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Series and sums – math for web site
Example:
S = 1 + 2 +3 + 4 + 5 = ? = 15
Consider:
S = a + (a + d) + (a + 2d) + (a + 3d) ……+ (a + (n-1)d) + l
S = l + (l – d) + (l -2d) + (l – 3d) + …… + (l – (n-1)d) + a
addition follows:
2S = (a + l) + (a + l) + (a + l) + ….. + (a + l) + ( a + l)
2S = n(a + l)
S = n(a + l)/2
where a = first # and l = last # and n = # of terms in series
Check: in above, we have a = 1, l = 5, and n = 5. Therefore,
Sum = 5(5 + 1)/2 = 30/2 = 15
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new sum = 1 + 2 + 3 + 4 + 5 + 6 = 21
and
new sum = 6(1 + 6)/2 = 42/2 = 21
Q.E.D
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Consider sum = 19 + 21 + 23 where a = 19 and n = 3 and l= 23
Therefore, sum = S = n(a + l)/2 or
sum = 3(19 + 23)/2 = 3(42)/2 = 3(21) = 63
Q.E.D.
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Sum new = 87 + 85 + 83 = 3(80) + 7 + 5 + 3 = 240 + 15 = 255
Other way, sum new = S = n(a + l)/2 or
sum new = 3(87 + 83)/2 = 3(170)/2 = 3(85) = 255
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New sum = 1 + 2 + 3 + 4 + 5 + …. + 99 + 100
Other way: New sum = 100 (1 + 100)/2 = 50(101) = 50(100 + 1) =
New sum = 5000 + 50 = 5050
Q,E.D.
Result: much easier in second approach!
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Special case where a = l = 1 and n = 1
Special sum = (1)(1 + 1)/2 = 1
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Mention binomial series, etc.
Consider algebraic equations in two variables
x – y = 2
2x + y = 19
add these two equations to get: 3x = 2 + 19 = 21
or x = 7
therefore,
7 – y = 2
so, 7 – 2 = y
or, 5 = y
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A more complicated case,
5x + 2y = 9 Eq(1)
3x – y = 1 Eq(2)
multiply second equation by 2 and get:
6x -2y = 2 Eq(3)
Add Eq(1) to Eq(3) to get:
5x + 6x + 2y – 2y = 9 + 2 = 11
(5 + 6)x + 0 = 11
so, x =1 and from equation #1 above, we get: 5(1) + 2y = 9
or 5 + 2y = 9
therefore 2y = 9 – 5 = 4
So, y = 2
QED
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QED = ????
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Farmer Jones on a Hill
Imagine that there are three similar barns on a farmer’s rolling green hill in Dracut, Massachusettts. Last week, Jack painted one of these barns in 5 days while Jill painted a similar barn in 4 days.
But, now, this Dracut farmer needs to get his third barn painted soon because the pleasant autumn days will soon turn into chilly winter temperatures. How long would it take to paint a third identical barn if Jack and Jill climbed up that hill to paint it together? Of course, we need to stress that our two painters don’t get into each others way in performing their jobs.
This type of decision making is often needed in the world of manufacturing and product distribution.
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Elements of Algebra – Leonhard Euler
This book, originally published in 1765, is a gentle introduction to algebra by one of history’s greatest mathematicians, Leonhard Euler. Starting with basic mathematical concepts such as signs, fractions, powers and roots, logarithms, infinite series, arithmetic and geometric ratios, and the calculation of interest, Euler then discusses how to solve equations of varying degrees, methods of rendering certain formulas rational, and more.
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Algebra
logaritms
roots
infinite series
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Word problems
Example 2. There are b boys in the class. This is three more than four times the number of girls. How many girls are in the class?
Solution. Again, let x represent the unknown number that you are asked to find: Let x be the number of girls.
(Although b is not known, it is not what you are asked to find.)
The problem states that “This” — b — is three more than four times x:
4x + 3 = b.
Therefore,
4x = b − 3
x = b − 3
4 .
The solution here is not a number, because it will depend on the value of b. This is a type of “literal” equation, which is very common in algebra.
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Example 4. The sum of two consecutive numbers is 37. What are they?
Solution. Two consecutive numbers are like 8 and 9, or 51 and 52.
Let x, then, be the first number. Then the number after it is x + 1.
The problem states that their sum is 37:
word problem = 37
2x = 37 − 1
= 36.
x = 36
2
= 18.
The two numbers are 18 and 19.
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Picasso didn’t learn to paint by watching lectures
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Consider again the sum of a series
Consider:
S = a + (a + d) + (a + 2d) + (a + 3d) ……+ (a + nd) + l
also,
S = l + (l – d) + (l -2d) + (l – 3d) + …… + (l – nd) + a
addition follows:
2S = (a + l) + (a + l) + (a + l) + ….. + (a + l) + ( a + l)
2S = n(a + l)
S = n(a + l)/2 = the sum of all n terms in a series where first # = a & last # = l
Example #1
a = 3 and n = 5 and d = 4, so, we have what for l?
Sum = 3 + (3 +d) + (3 + 2d) + (3 + 3d) + (3 + 4d) ; NOTE: l = 3 + 4(4) = 19
Sum = 5(3) + d(1 + 2 + 3 + 4)
Sum = 15 + 4(10)
Sum = 55
In general,
l = a + (n – 1)d
and, here, l = 3 + (5 – 1)(4) = 19
Again, in general, we have:
Sum(n) = n(a + l)/2 = n( a + a + (n – 1)d)/2 =
Sum(n) = n(2a + (n – 1)d)/2
Special case, d = 1
so,
Sum(n, d) = n(2a + (n – 1))/2
and if a = 1, Sum(7,1) = 7(2 + 6)/2 = 7(8)/2 = 7(4)
Sum(7,1) = 28 where d = 1 and a = 1
Long way follows:
Sum(7,1) = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 6 + 4 + 11 + 7
Sum(7,1) = 10 + 11 + 7 = 21 + 7 = 28
QED
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Special cases, check for n = 7, 14, 21, 28, etc. or at the end of week #1, #2, #3, #4, etc
Note,
Sum(7,1) = 28
Sum(14,1) =
Sum(21, 1) =
Sum(28,1) =
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n Sum(n)
7 28
14
At end of one week, n = 7 so, Sum(7,1) =
Other way gives:
S = 5(3 + 19)/2 = 5(22)/2 = 5(11) = 55
QED
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Number of doughnuts needed after 7, 14, 21, 28, days or 1, 2, 3, 4 weeks
Consider that Sum(n) = S = n(a + l)/2 where we do not know value of l = last number in a series.
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Math for Fun and Good Use
1) Easy Algebra
Consider the number of eggs on the kitchen counter. There are:
3 eggs plus 6 eggs, so the number of eggs equals 9
or 3 + 6 = 9.
If, instead, you were counting the number of lobsters on the counter, you would have:
3 lobsters plus 6 lobsters, so the total number of lobsters would equal 9. Pretty easy, no?
If you understand this bit of thinking, then you, already, are a good candidate for doing high school algebra!
Note, in passing: you also realize that the order of counting lobsters or eggs gives the same result, i.e., 3 + 6 = 6 + 3. The same rules hold in algebra for any two numbers, so that a + b = b +a where a and b are ordinary numbers.
Surprised? You should not be. Every day, you use this type of calculation in making decisions to run your life and business. You, also, use this type of thinking in preparing a nice batch of creamy batter using your fabulous Quebec crepe recipe.
Examples of using algebra in your daily life
Carpentry:
a) Joe needs two pieces of 2×4 boards to match in length. The longer piece is 96 inches (eight feet) long while the shorter one is 85 inches long. How much of the longer piece must he cut to get two equal length pieces? This calculation you could do in you head, but here is how it would be expressed in algebra:
96 – x = 85 or
96 – 85 = 11 = x; Answer: unknown length is 11 inches
b) A more difficult problem follows:
Joe is now building a rectangular picture frame for his daughter. The distance around the frame is called the periphery, P, which is given in general by:
P = 2 (l +w)
where “l” is the length and “w” is the width of the frame. If his daughter wants the the length, “l”, to be 3 times the width, “w”, how do we use algebra to express P, the periphery?
First, we can, now, write that
P = 2(3w + w) = 2(4w) = 8w
and, if she needs the distance around the frame to be 48 inches, then we conclude that P = 48 = 8w or w = 6 inches, so “l” must be:
l = 3w = 3(6) = 18 inches.
Check your answer:
In general, P = 2(l + w) and, in this case, we have P = 2(18 +6) = 2(24) = 48
All checks out!
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Contractor’s Problem: A typical word problem
Farmer Jones has three identical barns on his property in the woods of Dracut, Mass. Also, Jack and Jill are each independent contractors in the area.
At first, the good Farmer Jones asks Jack to paint the first barn, which he completes in 40 hours.
Next week, Jill is hired to paint the second barn, which she completes in only 35 hours.
Finally, the farmer needs to have his third barn also painted and he wishes to have both Jack and Jill work on this last project together without getting into each others way.
How long would it take to have this last barn painted?
Note:
Jack’s rate of work is the job done in 40 hours.
Jill’s rate of work is the job done in 35 hours.
The team rate of work is what?
Note that, in general,
Job cost = worker rate multiplied by hours on the job.
In this case, Job = (Job/40 + Job/35)x where x is the time in hours to finish the job while working together. Basically, this says that
Job = Job(1/40 + 1/35)x or
1 = (1/40 + 1/35)x and multiplying each side by 40, we find:
40 = (1 + 40/35)x = (1 + 8/7)x = ( 7 + 8)/7(x)
7(40) = 15x or 7(8) = 3x and finally, x = (7)(8)/3 = 56/3 = 18.6667 hours.
Note that if each painter worked Jill’s rate, then time to complete the job would be the following:
Job = Job(1/35 + 1/35)x = Job(2)(1/35)x so, therefore, 1 = 2(1/35)x or
35 = 2x, so x = 35/2 = 17.5 hours or half of Jill’s rate while she was working alone.
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Latin and Early Roman History
Background
The Roman Republic’s Mediterranean expanse from 509 B.C. to 27 B.C. is well documented in multiple, Latin texts, i.e., written records appearing in the form of historic scrolls (books did not exist at that time), which were later archived in libraries. Indeed, these tabulated, pre-Christian-era events form, even today, the primary source of information that remains regarding the attitudes, regulations, laws, wars, wines, agriculture and day-to-day customs of that period.
Latin was the lingua franca of everyday life but also the official, legal language of the government. It would be difficult, today, to understand and appreciate that pre-Empire history without the written works of Cincinnatus, Levy, Caesar, Cicero, Tacitus, and Augustus. For example, the valiant efforts made by the Gracchi brothers to make Rome a more plebs-centric form of government would never have reached our ears without the political xxxxx.
Of course, Latin was also the official language of the Church for many centuries after the fall of the Roman Empire in the west. Then, it began to slowly transform itself into variations around the main theme, i.e., local modes of communication, which became the Spanish, Portuguese, French, Belgian, Italian, Rumanian, etc., of present-day Europe.
It is curious, for example, to note that, even in Britain during the days of Sir Isaac Newton, Latin was still used in scientific circles to maintain a close network of communications. Newton’s treatise called “Principia mathematica” published in 1687 is simply a case in point. Although many tradesmen knew and used English at the time, scientists in all disciplines across Europe did not use or understand the King’s English. Being recognized as a competitor and contributor in research required then as it does now to establish as broad readership as possible.
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Daily Mass and everyday prayers said in Latin were commonplace events in my childhood world. My parents, relatives and neighbors, “Les canucks” of Centralville, Pawtucketville and Little Canada, were well-versed in the customs and beliefs of French Catholicism going back to France even before the reign of Louis XIV(1643 – 1715).
“Dominus vobiscum. Et cum spirito tuo.” – “The Lord be with you. And with your spirit”
“Gloria tibi, Domine.” – “Glory to you, Lord.”
“Sed libera nos a malo. Amen.” – “But deliver us from evil. Amen.”
“PATER NOSTER” – “Our Father”
“In nomine Patris et Filii et Spiritus Sancti” – “In the name of the Father, the Son and the Holy Spirit.”
The above incantations resonate strongly in my mind, even today, as I relive religious ceremonies of my youth. English words formed the common background of those youthful days, but adopted, ancient memories and beliefs, all in French and Latin, also often counter-resonated in my skull as an English-based reflection was examined.
A fact presented and examined in different language settings seems to change a little or a lot in meaning. There seems to be no direct, one-on-one correlation of word meanings across languages. Of course, sometimes, the respective meanings are quite similar, which is most fortunate if we ever hope to connect to one another across the planet.
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Latin and Early Roman History
Background
The Roman Republic’s Mediterranean expanse from 509 B.C. to 27 B.C. is well documented in multiple, Latin texts, i.e., written records appearing in the form of historic scrolls (books did not exist at that time), which were later archived in libraries. Indeed, these tabulated, pre-Christian-era events form, even today, the primary source of information that remains regarding the attitudes, regulations, laws, wars, wines, agriculture, and day-to-day customs of that period.
Latin was the lingua franca of everyday life but also the official, legal language of the government. It would be difficult, today, to understand and appreciate that pre-Empire history without the written works of Cincinnatus, Levy, Caesar, Cicero, Tacitus, and Augustus. For example, the valiant efforts made by the Gracchi brothers to make Rome a more plebs-centric form of government would never have reached our ears without the political xxxxx.
Of course, Latin was also the official language of the Church for many centuries after the fall of the Roman Empire in the west. Then, it began to slowly transform itself into variations around the main theme, i.e., local modes of communication, which became the Spanish, Portuguese, French, Belgian, Italian, Rumanian, etc., of present-day Europe.
It is curious, for example, to note that, even in Britain during the days of Sir Isaac Newton, Latin was still used in scientific circles to maintain a close network of communications. Newton’s treatise called “Principia mathematica” published in 1687 is simply a case in point. Although many tradesmen knew and used English at the time, scientists in all disciplines across Europe did not use or understand the King’s English. Being recognized as a competitor and contributor in research required then as it does now to establish as broad readership as possible.
St. Joseph High School from 1953 to 1957
Staff: Brothers Giles, Arthur, Fermin and Louis
Location: xxx Merrimack Street,
Lab Equipment: a Bunsen burner, ball and hoop thermal expansion experiment
Missing lab equipment: microscope, litmus paper, and tests, Periodic table? everything needed in a chem lab
English Literature: Tale of Two Cities, Edgar Allen Poe, Catholic English authors,
Latin: readings, missle texts, Julius Caesar, Cincinnatus, Romulus and Remus, elephants over the Alps, Colosseum, Circus Maximus, “Quo vadis, domine?”
geometry: plane geometry, Euclid’s statements,
algebra: linear equations, Pythagorean theorem, Gauss elimination method, SR(2), arithmetic and geometric formulae, conservation of energy (no friction) leads to mechanical advantage concept,
trigonometry: identities, using Pythagoras theorem for and knowing one other angle plus one other side, you could be a surveyor and determine the height of a wall across Dana St.
religion: miracles, Bernadette, Lourdes, cures, Notre Dame de Beau Pres,
Greek & Roman history: Greek gods like Zeus, Sysiphus, Athena,
Greek authors: Aristotle, Plato, Socrates
American history: “ … on a cross of gold”, movers and shakers that made us great
Church history: Benedictines, Trappists, Franciscans, Cistercians,
Quebec history: la dictee, Samuel Champlein, les jesuites,
Spelling bee of French words
Losing out to Jacqueline Ducharme on a tough word – shame!
Presentation in French before the class
Marching in and out of school in ranks of two by two and no talking allowed
Recess: outside on a gated paved basketball area
Eating my sandwich in the cellar and also outside
Occasional visits to variety school on the corner of Merrimack and Aiken
Taking the NY Regents exam for a three-hour test during senior year – disappointing
Victor Buisson – walking to school partner during freshman year
Drop by to see Aunt Lida at Moody and Austin across from Clermont’s meat market
Cousin Georges (Soap) Ouellette gave me $1.00 once for no reason on such a visit
Science fair exhibit re. nitrogen process in the atmosphere – nitrates for plants
Roger St-Louis & Donald Bergeron: driving to school in Roger’s truck in senior year
My Feelings
a) scared, confused and poor
b) inadequate, not quite good enough
c) socially not fitting in with girls; I was very shy.
d) quite poor, a little desperate at times, depressed
e) Attracted to girls but very shy and awkward
f) Clumsy at dancing; “Sesoeur” taught me how to dance the waltz in her kitchen
g) ashamed of the hand-me-down clothes that were nice but did not fit me well
h) many conflicts on television – here and abroad
i) concerned about Mom, Michelle, and Denise, but knew that Bob would do OK
j) Where was I going in life?
k) Career? but where and how?
l) The priesthood, a monk, maybe, a wage earner?
m) Missed the support that my father might have given me – advice and guidance
n) alone and empty but with a biting sense of humor
o) good scholastically in high school
p) the instructors all seemed to encourage my efforts
q) some conflict between French-Canadian beliefs and real-world facts in city
r) why was there so much pain and despair in our world?
s) where was the all-knowing and all-loving God when we were feeling beaten & lost?
t) how could the Church help us in being in the day-to-day world?
u) How might college change my attitude and make me hopeful?
v) how could my mother and my three siblings survive in a time of great stress and war?
w) is there life after Catholic school?
x) will be getting good grades at St-Joseph High School keep me out of the textile mills depression?
y) do the girls (they were children, too) find me interesting, attractive, xxx
z) I had no HS chums at St-Joseph but I did have George, Roger, and Richard as friends
St. Joseph High School from 1953 to 1957
Staff: Brothers Giles, Arthur, Fermin and Louis
Location: xxx Merrimack Street,
Lab Equipment: a Bunsen burner, ball and hoop thermal expansion experiment
Missing lab equipment: microscope, litmus paper, and tests, Periodic table? everything needed in a chem lab
English Literature: Tale of Two Cities, Edgar Allen Poe, Catholic English authors,
Latin: readings, missle texts, Julius Caesar, Cincinnatus, Romulus and Remus, elephants over the Alps, Colosseum, Circus Maximus, “Quo vadis, domine?”
geometry: plane geometry, Euclid’s statements,
algebra: linear equations, Pythagorean theorem, Gauss elimination method, SR(2), arithmetic and geometric formulae, conservation of energy (no friction) leads to mechanical advantage concept,
trigonometry: identities, using Pythagoras theorem for and knowing one other angle plus one other side, you could be a surveyor and determine the height of a wall across Dana St.
religion: miracles, Bernadette, Lourdes, cures, Notre Dame de Beau Pres,
Greek & Roman history: Greek gods like Zeus, Sysiphus, Athena,
Greek authors: Aristotle, Plato, Socrates
American history: “ … on a cross of gold”, movers and shakers that made us great
Church history: Benedictines, Trappists, Franciscans, Cistercians,
Quebec history: la dictee, Samuel Champlain, les jesuites,
Spelling bee of French words
Losing out to Jacqueline Ducharme on a tough word – shame!
Presentation in French before the class
Marching in and out of school in ranks of two by two and no talking allowed
Recess: outside on a gated paved basketball area
Eating my sandwich in the cellar and also outside
Occasional visits to variety school on the corner of Merrimack and Aiken
Taking the NY Regents exam for a three-hour test during senior year – disappointing
Victor Buisson – walking to school partner during freshman year
Drop by to see Aunt Lida at Moody and Austin across from Clermont’s meat market
Cousin Georges (Soap) Ouellette gave me $1.00 once for no reason on such a visit
Science fair exhibit re. nitrogen process in the atmosphere – nitrates for plants
Roger St-Louis & Donald Bergeron: driving to school in Roger’s truck in senior year
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Similarly, the Irish in Upper Centralville and in the Highlands had their own part of town with their churches, taverns, restaurants and a special holiday on St-Patrick’s Day. Their parades and festivities brightened everyone’s life at the time. It was easily understood that many people simply wanted to heed the advice: “Kiss me. Im Irish”, which those pretty young lassies happily displayed on colorful badges, which they wore.
A Portuguese Insert
The heart and soul of Lowell’s Portuguese community is centered around St Anthony of Padua Parish Church, located at 893 Central Street. A back-Central location in the parlance of the locals. This Portuguese-American Roman Catholic Church, established in 1901, still offers Mass in English on Saturday and Sunday plus Portuguese services on weekends and, also, on Wednesday evening.
It remains the source of many community connections to family-friendly shops and restaurants in the area. Some of these are described below.
A favorite locale, which is in business today, is called Nana’s Kitchen at 524-492 Central St. This is a family-owned business that claims to be looking forward to meeting new customers. Their menu includes: coffee, deserts, appetizers plus lunch and dinner.
But, for the visitor wishing to taste authentic, Portuguese cuisine, Nina’s Kitchen also offers its customers culinary treats that are colorfully described in the blurb below:
Bife a nana,bacalhau a posta,bacalhau de natas,camarão alhinho,abrótea frita,cherne frito,chicharros fritos,peixe vermelho frito,iscas de fígado,morcela frita,feijoada,bifanas no prato ou no pão,chicken wingrs ou fingers,hamburger ou chesseburger, sopas de caldo verde,canja de galinha,e couve repolho e feijão.
Hopefully, the manager of this establishment also provides on-the-spot translation services required to assist customers with only a marginal command of this Latin-based language.
So, in the hope of providing the reader a wide choice of authentic, culinary delights, the following establishments are presented for consideration:
1) Cavaleiro’s Restaurant: a familiar, neighborhood Portuguese fixture with up-market seafood & steak entrees & house-made sangria.
2) Marko’s Mediterranean Grill on Thorndike: Favorites include a falafel salad or sandwich plus a fire-grilled gyro
3) Santoro’s Subs & Pizza: Great for organic products and fresh produce
4) Valentina’s Portuguese Market: an authentic bakery and delicatessen
However, the Taj Mahal of dining extravaganza in this locale is the:
5) Portuguese-American CVC League located at 512 Central Street. Many customers claim that this elegant restaurant is an awesome place to have parties, and enjoy authentic, delicious Portuguese food!
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The Upper Highlands and Belvedere might house successful professionals of Russian-Jewish descent or long-term Yankees, who first established themselves in the region before the start of the Industrial Revolution. These old-time Yankees were mostly of Protestant, Scottish and Irish persuasion, whose ancestors had been the first to set foot on Plymouth Rock around 1620 when England still ruled the seas.
This short enumeration of typical ethnic varieties to be encountered in the city falls far short of the long list of immigrant types that had sought refuge in Lowell over the years.
Indeed, the local author, Rev. George F. Kenngott, who wrote “The Record of a City, Macmillan Company, 1912“ indicated that there were 40 or more ethnic groups in the city at the time of his publication. In seems that everyone had his/her personal viewpoint regarding life in our big city.
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A test AbiWord paper – Gauss sum formula expanded.
This is a test of the AbiWord Format and Font fcns.
y = (a + x)2 = (a + x)(a + x) = a2 + 2ax + x2
OK, good work!
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For a right triangle where a = b = 1, then the hypotenuse, c, is given by
c2 = (1)2 + (1)2 = 2
Therefore,
c = “#@lS¥§³¹áèò
c = √2 = ?
that is: c is the square root of 2.
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Another way: ???
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Is √2 the ratio of two integers?
Let √2 = n/m where both n and m are integers and n > m so n/m > 1.
Therefore, n = m + p, so √2 = n/m = (m + p)/m = 1 + p/m where p/m < 1.
Again, √2 = (1 + p/m) = (1 + r) or 2 = 1 + 2r + r2 , so r2 + 2r – 1 = 0.
What next?
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Reductio ad absurdum argument follows.
Reductio ad absurdum
Syllogisms
indirect logic
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Euclid was born in Megara,[1][b] but in Athens he became a follower of Socrates.
Socratic method
Aristotle
Plato – story of the cave
Plato’s Theaetetus
https://en.wikipedia.org/wiki/Gauss%27s_law
Gaussian method
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Gauss -Jordan Elimination Methods
Pythagorean Hypothesis
Forms and shapes do this later
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Consider sums of series
Consider the sum below:
Sum = 1 +2 + 3 + 4 + 5
and also
Sum = 5 + 4 + 3 + 2 + 1 = 15 and average, Ave = Sum/5 = 15/5 = 3
I general, n(Average) = Sum where n = number of terms in the sequence.
Ad these expressions together to get:
2Sum = (1 + 5) + (2 + 4) + (3 + 3) + (4 + 2) + (5 + 1)
or
2Sum = 2(1 + 5) + 2(2 +4) + (3 + 3) = 2(1 + 5) + 2(2 + 4) + 2(3)
Finally, Sum = (1 + 5) + (2 + 4) + 3 = Sum, a tautology?
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Consider another sum below:
Sum = a + (a +d) + (a + 2d) + (a + 3d) + (a + 4d) +…….. + (a + (n – 1)d) + l
but also,
Sum = l + (l – d) + (l – 2d) + (l – 3d) + (l -4d)
Finally, we get:
2Sum = (a + l) + (a + l) + (a + l) + (a + l) + (a + l) = 5(a + l)
Finally, Sum = 5(a + l)/2 this is sum for 5 terms
Sum for n terms = n(a + l)/2
Test:
Sum = 5(1 +5)/2 = 5(3) = 15 OK
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For Gauss’s classroom case, we have a = 1, l = 100 and n = 100
so, Sum = 100(1 + 100)/2 = 50(101) = 50( 100 + 1) = 5000 + 50 = 5050
QED
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In general, we have Sum = n(a + l)/2
Now express l as a fcn of a, d and n
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Sum = n(a + l)/2
Tests follow:
a = 1 and l = 4
Sum = 4(1 + 4)/2 = 2(5) = 10
In general, l = a + (n – 1)d
i.e. last # = first # + (n – 1) increments of size = d
and, if d = 1, then we have:
Test: if l = 5, then l = 1 + (5 – 1)(1) = 5
Again, in general, we have:
Sum = n(a + l)/2
and l = a + (n – 1)d
Therefore,
NOTE: (a + l) = a + a + (n – 1)d = 2a + (n – 1)d
Therefore,
Sum = n/2(2a + (n – 1)d) = n(2a + (n – 1)d)/2
examples follow:
a = 1 and d = 1 and n = 6 or Sum = 1 + 2 + 3 +4 + 5 + 6 = 6 + 4 + 11 = 10 + 11 = 21
or
Sum = 6(2(1) + (6 – 1)(1))/2 = 3(2 + 5(1) = 3(2 + 5) = 3(7) = 21
In general, Sum = n(2a + (n – 1)d)/2
If we eliminate the l (above), we get:
Sum = n(a + a + (n -1)d)/2 This is the generalized Gauss’s Sum function.
Make a table with n and Sum as fcn with d = 1 & a = 1
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where a = first term
n = number of terms
l = last term
In above calculation, we had a = 1, d = 1, and n = 5
A test AbiWord paper
This is a test of the AbiWord Format and Font fcns.
y = (a + x)2 = (a + x)(a + x) = a2 + 2ax + x2
OK, good work!
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For a right triangle where a = b = 1, then the hypotenuse, c, is given by
c2 = (1)2 + (1)2 = 2
Therefore,
c = “#@lS¥§³¹áèò
c = √2 = ?
that is: c is the square root of 2.
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Another way: ???
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Is √2 the ratio of two integers?
Let √2 = n/m where both n and m are integers and n > m so n/m > 1.
Therefore, n = m + p, so √2 = n/m = (m + p)/m = 1 + p/m where p/m < 1.
Again, √2 = (1 + p/m) = (1 + r) or 2 = 1 + 2r + r2 , so r2 + 2r – 1 = 0.
What next?
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Reductio ad absurdum argument follows.
Reductio ad absurdum
Syllogisms
indirect logic
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Euclid was born in Megara,[1][b] but in Athens he became a follower of Socrates.
Socratic method
Aristotle
Plato – story of the cave
Plato’s Theaetetus
https://en.wikipedia.org/wiki/Gauss%27s_law
Gaussian method
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Gauss -Jordan Elimination Methods
Pythagorean Hypothesis
Forms and shapes do this later
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Consider sums of series
Consider the sum below:
Sum = 1 +2 + 3 + 4 + 5
and also
Sum = 5 + 4 + 3 + 2 + 1 = 15 and average, Ave = Sum/5 = 15/5 = 3
I general, n(Average) = Sum where n = number of terms in the sequence.
Ad these expressions together to get:
2Sum = (1 + 5) + (2 + 4) + (3 + 3) + (4 + 2) + (5 + 1)
or
2Sum = 2(1 + 5) + 2(2 +4) + (3 + 3) = 2(1 + 5) + 2(2 + 4) + 2(3)
Finally, Sum = (1 + 5) + (2 + 4) + 3 = Sum, a tautology?
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Consider another sum below:
Sum = a + (a +d) + (a + 2d) + (a + 3d) + (a + 4d) +…….. + (a + (n – 1)d) + l
but also,
Sum = l + (l – d) + (l – 2d) + (l – 3d) + (l -4d)
Finally, we get:
2Sum = (a + l) + (a + l) + (a + l) + (a + l) + (a + l) = 5(a + l)
Finally, Sum = 5(a + l)/2 this is sum for 5 terms
Sum for n terms = n(a + l)/2
Test:
Sum = 5(1 +5)/2 = 5(3) = 15 OK
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For Gauss’s classroom case, we have a = 1, l = 100 and n = 100
so, Sum = 100(1 + 100)/2 = 50(101) = 50( 100 + 1) = 5000 + 50 = 5050
QED
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In general, we have Sum = n(a + l)/2
Now express l as a fcn of a, d and n
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where a = first term
n = number of terms
l = last term
In above calculation, we had a = 1, d = 1, and n = 5
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Algebra and math in high school
In algebra, we often run into equations and find roots to these equations.
Consider:
x + 4 = 9
x = 9 – 4 = 5
QED
2x + 3y = 17 ; Eq(1)
3x – 3y = 7 ; ; Eq(2)
Addition gives us: 5x = 17 + 7 = 24
so we have: x = 24/5
and, therefore, from Eq(2), we get: 3(24/5) -3y = 7
or, 3(24/5) – 7 = 3y
Therefore, we get: y = (3(24/5) -7)/3
Simplify first: Consider that numerator called Num = 3(24/5) – 7 = 3(24)/5 – 7 = 72/5 – 7 =
= (72- 7(5))/5 = (72 – 35)/5 = 37/5 = Num
So, we get: y = Num/3 or
y = (37/5)/3 = 37/15
Test in Eq(1)
2(24/5) + 3(37/15) = 17 (?)
or
(2(24) + 37)/5 = 17 (?)
or
48 + 37 = 5(17) (?)
85 = 5(17) = 85 yes!
QED
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Consider:
y = (x + a)(x + a)
Now, write this equation using a superscript.
y = (a + x)2(b + y)17 + 37(b + a) – 67
Ok, go to Format and then Font to get the superscript symbol.
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Consider: a2 + b2 = c2
Let
a = n
b = n + 1
c = n + 2
So, we have:
n2 + (n + 1)2 = (n +2)2
n2 + n2 + 2n + 1 = n2 + 4n + 4
Simplify asap:
n2 + 2n + 1 = 4n +4 = 4(n + 1)
therefore:
n2 – 2n + 1- 4 = 0
n2 -2n -3 = 0
(n – 3)(n + 1) =0
Roots are n = 3 and n = -1
From geometry, n >= 0,
So solution remains that only n = 3 is valid and so we have:
a = 3, b = 4, c = 5 which gives: 9 + 16 = 25
No other possible set of consecutive values for a, b and c are possible.
QED
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The Immigrant Way
If anyone of the readers experienced daily life among the many culturally different ethnic groups in Lowell in the 1930s or before, then you remain the living witnesses to the socioeconomic history of that period.
At that time, young people usually shared a house or a tenement (an apartment would be too fancy a word to use) with family members often covering three and, sometimes, even four generations. These tight and sparse living arrangements were simply practical solutions that were reluctantly adopted to manage the very limited, combined incomes of all the adults in the group. Children were exempt after child labor laws came into effect.
Those older group members, the elders one might say, carried the indelible, transparent stamp of a psychologically displaced, French-Canadian man or woman, who was seemingly still quite lost in a non-Catholic, industrial work environment where English was the language of the land.
In contrast, however, one also found that the younger members of such a family group had often slowly managed to acquire useful skills in conversational English of the area, the gruff, Scots-Irish parlance spoken by the so-called Yankees.
This latter group had first set up local communities such as the town of Chelmsford, Massachusetts based mostly on family farming starting back in the 1640s and 1650s. These Yankees were well-established as rulers of the land prior to the American War of Independence (~ 1776 to 1780) and even more so after New England experienced the beginnings of the Industrial Revolution in the 1830s.
The need for cheap, untrained and willing (desperate) laborers, who were willing to work ten to twelve hours per day on a Monday through Saturday schedule, established the lifestyle and expectations of thousands of adopted foreigners, the immigrants, who were then given a chance at US citizenship.
The cost remained quite easy to comprehend, i.e., 60 to 72 hours a week laboring in the humid, dangerous, and, usually, unhealthy environment required for successful textile fabrication.
However, since the ideal environment for the mass production of textiles required ambient temperatures of about 90 degrees Fahrenheit and 90 percent humidity, few of these immigrants expected an easy lifestyle in the years to come but it seemed preferable to starving on a barren plot of land in Ireland, Greece, Poland, Portugal, French-Canada, Russia and elsewhere.
The times were hard and only those laborers willing and able to keep up the industrial pace, year after year, could, at least, manage to survive.
As a French proverb would have you believe:
“La vie ne fait pas de cadeaux.”
This acerbic comment can be rendered into English as:
“There is no free lunch.”
Cultural Differences and Many Languages
As I grew up during the war years, I became aware that this world of ours supports a wide variety of peoples, who in turn spoke foreign languages, ate interesting, different foods, and believed in a confusing number of religious tenets.
My parents, the Lowell Sun and local WLLH radio broadcasts were my primary sources of information. Later, The Readers Digest, the Lowell City Library, and movie theater newsreel clips filled in the gaps for me.
These many different peoples were scattered all over the world, but, more importantly, some of them were also inhabitants of my hometown. Ethnic differences abounded throughout the city’s many ghetto neighborhoods from Little Canada to The Acre and into sections of Upper Centralville along Bridge Street where many Irish and Yankees lived. There also existed Polish and Portuguese parts of town that were separate from all others. In a sense, every ethnic group chose to live tightly packed together and away from all other ethnic influences.
But, why so many social separations in a city of about 100,000 people? Do people coming directly from a different culture and belief system only feel safe by first choosing to segregate themselves within their ethnic group, which remained mostly closed to outsiders? Maybe, there is a pearl of wisdom in the popular expression that:
“Birds of a feather flock together.”
Another expression along this line reminds us that:
“Qui se ressemblent, se rassemblent.”
as any observant “franco-canadien” or “franco-canadienne”might have told a visitor.
Words, Words of All Varieties and Forms
Recall that most people immigrating to America at the start of the previous century had absolutely no background or training in the English language or in our system of customs and laws. The lack of attainable opportunities in their mother country eventually had driven these people to abandon family, friends, traditions, and language too, hopefully, start a new life in these United States.
A few of these new arrivals came upon our shores with certain career advantages such as having certain technical skills or homespun millenery savoir-faire as examples. As an aside, it remains historically important to note that Canada’s early fur trade was largely built on the fashion for beaver hats in Europe, particularly top hats. Hat making was millinery magic at its best.
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One of my many early memories centered on those exciting, Saturday morning escapades walking, among and between, busy shoppers, who were traveling up and down Merrimack Street (near Central) in the commercial epicenter of the city.
The Eastern Massachusetts bus station, which was nestled between two five-and-ten-cent stores set the stage. Across the street, there loomed that impressive Page clock, maybe twenty feet tall, that dominated the scene. Busy shoppers were invited to relax, unwind, and, also, snack at our favorite malt shop located on the corner of Central and Merrimack. Sure, New York City had Times Square, but we, Lowellians, had our famous Kearney Square.
Kearney Square was the center of my boyhood universe. The world of local commerce was essentially centered at the intersection of two main streets, Merrimack and Bridge, which strongly defined the socioeconomic pulse of the city. However, Central Street where one found McQuades, The Strand, and The Rialto movie theaters, a malt shop or two plus Newman’s Men’s Store also defined the business and social ethos of the inner city.
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As a young boy wearing my knickerbocker shorts or corduroys, weaving through busy, downtown crowds was a delightful challenge. Usually, my mother and, sometimes, my father accompanied me and my brother, Bob, on these shopping adventures.
I recall such a walk from long ago. Men in suits and women wearing simple cotton dresses were walking in front of us as we pasted one of the two 5&10-cents stores to our right. I was overhearing conversations or eavesdropping on perfect strangers.
The older woman was actively soliciting her companion in a foreign tongue. Words that sounded like: “fereastră, uşă, lumina zilei, lipici, cleşte” had a strong effect on her friend’s composure.
Of course, all that I could gather was that some form of communication had transpired without my understanding of anything. My grasp of Rumanian was marginal at best but I made no effort to add that language to my to-do list.
Another couple, just ahead, was speaking German, I thought. Now, since German has many cognates in English, I might certainly be able to extract a message from that encounter. Words like: “Elektrizität, Tür, Tageslicht, kleben” poured out from the mouth of a well-dressed gentleman. Apparently, he was talking about “electricity, door and daylight” but I could not understand “kleben”. However, this experience taught me to focus only on French and German if I wished to make any real progress in the translation business.
My interest in foreign languages probably started with my family’s appreciation of the cuisine from all over the world, and Lowell had no scarcity of ethnic restaurants and tiny butcher shops. At the Olympia Restaurant, the visitor, who might be familiar with world-famous Hellenistic dishes like “Souvláki, Kóta Riganáti, Gyros and Spanakópita” had good reason to believe that he/she had stumbled into Greek heaven. Added to this delightful, culinary surprise, this visitor would also be treated to a quick overview of the staff’s speech repertoire.
In their native language, the waiters and waitresses bubbled with references to: “κουζίνα, κατσαρόλα, κοτόπουλο, κρασί, αρνάκι, σάλτσα”, which any well-bred Greek-American knew immediately to mean: “kitchen, casserole, chicken, wine, lamb, sauce”.
With a French-Canadian background like mine, this linguistic artistry was something to be greatly admired. It was early in life that I decided that a bilingual person sees and feels the world in more than one fashion. I enjoyed this diversity.
Polish, Portuguese and Irish Flavors
Slavic roots within the city included two, Polish enclaves, one in the vicinity of Lakeview Avenue and Bridge Street and another in the neighborhood setting near Hildreth and Bridge Streets where the beloved Polish Club proclaimed the traditional sounds of the polka, waltz, hasapico, sirtaki and, of course, the oberek. My mother always loved the music and dancing at the club and she enjoyed a beer or two within the company of friends there.
In this locale, the visitor might hear references to daily life with a certain, Slavic twist. Words like: “taniec, człowiek, kobieta, dziewczynka, chłopak, ładny” could be heard at all the tables where visitors sat to watch the dancers on the floor.
Again, any Polish-American person in the crowd easily knew their equivalent, English meaning. In this case, the savvy visitor knew that these Polish words meant: “dance, man, woman, girl, boy, pretty”. For me, words from all languages seem to invigorate the moment with a special zest giving the event a special flavor.
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&&&
If we wish to sum this sequence from n=1 to any positive integer, for example 100
, we would write ∑n=1100n=1+2+3+⋯+100
This gives the answer to the sum of the first 100
positive integers.
The mathematician, Karl Friedrich Gauss, discovered the following proof when he was only 8 years old. His teacher had decided to give his class a problem which would distract them for the entire day by asking them to add all the numbers from 1 to 100. Young Karl quickly realised how to do this and shocked the teacher with the correct answer, 5 050
. This is the method that he used:
- Write the numbers in ascending order.
- Write the numbers in descending order.
- Add the corresponding pairs of terms together.
- Simplify the equation by making Sn
- the subject of the equation.
S100+S100––––∴2S100∴2S100∴S100=1+2+3+⋯+98+99+100=100+99+98+⋯+3+2+1–––––––––––––––––––––––––––––––––––=101+101+101+⋯+101+101+101=101×100=10 100=10 1002=5 050
, or
, or
, or