ‘to explain,’ ‘to make quite clear,’ and the Latin 1st-and-2nd-declension adjectival suffix ‘-īva,    -īvus, -īvum,’ which means ‘of,’ ‘concerning,’ ‘pertaining to.’ From the Latin prefix ‘dē-’ which expresses intensive force, and the Latin 1st-conjugation verb, ‘clārō, clārāre, clārāvī, clārātum,’ which means ‘to brighten,’ ‘to illuminate,’ ‘to clarify.’

epistemology

[mass noun] [PHILOSOPHY] the theory of knowledge, especially with regard to its methods, validity, and scope, and the distinction between justified belief and opinion.

< DERIVATIVES> epistemological adjective. epistemologically adverb. epistemologist noun.

         < ORIGIN> mid 19th century: from Greek episteme ‘knowledge’, from epistathai ‘know, know how to do’.[2]

<ETYMOLOGY> From the Ancient-Greek Feminine noun, ‘hē épistḗmē,’ which means ‘knowledge,’ and the Ancient-Greek Masculine noun, ‘ho lógos,’ which denotes a ‘study.’ Therefore, the English term, ‘epistemology’ can be said, etymologically, to mean ‘the study of knowledge. The Ancient-Greek Feminine noun, ‘hē épistḗmē,’ which means ‘knowledge’ can be broken down, a little further, into the Ancient-Greek preposition, ‘epí,’ which means ‘above,’ or ‘over,’ and the Ancient-Greek verb, ‘hístēmi,’ which means ‘to stand.’ Hence, ‘hē épistḗmē,’ at root, means ‘above-standing,’ ‘over-standing.’ In English, the term ‘knowledge’ is more-or-less synonymous with the term, ‘understanding.’ The Ancient-Greeks did not “understand:” instead they were inclined to “above-stand;” they were inclined to “over-stand.” Etymologically, therefore, the English term, ‘epistemology’ can be said to mean ‘the study of over-standing,’ ‘the study of above-standing.’

 

 

 

 

imperative[3]

• adjective.

1. of vital importance; crucial: immediate action was imperative | [with clause] it is imperative that standards are maintained.
2. giving an authoritative command; peremptory: the bell pealed again, a final imperative call.
   • [GRAMMAR] denoting the mood of a verb that expresses a command or exhortation, as in come here!

• noun.
  1. an essential or urgent thing: free movement of labour was an economic imperative.
     • a factor or influence making something necessary: the biological imperatives which guide male and female behaviour.
  2. [GRAMMAR] a verb or phrase in the imperative mood.
     • (the imperative) the imperative mood.

<DERIVATIVES> imperatival adjective. imperatively adverb. imperativeness noun.

<ORIGIN> late Middle English (as a grammatical term): from Late Latin imperativus (literally ‘specially ordered’, translating Greek prostatikē enklisis ‘imperative mood’), from imperare ‘to command’, from in- ‘towards’ + parare ‘make ready’.[4]

<ETYMOLOGY> from the Latin 1st-and-2nd-declension adjective, ‘impĕrātīva, impĕrātīvus, impĕrātīvum,’ which means ‘pertaining to the command;’ ‘of the command.’ From the Latin 1st-conjugation verb, ‘imperō, imperāre, imperāvī, imperātum,’ which means ‘to command,’ ‘to order,’ and the Latin 1st-and-2nd-declension adjectival suffix ‘-īva, -īvus,              -īvum,’ which means ‘of,’ ‘concerning,’ ‘pertaining to.’ From the Latin prefix ‘in-’ which expresses the concept of ‘unto,’ ‘toward,’ and the Latin 1st-conjugation verb, ‘parō, parāre, parāvī, parātum,’ which means ‘to make ready,’ ‘to prepare.’ The etymological sense, therefore, of the English adjective, ‘imperative’ is: ‘concerning the command;’ ‘pertaining to the command;’ ‘of the command;’ ‘concerning the order;’ ‘pertaining to the order;’ ‘of the order;’ ‘concerning the making ready of;’ ‘pertaining to the making ready of;’ ‘of the making ready of;’ etc.

 

 

 

 

[1] Oxford University Press. Oxford Dictionary of English (Electronic Edition). Oxford. 2010. Loc178909.

[2] Oxford University Press. Oxford Dictionary of English (Electronic Edition). Oxford. 2010. Loc 234206

[3] ibid. Loc 345797

 

Hexadecimal

Figure 1:  Today’s Blog-post is brought to you by the number, 16.  I drew the above image in Microsoft Paint.

 

Figure 2:  On today’s show, we are going to teach you how to Count – geddit?  because the above is a picture of The Count from Sesame Street? – in sixteens.

The term, ‘hexadecimal¹,’ refers to a base-sixteen number system. A hexadecimal number system uses sixteen distinct symbols so as to represent numerical quantities. These 16 symbols are:

0_sixteen = 0 _ten
1_sixteen = 1 _ten
2_sixteen = 2 _ten
3_sixteen = 3 _ten
4_sixteen = 4 _ten
5_sixteen = 5 _ten
6_sixteen = 6 _ten
7_sixteen = 7 _ten
8_sixteen = 8 _ten
9_sixteen = 9 _ten
A_sixteen = 10 _ten
B_sixteen = 11 _ten
C_sixteen = 12 _ten
D_sixteen = 13 _ten
E_sixteen = 14 _ten
F_sixteen = 15 _ten

Let us now, for the sake of comprehension, equate the denary² whole numbers, 0_ten to 15_ten to their equivalents in Hexadecimal:

0 _ten = 0_sixteen
1 _ten = 1_sixteen
2 _ten = 2_sixteen
3 _ten = 3_sixteen
4 _ten = 4_sixteen
5 _ten = 5_sixteen
6 _ten = 6_sixteen
7 _ten = 7_sixteen
8 _ten = 8_sixteen
9 _ten = 9_sixteen
10 _ten = A_sixteen
11 _ten = B_sixteen
12 _ten = C_sixteen
13 _ten = D_sixteen
14 _ten = E_sixteen
15 _ten = F_sixteen

Sixteen is equal to two to the power of four:

16 _ten =2^4 _ten

In Computer science, a byte is a combination of 8 binary digits, or bits. In Computer Science, we humorously term a combination of 4 binary digits – or half a byte – a ‘nibble.’

Every possible nibble, or permutation of four binary digits, can be represented by a single hexadecimal symbol/number.

0000 _two = 0 _sixteen
0001 _two = 1 _sixteen
0010 _two = 2 _sixteen
0011 _two = 3 _sixteen
0100 _two = 4 _sixteen
0101 _two = 5 _sixteen
0110 _two = 6 _sixteen
0111 _two = 7 _sixteen
1000 _two = 8 _sixteen
1001 _two = 9 _sixteen
1010 _two = A _sixteen
1011 _two = B _sixteen
1100 _two = C _sixteen
1101 _two = D _sixteen
1110 _two = E _sixteen
1111 _two = F _sixteen

Let us now, for sake of comprehension, express each single-digit hexadecimal number, and equate it to its equivalent nibble:

0 _sixteen = 0000 _two
1 _sixteen = 0001 _two
2 _sixteen = 0010 _two
3 _sixteen = 0011 _two
4 _sixteen = 0100 _two
5 _sixteen = 0101 _two
6 _sixteen = 0110 _two
7 _sixteen = 0111 _two
8 _sixteen = 1000 _two
9 _sixteen = 1001 _two
A _sixteen = 1010 _two
B _sixteen = 1011 _two
C _sixteen = 1100 _two
D _sixteen = 1101 _two
E _sixteen = 1110 _two
F _sixteen = 1111 _two

A Byte, as alluded to above, is two nibbles, we can, therefore, express each byte as a combination of two hexadecimal digits.

There are 256 possible bytes, and 256 possible permutations of two-digit hexadecimal numbers.

0000-0000_two = 00_sixteen
0000-0001_two = 01_sixteen
0000-0010_two = 02_sixteen
0000-0011_two = 03_sixteen
0000-0100_two = 04_sixteen
0000-0101_two = 05_sixteen
0000-0110_two = 06_sixteen
0000-0111_two = 07_sixteen
0000-1000_two = 08_sixteen
0000-1001_two = 09_sixteen
0000-1010_two = 0A_sixteen
0000-1011_two = 0B_sixteen
0000-1100_two = 0C_sixteen
0000-1101_two = 0D_sixteen
0000-1110_two = 0E_sixteen
0000-1111_two = 0F_sixteen

0001-0000_two = 10_sixteen
0001-0001_two = 11_sixteen
0001-0010_two = 12_sixteen
0001-0011_two = 13_sixteen
0001-0100_two = 14_sixteen
0001-0101_two = 15_sixteen
0001-0110_two = 16_sixteen
0001-0111_two = 17_sixteen
0001-1000_two = 18_sixteen
0001-1001_two = 19_sixteen
0001-1010_two = 1A_sixteen
0001-1011_two = 1B_sixteen
0001-1100_two = 1C_sixteen
0001-1101_two = 1D_sixteen
0001-1110_two = 1E_sixteen
0001-1111_two = 1F_sixteen

0010-0000_two = 20_sixteen
0010-0001_two = 21_sixteen
0010-0010_two = 22_sixteen
0010-0011_two = 23_sixteen
0010-0100_two = 24_sixteen
0010-0101_two = 25_sixteen
0010-0110_two = 26_sixteen
0010-0111_two = 27_sixteen
0010-1000_two = 28_sixteen
0010-1001_two = 29_sixteen
0010-1010_two = 2A_sixteen
0010-1011_two = 2B_sixteen
0010-1100_two = 2C_sixteen
0010-1101_two = 2D_sixteen
0010-1110_two = 2E_sixteen
0010-1111_two = 2F_sixteen

0011-0000_two = 30_sixteen
0011-0001_two = 31_sixteen
0011-0010_two = 32_sixteen
0011-0011_two = 33_sixteen
0011-0100_two = 34_sixteen
0011-0101_two = 35_sixteen
0011-0110_two = 36_sixteen
0011-0111_two = 37_sixteen
0011-1000_two = 38_sixteen
0011-1001_two = 39_sixteen
0011-1010_two = 3A_sixteen
0011-1011_two = 3B_sixteen
0011-1100_two = 3C_sixteen
0011-1101_two = 3D_sixteen
0011-1110_two = 3E_sixteen
0011-1111_two = 3F_sixteen

0100-0000_two = 40_sixteen
0100-0001_two = 41_sixteen
0100-0010_two = 42_sixteen
0100-0011_two = 43_sixteen
0100-0100_two = 44_sixteen
0100-0101_two = 45_sixteen
0100-0110_two = 46_sixteen
0100-0111_two = 47_sixteen
0100-1000_two = 48_sixteen
0100-1001_two = 49_sixteen
0100-1010_two = 4A_sixteen
0100-1011_two = 4B_sixteen
0100-1100_two = 4C_sixteen
0100-1101_two = 4D_sixteen
0100-1110_two = 4E_sixteen
0100-1111_two = 4F_sixteen

0101-0000_two = 50_sixteen
0101-0001_two = 51_sixteen
0101-0010_two = 52_sixteen
0101-0011_two = 53_sixteen
0101-0100_two = 54_sixteen
0101-0101_two = 55_sixteen
0101-0110_two = 56_sixteen
0101-0111_two = 57_sixteen
0101-1000_two = 58_sixteen
0101-1001_two = 59_sixteen
0101-1010_two = 5A_sixteen
0101-1011_two = 5B_sixteen
0101-1100_two = 5C_sixteen
0101-1101_two = 5D_sixteen
0101-1110_two = 5E_sixteen
0101-1111_two = 5F_sixteen

0110-0000_two = 60_sixteen
0110-0001_two = 61_sixteen
0110-0010_two = 62_sixteen
0110-0011_two = 63_sixteen
0110-0100_two = 64_sixteen
0110-0101_two = 65_sixteen
0110-0110_two = 66_sixteen
0110-0111_two = 67_sixteen
0110-1000_two = 68_sixteen
0110-1001_two = 69_sixteen
0110-1010_two = 6A_sixteen
0110-1011_two = 6B_sixteen
0110-1100_two = 6C_sixteen
0110-1101_two = 6D_sixteen
0110-1110_two = 6E_sixteen
0110-1111_two = 6F_sixteen

0111-0000_two = 70_sixteen
0111-0001_two = 71_sixteen
0111-0010_two = 72_sixteen
0111-0011_two = 73_sixteen
0111-0100_two = 74_sixteen
0111-0101_two = 75_sixteen
0111-0110_two = 76_sixteen
0111-0111_two = 77_sixteen
0111-1000_two = 78_sixteen
0111-1001_two = 79_sixteen
0111-1010_two = 7A_sixteen
0111-1011_two = 7B_sixteen
0111-1100_two = 7C_sixteen
0111-1101_two = 7D_sixteen
0111-1110_two = 7E_sixteen
0111-1111_two = 7F_sixteen

1000-0000_two = 80_sixteen
1000-0001_two = 81_sixteen
1000-0010_two = 82_sixteen
1000-0011_two = 83_sixteen
1000-0100_two = 84_sixteen
1000-0101_two = 85_sixteen
1000-0110_two = 86_sixteen
1000-0111_two = 87_sixteen
1000-1000_two = 88_sixteen
1000-1001_two = 89_sixteen
1000-1010_two = 8A_sixteen
1000-1011_two = 8B_sixteen
1000-1100_two = 8C_sixteen
1000-1101_two = 8D_sixteen
1000-1110_two = 8E_sixteen
1000-1111_two = 8F_sixteen

1001-0000_two = 90_sixteen
1001-0001_two = 91_sixteen
1001-0010_two = 92_sixteen
1001-0011_two = 93_sixteen
1001-0100_two = 94_sixteen
1001-0101_two = 95_sixteen
1001-0110_two = 96_sixteen
1001-0111_two = 97_sixteen
1001-1000_two = 98_sixteen
1001-1001_two = 99_sixteen
1001-1010_two = 9A_sixteen
1001-1011_two = 9B_sixteen
1001-1100_two = 9C_sixteen
1001-1101_two = 9D_sixteen
1001-1110_two = 9E_sixteen
1001-1111_two = 9F_sixteen

1010-0000_two = A0_sixteen
1010-0001_two = A1_sixteen
1010-0010_two = A2_sixteen
1010-0011_two = A3_sixteen
1010-0100_two = A4_sixteen
1010-0101_two = A5_sixteen
1010-0110_two = A6_sixteen
1010-0111_two = A7_sixteen
1010-1000_two = A8_sixteen
1010-1001_two = A9_sixteen
1010-1010_two = AA_sixteen
1010-1011_two = AB_sixteen
1010-1100_two = AC_sixteen
1010-1101_two = AD_sixteen
1010-1110_two = AE_sixteen
1010-1111_two = AF_sixteen

1011-0000_two = B0_sixteen
1011-0001_two = B1_sixteen
1011-0010_two = B2_sixteen
1011-0011_two = B3_sixteen
1011-0100_two = B4_sixteen
1011-0101_two = B5_sixteen
1011-0110_two = B6_sixteen
1011-0111_two = B7_sixteen
1011-1000_two = B8_sixteen
1011-1001_two = B9_sixteen
1011-1010_two = BA_sixteen
1011-1011_two = BB_sixteen
1011-1100_two = BC_sixteen
1011-1101_two = BD_sixteen
1011-1110_two = BE_sixteen
1011-1111_two = BF_sixteen

1100-0000_two = C0_sixteen
1100-0001_two = C1_sixteen
1100-0010_two = C2_sixteen
1100-0011_two = C3_sixteen
1100-0100_two = C4_sixteen
1100-0101_two = C5_sixteen
1100-0110_two = C6_sixteen
1100-0111_two = C7_sixteen
1100-1000_two = C8_sixteen
1100-1001_two = C9_sixteen
1100-1010_two = CA_sixteen
1100-1011_two = CB_sixteen
1100-1100_two = CC_sixteen
1100-1101_two = CD_sixteen
1100-1110_two = CE_sixteen
1100-1111_two = CF_sixteen

1101-0000_two = D0_sixteen
1101-0001_two = D1_sixteen
1101-0010_two = D2_sixteen
1101-0011_two = D3_sixteen
1101-0100_two = D4_sixteen
1101-0101_two = D5_sixteen
1101-0110_two = D6_sixteen
1101-0111_two = D7_sixteen
1101-1000_two = D8_sixteen
1101-1001_two = D9_sixteen
1101-1010_two = DA_sixteen
1101-1011_two = DB_sixteen
1101-1100_two = DC_sixteen
1101-1101_two = DD_sixteen
1101-1110_two = DE_sixteen
1101-1111_two = DF_sixteen

1110-0000_two = E0_sixteen
1110-0001_two = E1_sixteen
1110-0010_two = E2_sixteen
1110-0011_two = E3_sixteen
1110-0100_two = E4_sixteen
1110-0101_two = E5_sixteen
1110-0110_two = E6_sixteen
1110-0111_two = E7_sixteen
1110-1000_two = E8_sixteen
1110-1001_two = E9_sixteen
1110-1010_two = EA_sixteen
1110-1011_two = EB_sixteen
1110-1100_two = EC_sixteen
1110-1101_two = ED_sixteen
1110-1110_two = EE_sixteen
1110-1111_two = EF_sixteen

1111-0000_two = F0_sixteen
1111-0001_two = F1_sixteen
1111-0010_two = F2_sixteen
1111-0011_two = F3_sixteen
1111-0100_two = F4_sixteen
1111-0101_two = F5_sixteen
1111-0110_two = F6_sixteen
1111-0111_two = F7_sixteen
1111-1000_two = F8_sixteen
1111-1001_two = F9_sixteen
1111-1010_two = FA_sixteen
1111-1011_two = FB_sixteen
1111-1100_two = FC_sixteen
1111-1101_two = FD_sixteen
1111-1110_two = FE_sixteen
1111-1111_two = FF_sixteen

Thus, as regards Computer Programming, Hexadecimal is sometimes dubbed “a shorthand for binary.” It is much easier, and it is much more conducive to accuracy, for a human being to low-level program by entering in permutations of two Hexadecimal Digits, than for him/her to low-level program by entering in permutations of eight binary digits. It is possible to low-level program in hexadecimal by using a hexadecimal editor, often referred to as a “hex editor.”

---

¹ The term ‘hexadecimal’ is derived from the Ancient-Greek Cardinal Number, ‘héx,’ which means ‘six,’ and the Ancient-Greek Cardinal Number, ‘déka,’ which means ‘ten.’ When we affix the Latin 3rd-declension suffix, ‘-ālis, -āle’ to the numbers, ‘héx,’ and ‘déka,’ – which, when combined make the Latin prefix, ‘hexadecima-‘ which denotes 16 – we get the Latin 3rd-declension adjective, ‘hexadecimālis, hexadecimāle,’ which means ‘of 16;’ ‘relating to 16;’ etc. Thus, the English adjective, ‘hexadecimal,’ possesses the etymological sense: ‘of [base] 16;’ ‘relating to [base] 16;’ etc.

² The base-10 or decimal numbers that we ordinarily use on a day-to-day basis.

I Have 7 Cats: Formatting Numbers in Python

---

Link To Pdf:

Below is a Link to a pdf version of this web post:

Formatting Numbers in Python

---

 

Formatting Numbers in Python.

Figure 1: I have 7 cats.

The table below shows us different ways that we may format a number in Python. In this instance, I have chosen the number, 7.

Syntax:	Output:
print(“I have {0:d} cats”.format(7,6,5,4))	I have 7 cats
print(“I have {0:3d} cats”.format(7,6,5,4))	I have 7 cats
print(“I have {0:03d} cats”.format(7,6,5,4))	I have 007 cats
print(“I have {0:f} cats”.format(7,6,5,4))	I have 7.000000 cats
print(“I have {0:.2f} cats”.format(7,6,5,4))	I have 7.00 cats

I will take every entry of the above table, individually, and shall explain what is going on.

1. print (“I have {0:d} cats”.format(7,6,5,4))
   

   
   

   Figure 2: The contents of the chain parenthesis analysed.
   

In the above command, we specify, to python, that we wish to format the zeroth¹ number-element in the listed sequence:

(7,6,5,4)

.

This is what the:

0

part of:

{0:d}

is for.

In this instance the zeroth number-element in the listed sequence is:

7

.

Therefore, it will be the number, 7, that will be formatted and printed by Python.

We use a

d

in the chain parenthesis, to let Python know that we wish to format the number:

7

as an ordinary decimal number.

When we give the command:

>>> print(“I have {0:d} cats”.format(7,6,5,4))

to Python, Python outputs:

I have 7 cats

.

Below are examples of what occurs when we give formatting commands such as these to a Python Interactive Window:

Figure 3: In the above example, we, systematically, format all of the number-elements in the sequence: (7,6,5,4). We do this by altering the value of the number before the colon in the chain parenthesis.

1. print (“I have {0:3d} cats”.format(7,6,5,4))

Figure 4: The contents of the chain parenthesis analysed.

In the above command, we specify, to python, that we wish to format the zeroth number-element in the listed sequence:

(7,6,5,4)

.

This is what the

0

part of:

{0:3d}

is for.

In this instance the zeroth number-element in the listed sequence is:

7

.

Therefore, it will be the number, 7, that will be formatted and printed by Python.

We use a

d

in the chain parenthesis, to let Python know that we wish to format the number:

7

as an ordinary decimal number.

The

3

character tells python that we wish the decimal number, i.e. 7, to be the third character after leading² characters. As we do not specify what form that we wish for these leading characters to take, then:

7

will be the third character after two leading spaces.

When we give the command:

>>> print(“I have {0:3d} cats”.format(7,6,5,4))

to Python, Python outputs:

I have 7 cats

.

Below are examples of what occurs when we give formatting commands such as these to a Python Interactive Window:

Figure 5: In the above example, we, systematically, format all of the number-elements in the sequence: (7,6,5,4). We do this by altering the value of the number before the colon in the chain parenthesis.

1. print (“I have {0:03d} cats”.format(7,6,5,4))

Figure 6: The contents of the chain parenthesis analysed.

In the above command, we specify, to python, that we wish to format the zeroth number-element in the listed sequence:

(7,6,5,4)

.

This is what the

0

part of:

{0:03d}

is for.

In this instance the zeroth number-element in the listed sequence is:

7

.

Therefore, it will be the number, 7, that will be formatted and printed by Python.

We use a

d

in the chain parenthesis, to let Python know that we wish to format the number:

7

as an ordinary decimal number.

The

3

character tells python that we wish the decimal number, i.e. 7, to be the third character after leading characters³.

The:

0

prior to the:

3

and following the:

:

in:

{0:03d}

signifies the leading character:

zero

.

Therefore:

7

will be the third character after two leading zeros.

When we give the command:

>>> print(“I have {0:03d} cats”.format(7,6,5,4))

to Python, Python outputs:

I have 007 cats

.

Below are examples of what occurs when we give formatting commands such as these to a Python Interactive Window:

Figure 7: In the above example, we, systematically, format all of the number-elements in the sequence: (7,6,5,4). We do this by altering the value of the number before the colon in the chain parenthesis.

1. print (“I have {0:f} cats”.format(7,6,5,4))

Figure 8: The contents of the chain parenthesis analysed.

In the above command, we specify, to python, that we wish to format the zeroth number-element in the listed sequence:

(7,6,5,4)

.

This is what the

0

part of:

{0:f}

is for.

In this instance the zeroth number-element in the listed sequence is:

7

.

Therefore, it will be the number, 7, that will be formatted and printed by Python.

We use an:

f

in the chain parenthesis, to let Python know that we wish to format the number:

7

as a floating-point number⁴.

When we give the command:

>>> print(“I have {0:f} cats”.format(7,6,5,4))

to Python, Python outputs:

I have 7.000000 cats

.

As we can see, the number,

7,

its being a float is followed by a decimal point and six trailing zeros.

Below are examples of what occurs when we give formatting commands such as these to a Python Interactive Window:

Figure 9: In the above example, we, systematically, format all of the number-elements in the sequence: (7,6,5,4). We do this by altering the value of the number before the colon in the chain parenthesis.

1. print (“I have {0:.2f} cats”.format(7,6,5,4))

Figure 10: The contents of the chain parenthesis analysed.

In the above command, we specify, to python, that we wish to format the zeroth number-element in the listed sequence:

(7,6,5,4)

.

This is what the

0

part of:

{0:.2f}

is for.

In this instance the zeroth number-element in the listed sequence is:

7

.

Therefore, it will be the number, 7, that will be formatted and printed by Python.

We use a

f

in the chain parenthesis, to let Python know that we wish to format the number:

7

as a float.

The

.2

characters tell python that we wish the floating-point number, i.e. 7, to be followed, after a decimal point, by two trailing characters, in this instance, zeros.

When we give the command:

>>> print(“I have {0:.2f} cats”.format(7,6,5,4))

to Python, Python outputs:

I have 7.00 cats

.

As we can see, from the above example, the number:

7

, is now followed by a decimal point and two trailing zeros, as per our command.

Below are examples of what occurs when we give formatting commands such as these to a Python Interactive Window:

Figure 11: In the above example, we, systematically, format all of the number-elements in the sequence: (7,6,5,4). We do this by altering the value of the number before the colon in the chain parenthesis.

Figure 12: In this instance, the characters that trail after the decimal point are significant, i.e. not zero. Python rounds up 7.76543 to 7.77.

---

¹ In programing, it is conventional to begin counting beginning at 0, not beginning at 1. Therefore, zeroth, or 0th, is an ordinal number. Hence: Zeroth, First, second … Hence: 0th, 1st, 2nd …

^{2 In Mathematics, the two zeros that precede the number, 7, in a number such as: 007 , are termed ‘leading zeros.’ In Mathematics, the two zeros that follow a number such as: 0.700 , are termed ‘trailing zeros.’}

³ In Mathematics, the two zeros that precede the number, 7, in a number such as: 007 , are termed ‘leading zeros.’ In Mathematics, the two zeros that follow a number such as: 0.700 , are termed ‘trailing zeros.’

⁴ In programming, this is generally termed: ‘float.’

 

More than One Way to Draw a Cat.

I was watching the musical, Cats, by Andrew Lloyd Webber on Youtube.  Crazy Maniacal nonsense, but it works as a musical.  The tunes are very catchy.

The performance was an outdoor one in California.

How many of the audience,

I wondered to myself,

are on acid watching this?

To loosely quote the character, Otto Mann, the stoner Bus Driver from The Simpsons: one does not need L.S.D. to enjoy Cats, only to enhance it!

Something else cat related:

I am teaching myself the Extensible Markup Language, SVG.  ‘SVG’ stands for ‘Scalable Vector Graphics.’  With SVG, it is possible to script an illustration using markup language that is extremely similar in syntax to HTML 5.  The book that I am using as a study aid is SVG Essentials.

.

Figure 1:  SVG Essentials (2nd edition) 2014, by David Eisenberg.

The first image that one learns to code is a cat.

Fig 2:  When coding an SVG document, the scripted SVG image is able to run in the browser if saved as a .svg extension.  Also, you can simply code the SVG as an inline part of an HTML-5 document by simply using the <svg></svg> ekement.  I called my above image of a cat, “cat_face_text.svg” and, as you can see, it runs successfully in my browser.

Figure 3:  A closeup of the Cat image.  Lamentably, WordPress does not allow me to upload the SVG image directly, so I had to convert it to a .png file using Microsoft Paint.

Figures 4-7:  Above is the code needed to create the Cat SVG image.

 

The following is a link to a pdf file that contains the code depicted above in Figures 4-7.  Copy the code, and, then, paste it into Notepad.  Save the copy-and-pasted text as a .svg file extension, and the same image of a cat ought to run in your browser as well.

cat_face_pdf_code

 

The Geometry of Time Travel.

 

Figure 1:  I drew this image of a melting clock with pencils.

The Geometry of Time Travel pdf

The above links to a pdf of this blog post.

 

I am reading The Time Machine by H. G. Wells at present.

The first chapter is fascinating, but requires several readings in order to begin to comprehend it.

The Time-traveller and his after-dinner guests begin to talk about Geometry.

The Time-traveller begins in the manner of Euclid.

He defines something of zero dimensions, i.e. a point: that which hath no part[1] in the words of Euclid in his Elements.

He defines a line as being of one dimension, i.e. that of length.

He defines a square as being of two dimensions, i.e. those of length, and breadth.

He defines a cube as being of three dimensions, i.e. those of length, breadth and height.

 

—

 

The Ancient-Greek word for ‘dimension,’ is ‘diástasis.’  The word ‘diástasis’ can be further broken down into the adverb ‘diá’ which means ‘apart,’ and the noun, ‘stásis’ which means ‘standing.’  The etymological sense is ‘a standing apart.’

A point has zero dimensions.

A line can be conceived as a point standing apart from another point.

A square can be conceived as a line standing apart from another line.

A cube can be conceived as a square standing apart from another square.

 

—

 

 

The Time-traveller begins to ask whether a fourth dimension of space exists?  Is there such a thing as a four-dimensional cube?

The answer may surprise you:  a cube that occupies four dimensions of space does exist and is termed a ‘tesseract.’

The term ‘tesseract’ is derived from the Ancient-Greek cardinal number, ‘téssares, téssara,’ which means ‘four.’

 

 

Figure 2:  Four (IIII) in Roman Numerals is equal to four (Δ) in Ancient-Greek numerals.  I hope that my Latin and Greek be correct.

Figure 3:  I lifted this illustration of a tesseract from Wikipedia.  According to Wikipedia, the etymology of the word ‘tesseract’ is derived from the Ancient-Greek Cardinal Number, ‘téssares, téssara,’ which means ‘four,’ and the Ancient-Greek noun, ‘aktís,’ which means ‘ray.’  In Geometry, a line extends infinitely in two directions.  In Geometry, a ray terminates at a point, and extends infinitely in one direction.  In Geometry, a line segment terminates at two points.

---

[1] “A point is that which has no part.”  Euclid.  The Elements.  Book I.  Definitions.  Euclid: The Thirteen Books of The Elements.  Translated with introduction and commentary by sir Thomas L. Heath.  Second Edition.  Dover Publications, Inc.  New York.  (1956.  Kindle Edition.)  Loc 5106.

 

 

Three Mathematicians and a Pathological Liar.

Three Mathematicians and a Pathological Liar

The above link provides the pdf version of this following blog post.

Three Mathematicians and a Pathological Liar

The above link provides the Microsoft-Word (2007) version of this following blog post.

.

I have draughted three pencil drawings of 3 famous mathematicians.  Mathematicians born in the 19^th century seem to be very photogenic, especially in the case of their being portrayed in monochrome photographs.  I thought that I would write a little about these drawings.

 

 

 

Bram Stoker

Bram Stoker was born in Clontarf, Dublin, in 1847.  He received a Baccalaureus Artium (B.A.) in Mathematics from Trinity College, Dublin, in 1870.  The Novel, Dracula, is peppered with references to Mathematics.  Even the lunatic of the piece, Renfield, “has a good understanding of formal logic,” as Doctor Van Helsing remarks!

Bram Stoker died in London in 1912.

 

Figure 1:  The above photograph is what I based my pencil sketch on.  See Figure 2.

 

Figure 2:  This is my first attempt at pencil-sketching a person.  I got an ordinary A4 Sheet of paper, and shaded it in.  After that I used my pencil and eraser in trying to capture Bram’s image.

 

George Boole

George Boole was born in Lincoln, England, 1815.  Although not strictly speaking Irish, we tend to regard him as being Irish, as he lived in Ireland, and did some of his best work here… kind of like Saint Patrick!  He is best known for his writing of Investigation of Laws of Thought.

George Boole died – prematurely – in Cork, Ireland, in 1864, at the age of 49.  He was outside in the rain and caught his death from pneumonia.

 

Figure 3:  The image that I based my pencil drawing on.

 

Figure 4:  My second attempt at pencil drawing went a lot better, I feel.  I think that I captured a sufficient amount of Boole’s essence.  My Boole seems to be cheerier, somehow, than how he comes across in Figure 3.  He seems to be wryly smiling about something.  I wish that I could say that this was intentional on my part, but that would be lying!  In Figure 3 Boole looks very cheesed off.  I must say that I would feel rather annoyed if the Irish Weather had killed me in my intellectual prime!

 

 

 

Bertrand Russell

A formal logician as Boole was.  Whereas Boole wrote a book on logic that went by the name of Investigation of the Laws of Thought Bertrand Russell’s book on formal logic was entitled Principia Mathematica.  Bertrand Russell was born in 1872 and died in 1970.

 

Figure 5:  This is the black-and-white photograph of Russell upon which I based my sketch.

 

Figure 6: My sketch of Bertrand Russell.

 

 

My pencil drawings seem to be getting more refined with every attempt.  As I always say: Practice makes passable.

 

Figure 7:  Oakie Doke… because he is an oak tree, geddit?  This was a stock-motion cartoon that used to be shown on CBBC in 1995, when I was nine.  Oakie Doke, lived in an Oak Tree.  He used to be plagued by moles and mice and other rodent vermin who would come to him looking for assistance in some bother that they had gotten themselves into… as per the theme tune:

“Cross the Dell and ring the bell;

He’ll understand!

The friendliest of folk it’s Mr Oakie Doke…”

I will now cite Wikipedia regarding Oakie Doke:

“Towards the end of each episode, after Oakie had helped solve the problem, he would state: “Well, it’s like I always say…”, followed by a rhyming phrase. This phrase would be in relation to the solution of the problem. This was greeted with approving laughter and applause from whoever was present at the time.[1]”

I can tell you one thing: I was not applauding Mr Doke for this.  As a nine year old, I was angered to the point of rage by this.  I had NEVER heard oakie say this rhyming couplet before, and I had seen every previous episode.  Mr Doke was having us on.

Telling lies ain’t too “friendly.”

Figure 8:  You’re such a liar, Oakie!  You never said that before in your life!

[1] http://en.wikipedia.org/wiki/Oakie_Doke

Boolean Algebra Series II: Ohm’s Law & Expressing Algebra in English

Ohm’s Law & Expressing Algebra in English(i)

The above link is the Microsoft Word (2007) version of this blog post.

I hope to collate these essays of mine treating of Boolean Algebra and Electronics into an electronic book.  It will be a study aid to complement Code (2000) Microsoft Press, Charles Petzold.  Although Code is an excellent book, I feel that there are certain things that he deals with hastily, or not at all, leaving the non-mathematician, and the non-electrician – who are the target readership – a bit lost.  An understanding of the terms of basic algebra is necessary before one may deal with Boolean Algebra.  To this, Petzold devotes half a page!

Ohm’s Law & Expressing Algebra in English.

Ohm’s Law expresses the relationship that exists between Electromotive Force; Intensity and Resistance.

It states that Electromotive Force is equal to Intensity multiplied by Resistance.

We can state, then, that Ohm’s law is expressed by way of an equation.

This equation is algebra in the sense that it uses symbols to represent varying magnitudes of Voltage, Amperage and Resistance.  It uses the operator:

=

so as to imply a state of equality between the magnitudes of Voltage, Amperage and Resistance.

It is the use of the sign of equality:

that makes Ohm’s Law an equation.

Algebraic Symbols and What they Stand For:
Symbol Name:	Stands For:	Also Known As:	A Variable Quantity of:
E	Electromotive Force	Voltage.	Volts.
I	Intensity	Current; Amperage.	Amps; Amperes.
R	Resistance	n/a	Ohms; Ω.

We can express this relationship with English Words prior to our putting it forth in formula.

This is an excellent practice recommended by my 1st-year Mathematics Textbook, Text and Tests 1.  Unless you can express the symbols of a formula in English, you do not understand the formula!

The same holds true for equations.

 

I :

We can phrase the equation:

x = x

AS:

A thing is equal to the same thing.

OR:

A thing is equal – in quantity or magnitude, etc. – to the quantity or magnitude of the same thing.

OR:

A thing is equal – in quantity or magnitude, etc. – to the quantity or magnitude of an identical thing.

II :

We can phrase the equation:

x + y = x

AS:

A thing plus a different thing is equal to the former thing.  [y must equal zero.]  [The latter thing must equal zero.]

OR:

The quantity or magnitude of a thing plus the quantity or magnitude of another thing is equal to the quantity or magnitude of the former thing.  [The quantity, or magnitude, of y must equal zero.]  [The quantity or magnitude of the latter thing must equal zero.]

OR:

The quantity or magnitude of a thing plus the quantity or magnitude of another thing is equal to the quantity or magnitude of a thing identical – in quantity, or magnitude – to the former thing.  [The quantity, or magnitude, of y must equal zero.]  [The quantity or magnitude of the latter thing must equal zero.]

III :

We may phrase the equation:

x + y = y

AS:

A thing plus a different thing is equal to the latter thing.  [x must equal zero.]  The former thing must equal zero.]

OR:

The quantity or magnitude of a thing plus the quantity or magnitude of a different thing is equal to the quantity or magnitude of the latter thing. [ The quantity or magnitude of x must equal zero.]  [The quantity or magnitude of the former thing must equal zero.]

OR:

The quantity or magnitude of a thing plus the quantity or magnitude of a different thing is equal to the quantity or magnitude of a thing identical – in quantity or magnitude – to the quantity or magnitude of the latter thing.  [The quantity or magnitude of x must equal zero.]  [The quantity or magnitude of the former thing must equal zero.]

IIII :

We may phrase the equation:

x + y = z

AS:

A thing plus another thing is equal to a thing different to both the former thing and the latter thing.

OR:

A quantity or magnitude plus another quantity or magnitude is equal to a quantity or magnitude different to the former quantity or magnitude and the latter quantity or magnitude.

OR:

The quantity or magnitude of a thing plus the quantity or magnitude of another thing is equal to the quantity, or magnitude, of a thing different – in quantity or magnitude – to both the quantity or magnitude of the former thing and the quantity or magnitude of the latter thing.

—

Which is easier to Say:

x + y = z

or the long-winded paragraph above?  The signs and symbols of algebra serve as an excellent shorthand!

According to a program that I listened to on B.B.C. Radio 4, Bertrand Russel’s book, Principia Mathematica[1] concerns rendering simple – but then progressively more difficult! – algebraic equations in English so as to describe and prove the logic of said equations.  It is said that Bertrand Russel’s Principia Mathematica is a book notoriously difficult to comprehend, but I would like to have a stab at understanding it.  This book is on my to-read list!

Figure 1: I drew the above picture of the Mathematician, Lord Bertrand Russel 1872 – 1970, with pencils.

Let us restate the formula for Ohm’s Law:

E = IR

WHICH IN ENGLISH IS:

The magnitude of Electromotive Force – measured in volts – is equal to the magnitude of Intensity, or Current – measured in amperes – multiplied by the magnitude of Resistance – measured in ohms.

The above formula is easily remembered by way of this mnemonic: think of Eirtricity, an Irish Electricity provider.  The first three letters of its name, eir-tricity, spells out ohms law.

Figure 2: Eirtricity’s logo.  I drew this with pencils, a ruler, and a compass.

We can manipulate the formula for Ohm’s Law so as to make Intensity the subject of the formula:

E=IR

E(1/R)=IR(1/R)

E(1/R)=IR(1/R)

E(1/R)=I(1)

E(1/R)=I

E/R=I

I=E/R

So, the formula for Ohm’s Law, rewritten so as to make Intensity, or Current, the subject of the equation is:

I= E/R

WHICH IN ENGLISH IS:

The magnitude of Intensity, or Current – measured in amperes – is equal to the magnitude of Electromotive Force – measured in volts – divided by the magnitude of Resistance – measured in ohms.

We can – again – manipulate the formula for Ohm’s Law so as to make Resistance the subject of the formula:

E=IR

E(1/I)=IR(1/I)

E(1/I)=IR(1/I)

E(1/I)=R(1)

E(1/I)=R

E/I=R

R=E/I

WHICH IN ENGLISH IS:

The magnitude of Resistance – measured in ohms – is equal to the magnitude of Electromotive Force – measured in volts – divided by the magnitude of Intensity, or Current – measured in amperes.

Note:

The name of the Electricity Provider, mentioned above, is actually Airtricity, but I did not know this prior to my researching this article.  However, thinking of Eirtricity as in Electricity Éire has done me great service in remembering Ohm’s Law.  A mnemonic is not rendered useless by mere facts!

Glossary:

algebra /’aldʒɪbrə/

• noun. [mass noun] the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations.
  • A system of algebra based on given axioms.

<DERIVATIVES> algebraist /aldʒɪ’breɪɪst/ noun.

<ORIGIN> late Middle English: from Italian, Spanish and medieval Latin, from Arabic al-jabr ‘the reunion of broken parts’, ‘bone-setting’, from jabara ‘reunite, restore’.  The original sense, ‘the surgical treatment of fractures’, probably came via Spanish, in which it survives; the mathematical sense comes from the title of a book, ‘ilm al-jabr wa’l-muqābala ‘the science of restoring what is missing and equating like with like’, by the mathematician al-ḵwārizmi (see ALGORITHM).

amperage /’amp(ǝ)rɪdʒ/

the strength of an electric current in amperes.

equation /ɪ’kweɪʒ(ǝ)n/

• • [MATHEMATICS] a statement that the values of two mathematical expressions are equal (indicated by the sign =).

<ORIGIN> late Middle English: from Latin aequatio(n-), from aequare ‘make equal,’ ‘see EQUATE.’

Etmyology:  The term equation comes from the Latin 1^st and 2^nd declension adjective, ‘aequus, aequa, aequum,’ ‘equal;’ and the 3^rd-declension termination ‘-tiō’ which denotes a state of being, in this instance a state of being equal.  Hence ‘aequātiō, aequātiōnis’ is a feminine, 3^rd-declension noun, which means ‘the state of being equal.’

mnemonic /nɪ’mɒnɪk/

• noun. a device such as a pattern of letters, ideas or associations which assists in remembering something.
• adjective. aiding or designed to aid the memory.
  • relating to the power of memory.

<DERIVATIVES> mnemonically adverb.  mnemonist / ‘niːmənɪst/ noun.

<ORIGIN> mid 18^th century (as an adjective): via medieval Latin from Greek mnēmonikos, from mnēmōn ‘mindful.’

mnemonics

• plural noun. [usually treated as singular.]  the study and development of systems for improving and assisting the memory.

Ohm /ǝʊm/

Proper Noun.

• Georf Simon (1789-1854), German physicist. The units ohm and mho are named after him, as is Ohm’s law on electricity.

 

ohm

• noun. the SI unit of electrical resistance. transmitting a current of one ampere when subjected to a potential difference of one volt.  (Symbol: Ω)

Ohm’s Law

• [PHYSICS] a law stating that electric current is proportional to voltage and inversely proportional to resistance.

operation

• noun. [Mathematics.]  a process in which a number, quantity, expression, etc., is altered or manipulated according to set formal rules, such as those of addition, multiplication, and differentiation.

<ORIGIN> late Middle English: via Old French from Latin operatio(n-), from the verb operari ‘expend labour on’ (see OPERATE).

operator

• noun. [MATHEMATICS] a symbol or function denoting an operation (e.g. x, +).

 

The above definitions are from:

Oxford University Press.  Oxford Dictionary of English (Electronic Edition).  Oxford.  2010

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[1] ‘prīncipium, prīncipiī,’ neuter, 2^nd-declension Latin noun, ‘beginning.’  ‘Prīncipia,’ nominative plural, ‘beginnings.’  ‘mathēmaticus,  mathēmatica, mathēmaticum,’ 1^st-and-2^nd-declension adjective, ‘mathematic,’ ‘mathematical.’  ‘mathēmatica,’ nominative neuter plural form of the adjective, agreeing in case, gender and number with ‘Prīncipia.’  Hence, a good translation of ‘Prīncipia Mathēmatica’ would be ‘Mathematical Beginnings.’