- *G5490 *1 χάσμα (khas'-mah) : from a form of an obsolete primary chao (to "gape" or "yawn"); a "chasm" or vacancy (impassable interval):--gulf.
- χασμα
- Luke 16:26 ... and you there is a great gulf fixed: so that they which ...
36. Strongs - chasm
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Words: χασμα
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*G5490 *1 χάσμα (khas'-mah) : from a form of an obsolete primary chao (to "gape" or "yawn"); a "chasm" or vacancy (impassable interval):--gulf.
37. Strongs
- *H1419 גָּדוֹל גָּדוֹל (gaw-dole') : or (shortened) gadol {gaw-dole'}; from H1431; great (in any sense); hence, older; also insolent:--+ aloud, elder(-est), + exceeding(-ly), + far, (man of) great (man, matter, thing,-er,-ness), high, long, loud, mighty, more, much, noble, proud thing, X sore, (X ) very.
38. 2 Samuel 18:17
39. 2 Samuel 18:17
KJV: And they took Absalom, and cast him into a great pit in the wood, and laid a very great heap of stones upon him: and all Israel fled every one to his tent.
Hebrew: ויקחו את אבשלום וישליכו אתו ביער אל הפחת הגדול ויצבו עליו גל אבנים גדול מאד וכל ישראל נסו איש לאהלו׃
Greek: και ελαβεν τον αβεσσαλωμ και ερριψεν αυτον εις χασμα μεγα εν τω δρυμω εις τον βοθυνον τον μεγαν και εστηλωσεν επ αυτον σωρον λιθων μεγαν σφοδρα και πας ισραηλ εφυγεν ανηρ εις το σκηνωμα αυτου
Latin: et tulerunt Absalom et proiecerunt eum in saltu in foveam grandem et conportaverunt super eum acervum lapidum magnum nimis omnis autem Israhel fugit in tabernacula sua
Wycliffe: And thei token Absolon, and castiden forth him in to a greet dich in the forest, and baren togidere a ful greet heep of stoonys on hym; forsothe al Israel fledde in to his tabernaclis.
Luther: Und sie nahmen Absalom und warfen ihn in den Wald in eine große Grube und legten einen sehr großen Haufen Steine auf ihn. Und das ganze Israel floh, ein jeglicher in seine Hütte.
40. Chaos theory and snowflakes
The discovery that small roundoff errors in computations can cascade to cause very noticeable effects resulted in the field that is today called "
chaos theory".
A common, though inaccurate in a sense, saying about chaos theory is that "
A butterfly flapping its wings in China might cause a typhoon in California".
41. Exact math only approximates reality
42. Exact math only approximates reality
A common programming fallacy taught to beginning students is to use floating point variables when it is not appropriate.
An exact mathematical solution involving real numbers (which are not really real, they are assumed to be real) is only an approximation.
Any floating point number (e.g., that represent real numbers) has an inherent rounding error.
43. Chaos theory
Chaos theory: sensitive dependency on initial conditions of modeling of dynamical systems.
Analogy: It is
as if a butterfly flapping it's wings resulted in a tornado (1972). His original
analogy used a seagull and a storm.
Some pastors will confuse "
potentiality" or "
as if" or "
analogy" with an "
actuality" or "
fact" and preach how a butterfly can create this effect and then preach that you should be like that butterfly, etc. The "
liken" or "
as if" word of
analogy is used
16 times in Matthew (more times in other Gospels and by Paul, etc.).
Other
chaos meanings: (KAOS is from the 1960's comedy
Get Smart)
- English: informal meaning of disorder as in an informal entropy.
- Greek myth: Universe started as chaos.
An
exact mathematical model is only an
approximation of reality.
- Question: What is the sum of ten one-tenths? That is, ten times one-tenth.
Some pastors will preach Greek mythology chaos as if it were the statements made in Genesis.
Note: We are here ignoring discrete mathematics that involve only integers.
44. Slide rules
A slide rule (not slide ruler) was once used to do manual computations.
Whenever a slide rule is used, it is very evident that any computation involving real numbers is an approximation.
45. Mathematics and computation
What are the values of the following (using a computer program)?
1/10
1/10 + 1/10
1/10 + 1/10 + 1/10
1/10 + 1/10 + 1/10 + 1/10
1/10 + 1/10 + 1/10 + 1/10 + 1/10
1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10
1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10
1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10
1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10
1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10
Note: These are rational numbers, not irrational like
√2 or transcendental numbers like
π or
e.
46. Mathematical point of view
Here is the mathematics result expressed as real (rational) numbers.
1/10 = 0.1
2/10 = 0.2
3/10 = 0.3
4/10 = 0.4
5/10 = 0.5
6/10 = 0.6
7/10 = 0.7
8/10 = 0.8
9/10 = 0.9
10/10 = 1.0
47. Computation point of view
48. Lua program
Here is a simple Lua program to add values of
1/10 as
0.1.
Here is the output of the Lua code.
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The 17 places in the output is important to being exactly precise.
An exact mathematical model is only an approximation of reality.
|
49. Error accumulation
Even when symbolic math can solve a problem, any attempt to compute real answers involves the same approximations
The difference is small at each step but the difference can add up to a significant amount over many iterations.
Note: One can fix this specific instance by using a fixed decimal notation, but that only works, in base 10, for numbers and increments that are multiples of 2 or 5, the prime factors of 10.
50. Chaos theory
Rounding errors need to be addressed in fields of computer science such as numerical analysis.
51. Chaos theory
James Gleick (American author and historical scientist) has written a very interesting book on the field known as "
chaos theory" - a sensitive dependency on initial conditions. The field was accidentally discovered by the young French mathematician Henri Poincaré while attempting to find an exact mathematical solution for the three body problem.
Lorentz, in the weather field, rediscovered this idea and provided a name for it. The
analogy is that, due to round-off errors in computations, it is "
as if" a butterfly flapped it's wings in California and a typhoon resulted in China.
Gleick, J. (1988).
Chaos: making a new science. New York: Penguin Books..
52. Two and three body problem
Exact mathematics can solve the (idealized) two body problem such as the sun and earth or the earth and moon.
An exact solution for the (idealized) three body problem such as the sun, earth and moon has not been found.
Accurate weather prediction requires solving the (idealized) almost infinite particle system.
53. Quantum computing in brief
Quantum computing:
- much faster than conventional computers
- best for problems that allow probabilistic solutions
- cannot solve all problems - despite the hype
[exponential speedup not clearly defined, like entanglement]
54. Quantum computing analogies
Quantum computing analogies: pick the best way
- walk on foot: pencil and paper
- drive by car : conventional computer (go most anywhere)
- fly by jet : quantum computers (does not go anywhere, sometimes impractical)
- no way to get to Mars, nearest star, etc.
55. Dollars and cents
Fallacy:
Dollars and cents should be represented using floating point variables.
The general rule, sometimes taught, that if the number has a decimal point, it should be represented as a floating point variable does not hold for dollars and cents.
- The amount $123.56 is not a floating point value. It an integer value. That is 12345¢ cents.
Rule: Convert dollars and cents to cents, do the arithmetic, then convert back into dollars and cents (for display purposes). This distinction needs to be carefully handled in languages such as JavaScript and Lua which do not have integer variables.
The approximation issue involves floating point division so every such division needs to result in an integer value.
56. Numbers
Fallacy:
Social security numbers and phone numbers should be represented using integers.
How should social security numbers and phone numbers be represented? They appear to be numbers. That is, integers.
Are you ever going to add, subtract, multiply or divide these values? If not, use text.
- SSN: 999-99-9999 (1 billion values, originally geographically located)
Ask your self the following question.
- Are you ever going to add, subtract, multiply or divide these values?
If not, represent them using text. Note:
- Leading zeros are lost using integers.
- If an ordering is present, care must be taken when sorting lists of such values.
57. Helix
Hebrews 1:12 And as a vesture shalt thou fold them up, and they shall be changed: but thou art the same, and thy years shall not fail. [kjv]
και ωσει περιβολαιον ελιξεις αυτους ως ιματιον και αλλαγησονται συ δε ο αυτος ει και τα ετη σου ουκ εκλειψουσιν [gnt]
A helix is a spiral from the Latin word
"helix" ≈ "spiral" from the ancient Greek word
"ἕλιξ" ≈ "something twisted or spiral". The modern Greek word
"έλιξ" (EH-leeks) ≈ "helix, spiral".
A famous helix is the double helix formed by
DNA (Deoxyribonucleic Acid). The English word "
helicopter" comes from the Greek word for "
helix".
A Greek word based on "
helix" appears twice in the Bible.
58. Chaos theory and snowflakes
The discovery that small roundoff errors in computations can cascade to cause very noticeable effects resulted in the field that is today called "
chaos theory".
A common, though inaccurate in a sense, saying about chaos theory is that "
A butterfly flapping its wings in China might cause a typhoon in California".
59. Strongs - trench
- *H8585 תְּעָלָה (teh-aw-law') : from H5927; a channel (into which water is raised for irrigation); also a bandage or plaster (as placed upon a wound):--conduit, cured, healing, little river, trench, watercourse.
60. Chaos and dust
1 Kings 18:38 Then the fire of the LORD fell, and consumed the burnt sacrifice, and the wood, and the stones, and the dust, and licked up the water that was in the trench. [kjv]
… בתעלה … [he]
και επεσεν πυρ παρα κυριου εκ του ουρανου και κατεφαγεν το ολοκαυτωμα και τας σχιδακας και το υδωρ το εν τη θααλα και τους λιθους και τον χουν εξελιξεν το πυρ [lxx]
The Greek phase at the end of the verse means something to the effect "
and the dust turned/folded the fire".
The modern Greek word
"πυρ" (peer) ≈ "fire" is related to the English word
"fire", where the sounds "
p" and "
f" are related due to Grimm's Law (of language sound change).
The modern Greek word
"χάος" (HA-os) ≈ "chaos, disorder" is related to the ancient Greek word
"χους" ≈ "dust".
61. Hebrew
1 Kings 18:38 Then the fire of the LORD fell, and consumed the burnt sacrifice, and the wood, and the stones, and the dust, and licked up the water that was in the trench. [kjv]
… בתעלה … [he]
και επεσεν πυρ παρα κυριου εκ του ουρανου και κατεφαγεν το ολοκαυτωμα και τας σχιδακας και το υδωρ το εν τη θααλα και τους λιθους και τον χουν εξελιξεν το πυρ [lxx]
*H8585 תְּעָלָה (teh-aw-law') : from H5927; a channel (into which water is raised for irrigation); also a bandage or plaster (as placed upon a wound):--conduit, cured, healing, little river, trench, watercourse.
62. 1 Kings 18:38
KJV: Then the fire of the LORD fell, and consumed the burnt sacrifice, and the wood, and the stones, and the dust, and licked up the water that was in the trench.
Hebrew: ותפל אש יהוה ותאכל את העלה ואת העצים ואת האבנים ואת העפר ואת המים אשר בתעלה לחכה׃
Greek: και επεσεν πυρ παρα κυριου εκ του ουρανου και κατεφαγεν το ολοκαυτωμα και τας σχιδακας και το υδωρ το εν τη θααλα και τους λιθους και τον χουν εξελιξεν το πυρ
63. End of page
Genesis 1:1 In the beginning God created the heaven and the earth. [kjv]
1:2 And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters. [kjv]
Genesis 1:13 And the evening and the morning were the third day. [kjv]
When exactly does the day start? And when does it end? There have been two primary views during history.
Genesis 1:1 In the beginning God created the heaven and the earth. [kjv]
εν αρχη εποιησεν ο θεος τον ουρανον και την γην [lxx]
1:2 And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters. [kjv]
Genesis 1:2 And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters. [kjv]
The chemical element "Neon", atomic number 10, was named for the Greek «νεο» because it was newly discovered (by Sir William Ramsay and Morris W. Travers in 1898).
Genesis 1:6 And God said, Let there be a firmament in the midst of the waters, and let it divide the waters from the waters. [kjv]
1:7 And God made the firmament, and divided the waters which were under the firmament from the waters which were above the firmament: and it was so. [kjv]
1:8 And God called the firmament Heaven. And the evening and the morning were the second day. [kjv]
Luke 16:26 And beside all this, between us and you there is a great gulf fixed: so that they which would pass from hence to you cannot; neither can they pass to us, that would come from thence. [kjv]
και εν πασιν τουτοις μεταξυ ημων και υμων χασμα μεγα εστηρικται οπως οι θελοντες διαβηναι ενθεν προς υμας μη δυνωνται μηδε εκειθεν προς ημας διαπερωσιν [gnt]
2 Samuel 18:17 And they took Absalom, and cast him into a great pit in the wood, and laid a very great heap of stones upon him: and all Israel fled every one to his tent. [kjv]
… הפחת … [he]
και ελαβεν τον αβεσσαλωμ και ερριψεν αυτον εις χασμα μεγα εν τω δρυμω εις τον βοθυνον τον μεγαν και εστηλωσεν επ αυτον σωρον λιθων μεγαν σφοδρα και πας ισραηλ εφυγεν ανηρ εις το σκηνωμα αυτου [lxx]
