Here is a problem I am given Using Diophantus method, find four s.pdftradingcoa
Here is a problem I am given: Using Diophantus\' method, find four square numbers such that
their sum added to the sum of their sides is 73. Why does Diophantus\' method always work?
or alternatively, another problem is:
Using Diophantus\' method, show that 73 can be decomposed into the sum of 2 squares in two
different ways. Show that Diophantus\' method always works.
Here is an example of Diophantus\' method, on which I have questions:
Example:
To find four square numbers such that their sum added to the sum of their sides makes a given
number.
Given number 12.
Now x^2 + x + ¼ = a square.
Therefore the sum of four squares + the sum of their sides + 1 = sum of four other squares = 13,
by hypothesis.
Therefore we have to divide 13 into four squares; then if we subtract ½ from each of their sides,
we shall have the sides of the required squares.
Now 13 = 4 + 9 = (64/25 + 36/25) + (144/25 + 81/25),
and the sides of the required squares are 11/10, 7/10, 19/10, 13/10, and the squares themselves
being 121/100, 49/100, 361/100, 169/100.
Where does the x^2+x+1/4 come from?
What does subtracting 1/2 do, where is this evident in the problem?
How would I or anyone else get from (64/25 + 36/25) + (144/25 + 81/25) to 11/10,7/10,19/10,
and 13/10??? This is my biggest question of them all. I would actually like to understand how to
do this problem, so some explanation and an answer as well would be great so I can check the
work myself, and be sure I\'m correct as well.
Solution
The known personal information of Diophantus is best summed up in the following
quote: “Here you see the tomb containing the remains of Diophantus, it is remarkable: artfully it
tells the measures of his life. The sixth part of his life God granted him for his youth. After a
twelfth more his cheeks were bearded. After an additional seventh he kindled the light of
marriage, and in the ?fth year he accepted a son. Elas, a dear but unfortunate child, half of his
father he was and this was also the span a cruel fate granted it, and he consoled his grief in the
remaining four years of his life. By this device of numbers, tell us the extent of his life.” With
our modern notation and use of algebra, it is rather easy to see that if Diophantus lived for x
years, then the problem is equivalent x to solving the equation x + 12 + x + 5 + x + 4 = x, which
has the solution x = 84. 6 7 2 It is hard to pinpoint when exactly Diophantus lived. The Arabian
historian Ab¯’lfaraj mentions u in his History of the Dynasties that Diophantus lived during the
time of Emperor Julian (361 - 363 A.D.), whereas Rafael Bombelli says authoritatively in his
book, Algebra, that Diophantus lived during the reign of Antoninus Pius (138 - 161 A.D.). There
is little or no con?rmation of either of these dates. However, in his On Polygonal Numbers,
Diophantus de?nes a polygonal number by quoting Hypsicles. Hypsicles was the writer of the
supplement to Book XIII of Euclid’s Elements, and so Diophantus must have written aft.
Science and contribution of mathematics in its developmentFernando Alcoforado
Mathematics is the science of logical reasoning that has its development linked to research, interest in discovering the new and investigate highly complex situations. The escalation of Mathematics began in ancient times when it was aroused the interest by the calculations and numbers according to the need of man to relate the natural events to their daily lives. Today, Mathematics is the most important science of the modern world because it is present in all scientific areas.
Please like this if you find this helpful.
Db
Jtjrhrgegdv
Xgfvtfej
Dear sister god bless 6you always stay blessed al ways you find in the presence is valued for sale me your phone number please samjo send karo aapna to karo na ji koi ni hai na to bata de mere ko jante bhi nahi bol rahi thi to nahi na to koi baat kar rahe the kya bola tha na to bol rahi rahe ki main aap se baat hi he bola tha na to ki main aap ke se par hi raheta he mere kya bola aap ki sagai ho nahi he karti hu aap to nahi ye sab batao aapne abhi kya he aapke kya he aap ka house name radha to nahi ye sab bol rahe sakte kya ho gya gaya Rajasthan se aapka kya face tha kesa chal time kayo ketla thya chhe ne ke bija sathe ne gare aaj 4try kar karo na pavan sing bhai please share your karte ho to kya kar he isliye na to bola.
Here is a problem I am given Using Diophantus method, find four s.pdftradingcoa
Here is a problem I am given: Using Diophantus\' method, find four square numbers such that
their sum added to the sum of their sides is 73. Why does Diophantus\' method always work?
or alternatively, another problem is:
Using Diophantus\' method, show that 73 can be decomposed into the sum of 2 squares in two
different ways. Show that Diophantus\' method always works.
Here is an example of Diophantus\' method, on which I have questions:
Example:
To find four square numbers such that their sum added to the sum of their sides makes a given
number.
Given number 12.
Now x^2 + x + ¼ = a square.
Therefore the sum of four squares + the sum of their sides + 1 = sum of four other squares = 13,
by hypothesis.
Therefore we have to divide 13 into four squares; then if we subtract ½ from each of their sides,
we shall have the sides of the required squares.
Now 13 = 4 + 9 = (64/25 + 36/25) + (144/25 + 81/25),
and the sides of the required squares are 11/10, 7/10, 19/10, 13/10, and the squares themselves
being 121/100, 49/100, 361/100, 169/100.
Where does the x^2+x+1/4 come from?
What does subtracting 1/2 do, where is this evident in the problem?
How would I or anyone else get from (64/25 + 36/25) + (144/25 + 81/25) to 11/10,7/10,19/10,
and 13/10??? This is my biggest question of them all. I would actually like to understand how to
do this problem, so some explanation and an answer as well would be great so I can check the
work myself, and be sure I\'m correct as well.
Solution
The known personal information of Diophantus is best summed up in the following
quote: “Here you see the tomb containing the remains of Diophantus, it is remarkable: artfully it
tells the measures of his life. The sixth part of his life God granted him for his youth. After a
twelfth more his cheeks were bearded. After an additional seventh he kindled the light of
marriage, and in the ?fth year he accepted a son. Elas, a dear but unfortunate child, half of his
father he was and this was also the span a cruel fate granted it, and he consoled his grief in the
remaining four years of his life. By this device of numbers, tell us the extent of his life.” With
our modern notation and use of algebra, it is rather easy to see that if Diophantus lived for x
years, then the problem is equivalent x to solving the equation x + 12 + x + 5 + x + 4 = x, which
has the solution x = 84. 6 7 2 It is hard to pinpoint when exactly Diophantus lived. The Arabian
historian Ab¯’lfaraj mentions u in his History of the Dynasties that Diophantus lived during the
time of Emperor Julian (361 - 363 A.D.), whereas Rafael Bombelli says authoritatively in his
book, Algebra, that Diophantus lived during the reign of Antoninus Pius (138 - 161 A.D.). There
is little or no con?rmation of either of these dates. However, in his On Polygonal Numbers,
Diophantus de?nes a polygonal number by quoting Hypsicles. Hypsicles was the writer of the
supplement to Book XIII of Euclid’s Elements, and so Diophantus must have written aft.
Science and contribution of mathematics in its developmentFernando Alcoforado
Mathematics is the science of logical reasoning that has its development linked to research, interest in discovering the new and investigate highly complex situations. The escalation of Mathematics began in ancient times when it was aroused the interest by the calculations and numbers according to the need of man to relate the natural events to their daily lives. Today, Mathematics is the most important science of the modern world because it is present in all scientific areas.
Please like this if you find this helpful.
Db
Jtjrhrgegdv
Xgfvtfej
Dear sister god bless 6you always stay blessed al ways you find in the presence is valued for sale me your phone number please samjo send karo aapna to karo na ji koi ni hai na to bata de mere ko jante bhi nahi bol rahi thi to nahi na to koi baat kar rahe the kya bola tha na to bol rahi rahe ki main aap se baat hi he bola tha na to ki main aap ke se par hi raheta he mere kya bola aap ki sagai ho nahi he karti hu aap to nahi ye sab batao aapne abhi kya he aapke kya he aap ka house name radha to nahi ye sab bol rahe sakte kya ho gya gaya Rajasthan se aapka kya face tha kesa chal time kayo ketla thya chhe ne ke bija sathe ne gare aaj 4try kar karo na pavan sing bhai please share your karte ho to kya kar he isliye na to bola.
b.) GeometryThe old problem of proving Euclid’s Fifth Postulate.pdfannaindustries
b.) Geometry:
The old problem of proving Euclid’s Fifth Postulate, the \"Parallel Postulate\", from his first
four postulates had never been forgotten. Beginning not long after Euclid, many attempted
demonstrations were given, but all were later found to be faulty, through allowing into the
reasoning some principle which itself had not been proved from the first four postulates. Though
Omar Khayyám was also unsuccessful in proving the parallel postulate, his criticisms of
Euclid\'s theories of parallels and his proof of properties of figures in non-Euclidean geometries
contributed to the eventual development of non-Euclidean geometry. By 1700 a great deal had
been discovered about what can be proved from the first four, and what the pitfalls were in
attempting to prove the fifth. Saccheri, Lambert, and Legendre each did excellent work on the
problem in the 18th century, but still fell short of success. In the early 19th century, Gauss,
Johann Bolyai, and Lobatchewsky, each independently, took a different approach. Beginning to
suspect that it was impossible to prove the Parallel Postulate, they set out to develop a self-
consistent geometry in which that postulate was false. In this they were successful, thus creating
the first non-Euclidean geometry. By 1854, Bernhard Riemann, a student of Gauss, had applied
methods of calculus in a ground-breaking study of the intrinsic (self-contained) geometry of all
smooth surfaces, and thereby found a different non-Euclidean geometry. This work of Riemann
later became fundamental for Einstein\'s theory of relativity.
William Blake\'s \"Newton\" is a demonstration of his opposition to the \'single-vision\' of
scientific materialism; here, Isaac Newton is shown as \'divine geometer\' (1795)
It remained to be proved mathematically that the non-Euclidean geometry was just as self-
consistent as Euclidean geometry, and this was first accomplished by Beltrami in 1868. With
this, non-Euclidean geometry was established on an equal mathematical footing with Euclidean
geometry.
While it was now known that different geometric theories were mathematically possible, the
question remained, \"Which one of these theories is correct for our physical space?\" The
mathematical work revealed that this question must be answered by physical experimentation,
not mathematical reasoning, and uncovered the reason why the experimentation must involve
immense (interstellar, not earth-bound) distances. With the development of relativity theory in
physics, this question became vastly more complicated.
c.) Algebra:
1. Babylonian algebra
The origins of algebra can be traced to the ancient Babylonians, who developed a positional
number system that greatly aided them in solving their rhetorical algebraic equations. The
Babylonians were not interested in exact solutions but approximations, and so they would
commonly use linear interpolation to approximate intermediate values. One of the most famous
tablets is the Plimpton 322 tablet, .
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
Here is something new! In our next Connector Corner webinar, we will demonstrate how you can use a single workflow to:
Create a campaign using Mailchimp with merge tags/fields
Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
But there’s more:
In a second workflow supporting the same use case, you’ll see:
Your campaign sent to target colleagues for approval
If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” button is pushed, colleagues will be alerted via Slack message
Join us to learn more about this new, human-in-the-loop capability, brought to you by Integration Service connectors.
And...
Speakers:
Akshay Agnihotri, Product Manager
Charlie Greenberg, Host
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
2. Introduction
Negative numbers were not generally
accepted until a few hundred years
ago.
Negative numbers first appeared
when people began to solve
equations.
3. Lets try a problem…
I am 18 years old and my sister is 11.
When will I be exactly twice as old as
my sister?
How would you react to that answer if
you did not know about negative
numbers?
4. Main Topics
Development of concepts of negative
numbers in…
China
Greece
India
Middle East
Europe
5. China 100BCE – 50BCE
In the “Nine Chapters of Mathematical Art”
they used red rods as positive coefficients
and black rods for negative coefficients to
explain methods for finding area of figures.
The Nine Chapters also included rules for
dealing with negative numbers.
6. Greece 570BCE – 300BCE
Greeks ignored negative numbers
completely.
Aristotle made a distinction between
numbers and magnitude, but gave no
indications of the concept of negative
numbers.
Euclid continued this distinction in his
work Elements.
7. Greece 3rd century CE
Diophantus did not deal with negative
numbers but he was aware of rules for
multiplying with the minus and solving
equations.
In book V of his Arithmetica, he
encounters the equation 4x+20 = 4
He believes that this problem is absurd,
since to him 4x + 20 meant adding
something to 20 to equal 4.
8. India 7th century CE
Brahmagupta recognized and worked with
negative numbers.
Positive numbers were possessions and negative
numbers were debts
Stated rules for adding, subtracting,
multiplying, and dividing negative numbers in
his work Correct Astronomical System of
Brahma.
Expanded on Diophantus concepts of the
quadratic equations (ax2 + bx = c, bx + c = ax2,
ax2 + c = bx) using negative numbers forming
the general form of the quadratic equations.
9. India 12 century CE
th
Bhaskara gives negative roots, but rejects
the negative root since it was
inappropriate in the context of the
problem.
“…For people have no clear understanding in
the case of a negative quantity”
10. Middle East 9th century CE
Arabs were familiar with negative numbers
from the work of India mathematicians, but
they rejected them.
Muhammad Ibn Musa Al-Khqarizimi did not use
negative numbers or negative coefficients in his
two books.
Knew how to expand products such as
(x – a)(x – b), but they only used this concept
when the problems involved subtractions
whose answers are positive.
11. Europe 16th Century
Negative numbers were still being ignored
and considered as “fictitious solutions.”
Mathematicians of this time still resisted
negative numbers and thought of them as
“fictitious” or “absurd.”
Some of the mathematicians of this time
were:
Cardano from Italy
Stifel from Germany
Viete from France
12. Europe 17th Century
Negative numbers started to become
accepted.
Along with the acceptance, came the
realization of problems with negative
numbers.
I.e. square roots of negatives
Rene Descartes partially accepted
these numbers.
15. 17th century continued…
Many mathematicians who started
accepting negatives didn’t know where
they belonged in relation to positives.
One math guy, John Wallis, thought that
negatives were larger than infinity.
Isaac Newton wrote a book in 1707
called Universal Arithmetick. In this
book he states, “Quantities are either
Affirmative or greater than nothing, or
Negative, or less than nothing.”
16. Questions for thought…
How can a quantity of something be
negative and less than nothing?
Can you have a negative quantity of
books, food, clothing, or money?
It was hard for people to grasp the
concept of negative numbers being
debt.
17. Europe Middle 18th century
Negatives are officially accepted as real
numbers!!
Euler was fine with negatives during the
writing of his book Elements of Algebra.
Even though negative numbers were known
and used, it was common for people to still
ignore them as results to equation systems.
It was still common practice to ignore a
negative results in any system of equations.
18. Europe 19th century
Negatives finally become important
enough to not be ignored.
The works of Gauss, Galois, and
Abel really had a big impact on
equation systems with negative
results.
Doubts of negative numbers finally
disappear.
19. Summary
Although negative numbers were
“discovered” in BCE, negative numbers
were not completely accepted until the
1800’s.
Still, generally, mathematicians used
negative numbers in computations, but did
not understand the concept of them.
20. Timeline
4th century BCE- Aristotle made a distinction
between numbers and magnitude.
100 BCE- In the Nine Chapters of Mathematical
Art, the Chinese used negative numbers in
solving systems of equations.
3rd century CE- Diophantus solved equations with
negative numbers in Arithmetica, but then
rejected the equation itself.
7th century CE- Indians used negative numbers to
represent debt.
9th century CE – Arabs were familiar with negative
numbers, but rejected them.
12th century CE- Bhaskara (Indian) gives negative
roots for quadratic equations, but rejects the
roots because people do not approve of negative
roots.
21. Timeline continued…
16th Century CE- European Mathematicians
thought of negative numbers as “fictitious” or
“absurd.”
17th Century CE- Rene Descartes claims the
result of negative square roots as
“imaginary.”
18th Century CE- Negatives start to become
accepted in Europe even though they are still
commonly ignored.
19th Century CE- Doubts of negative numbers
finally disappear and negatives are known
now as real numbers.
22. References
Berlinghoff, William P. , and Fernando Q. Gouvea.
Math through the Ages A Gentle History for Teachers
and Others. 1st ed. Farmington, Maine: Oxton House
Publishers, 2002.
Katz, Victor J.. A History of Mathematics. New York:
Pearson/Addison Wesley, 2004.
Negative and non-negative numbers." Wikipedia.
2006. 7 Sep 2006
<http://en.wikipedia.org/wiki/Negative_numbers>.
"Number." Wikipedia. 2006. 7 Sep 2006
<http://en.wikipedia.org/wiki/Number>.
Smith, Martha K.. "History of Negative Numbers."
University of Texas at Austin. 19 Feb 2001. 9 Sep
2006
<http://www.ma.utexas.edu/users/mks/326K/Negnos.
html>.