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[tutoroot]

What are the Applications of Pythagoras Theorem? 

The Pythagoras Theorem is a fundamental principle in geometry. It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Named after the ancient Greek mathematician Pythagoras, this theorem is one of the earliest mathematical discoveries and is still widely used today.  

Brief History and Origins  

The theorem dates to 570-495 BCE, attributed to Pythagoras, a Greek philosopher and mathematician. However, evidence suggests that civilizations like the Babylonians and Indians knew about the relationship long before Pythagoras formally proved it. Despite its ancient origins, the theorem’s practical value remains timeless.  

Understanding the Key Terms  

To grasp the theorem better, it’s crucial to understand the key terms involved:  

Hypotenuse: The longest side of a right triangle, opposite the right angle.  

Perpendicular: The side forming the right angle, perpendicular to the base.  

Base: The side forming the right angle is usually considered the bottom side.  

Visualizing these terms in a right triangle makes it easier to see how they relate to each other.  

Geometric Proof of Pythagoras Theorem  

One of the simplest proofs of the theorem involves geometry:  

Construct two identical squares, each containing four identical right triangles.  

In the first square, arrange the triangles so that their hypotenuses form a smaller square in the center.  

In the second square, rearrange the triangles to form two smaller rectangles.  

Algebraic Proof of Pythagoras Theorem  

For a more mathematical approach:  

Consider a right triangle with sides a, b, and c.  

Create a square with side length a+b containing four identical right triangles.  

Calculate the total area of the square in two ways:  

Applications of Pythagoras Theorem  

The theorem has countless practical uses, including:  

Architecture and Construction: Ensuring structures are level or calculating diagonal distances.  

Navigation: Determining the shortest path between two points.  

Surveying: Measuring inaccessible distances indirectly.  

Its applications extend to science, engineering, and even art!  

Fun Facts and Historical Insights  

Pythagoras founded a philosophical school, where numbers and geometry were seen as divine.  

Variations of the theorem exist in other cultures, like the Baudhāyana Sulba Sūtra in ancient India.  

Over 400 unique proofs of the theorem exist, showcasing its universal appeal.  

The Pythagoras Theorem is more than a mathematical principle; it bridgesit’s a bridge between abstract theory and practical application. From solving everyday problems to advancing scientific discoveries, its significance cannot be overstated. Whether you’re a student, architect, or engineer, understanding this theorem equips you with a tool that transcends time and discipline.  

The Pythagoras Theorem, a cornerstone of mathematics, establishes the relationship between the sides of a right triangle. Understanding such fundamental concepts is easier with Tutoroot online Maths tuition, where expert tutors provide interactive, personalised learning guidance. Unlock your mathematical potential and master topics like Pythagoras Theorem with ease! 

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Info about applying to PhD programs in pure math

This is... basically what it sounds like. I’m sure a lot of this is applicable to other PhD applications but I’m only very familiar with pure math. This is aimed at current seniors about to apply, but there is a section about prepping for applications in advance.

The highlights:

Recommendation letters are the most important thing. Most schools ask for 3ish. Try to get people who know you well, not just a student in the class. Someone you've conducted research with and one from a different institution are ideal if possible

Ask for rec letters at least a month before the due date is a good rule of thumb.

Research experience is probably the second most important.

Get the opinion of multiple professors who know you in order to build a list of potential schools. Then widdle it down to your will-actually-apply list (probably 8-14 ish). 

My opinion but please apply to at least 3-4 safety/match schools. Even when you're fully qualified, acceptance rates are simply low enough that a bit of bad luck means getting rejected or waitlisted from a few of them.

Most pure math due dates are in early-mid Dec but a few schools are in Nov and some are as late as mid-Jan. 

Schools will generally have their own graduate application portals. Some are better organized than others. Some require you to submit all your material before you can send a request for submitting rec letters so plan accordingly.

Acceptance letters will very slowly start going out in mid-Feb but the vast majority of programs won't send out anything until like, March and not be done until later than that. Accordingly, wait until at least mid-March to begin freaking out if you haven't been accepted anywhere.

You should 100% be expecting a tuition waiver and stipend from a program if you're applying for a PhD.

The rest of the posts is.... ridiculously long so I’m putting it under a cut. I mention things to do in advance to help you decide if grad school is right for you and things that make your application look good, give a full time line of the process, a list of things applications commonly ask for, and some miscellaneous notes. (The points above are repeated in more detail).

In addition, some links to other resources math students may appreciate:

an old post of mine about grad school apps (overlaps a lot and features some ranting from during the application process)

about REUs including my addition specifically about math ones

summer programs for undergrads that aren’t REUs by @counter-example and @jungleuniversity

Tips for prospective grad student visits 

Also about prospective grad student visits by @thisurlhasbeenleftasanexercise

Also for context, I went to a large state school in the US for undergrad. I started as a CS major and added on math as a secondary major after my first year and dropped CS during third year. I’m primarily interested in discrete and algebra, though I have a significant topology background from undergrad too. I got most of my advice from people around the department, as I became pretty involved during my third year. Now, I’m a first year grad student at another large state school in the US, generally considered pretty decent though not a “top math program” at all. Not that much else has happened so far.

Things in advance (aka things to help you decide if grad school is for you and things that look good on an application)

Take the standard classes. For pure math, this is at least one semester of linear alg, abstract alg, and analysis each. Linear and analysis are also good for applied math but I'm not sure what else if anything is considered standard.

Take some grad classes if you have the option. Most people are not ready for this until senior year, but some do manage as juniors. Talk to people who know you well and the prof teaching the class before you do this though.

Try to get involved with research whether this is through independent studies at your home institution, REUs, internships, or other stuff.

Be involved in your department. This helps with getting you more personalized advice for applying.

The rough suggested timeline (assuming junior yr is your second to last year and senior is your last of undergrad)

Junior April: Take the math subject GRE so you can take it again in Sep or Oct if desired (perhaps not applicable atm). The general can be taken kinda whenever; I suggest fall of senior year.

Junior April/May: Start talking to professors/post docs/mentors/etc. about programs you may be interested in. Write/type it down. Don't worry if it gets long, you will shorten again later.

Summer: Do some research if possible; an REU or research at your institution (if an REU, also get your mentor's opinion on potential schools towards the end as well)

Senior Sep: Start whittling down your list. 8-14 seems to be the "normal" range of schools to apply to but some people panic and do more. Remember that asking for waivers is completely acceptable but applying is still just generally expensive (I spent around $800 for 10 schools)

Senior Sep: Apply for the NSF GRFP. You can apply as an undergrad senior and once during your first or second year of grad school if you didn't already get it. The due date is in mid-late OC but ideally you'll have a draft of your essays and ask for rec letters by the end of Sep, if not earlier.

Senior early Nov: Ask for rec letters if you haven't already. The rule of thumb is a month before the due date. Provide them a list of schools you want to apply to including due date and where/how to submit as soon as possible (as well as anything else they request of course; many ask for a resume and a draft of your personal statement).

Senior Dec-Jan: Submit stuff! Pure math programs typically have deadlines in Dec or early Jan. I think the big days are Dec 10th, Dec 15th, and Jan 15th but some are earlier or later. (applied math masters tend to be earlier I think; in Nov). I suggest putting them all into a list or calendar. In addition, some schools won't let letter writers submit until all of your stuff is submitted so start applications early, even if you don't finish them immediately.

Senior Feb: Programs will slowly start sending out offers in early Feb and pick up in mid Feb, but don't fret until AT LEAST the beginning of March! Grad programs are just way too slow at getting out offers for it to be worth worrying until then (and even then, it's definitely not time to panic but mathematicians are frequently anxious people so I get it). Waitlists are slower to come out; usually starting in early March. Also note, there are many programs that don't actually send out replies to everyone unfortunately.

Senior late Feb-early April: prospective student days! They might be online in 2021 unfortunately but try to attend whatever form they're in if you can (only one of my visits during spring 2020 was online since the others happen to be very early and safely beat covid in the US). Be warned, it's very possible to get offers of admissions and to visit very last minute. I do not have advice for how to make that less stressful.

Senior April 15th: Common reply deadline. If you got your offer in the first round or two, this is probably your deadline to accept. In addition, this means more offers will likely come out shortly after once more people have declined. 

Senior summer: graduate. Send a completed, official transcript to your new institution. Check your new email account for stuff you're suppose to do. Some programs have some sort of program during the summer for in-coming students. Most places have graduate student training of some sort for a week or two before semester starts. 

Some common things to be asked for in applications

Not actually a thing asked for but many graduate schools have their own portal for which you will have to make an account to submit an application. A few use a common system that kinda sort shares a database of accounts? Some are fine and some massively suck.

Personal Statement/Statement of Purpose: Occasionally called something else and once in a while actually separate things; will usually have a prompt of wildly differing specificity. Sometimes, the prompts come from the department itself and sometimes from the university's graduate school. I suggest having one or two "base" essays then tweaking them for each school. Sometimes a word/page limit is specified but if it's not, around 2 pages/1000 words is pretty reasonable.

Transcript. Some accept unofficial but some require official but generally not an unsealed one. I ordered myself one official transcript and sent it to multiple schools instead of paying for them to be sent to each school during the application process.

Resume or CV: Most ask for either a CV or is fine with either, in which case I give them my CV. I sent more or less the same one everywhere.

Some other notes

Yes, ask for application waivers. Just be polite about it.

Your goals for your essays are primarily to show that you're interested in math and math research and are capable of like…. writing things that make sense

Do not start out an essay with either "I loved math since I was little" or "I actually didn't like math when I was young" or any variations of those. (I had one essay that started with a mildly humourous anecdote from undergrad combinatorics and another that talked about how my undergrad department has greatly affected me).

You should 100% expect to get a tuition waiver and living stipend as part of a TA fellowship (or more rarely, an research fellowship) as part of your offer of acceptance for a math PhD program (pure or applied). Health insurance is also frequently part of the package. This is not true of masters programs unfortunately.

How schools do waitlists depend wildly though most don't have super long ones like prestigious undergrads do. If you're still interested in a place you're waitlisted at, follow their instructions to confirm your placement on the waitlist then wait until April before following up again, expressing your continued interest and asking for an update. You might even want to wait until around the common deadline, April 15th. The number of people who declined before April is just really really low so nothing really happens until then.

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Are You Seeking Help With Linear Algebra Problems? You’re At The Right Place!

We all have faced multiple problems with maths examinations at some point in our life. Linear algebra is not new and students face similar problems with this topic as well. Whether it is elementary algebra, pre algebra or even abstract algebra, it is common for every student to face problems with mathematics. This is where they look up for the best maths teacher near me on the internet.

Linear algebra problems can be pretty challenging for students as it can become challenging to visualise. It is simple for the student to get demotivated when there are concepts or problems that they do not understand. But the problem with linear algebra can easily be defeated with the help of the best maths tutor in Kolkata.

Understanding The Signs

Initially, it is essential to understand when to look for help with linear algebra. If you have hit the subject of linear algebra in particular, and keep coming to teachers for algebra help, you might be having problems understanding the concept entirely. If that is the situation, looking for a maths tutor near me and understanding the problem can help you look at things in a different manner.

You all have your weaknesses and strengths when it comes to education and people look at every problem differently. Some students consider their strength to be Social studies while others have their strength in solving complex algebra equations. Most of the students have experienced challenging movements when they are unable to solve a particular problem in an examination due to the lack of understanding. This is where the best maths tuition professional can help you.

Other than that, it is essential to understand what your needs are. If you are struggling, ask yourself how you can get help. The best would be looking for a maths tutor in Kolkata. Algebra can be essential as it forms the base of future mathematics and will be referred to often in higher studies. Therefore, mastering linear algebra can help you in the future studies as well.

How To Get Help For Linear Algebra?

Algebra is usually used as a byword for mathematics that no one likes. It does not need to be that way and for solving linear algebra, there are particular things that you can do for helping to understand mathematics in a better way.

Students often learn differently then elders and even if the elders try to help the students sometimes they still would not understand the subject which can lead to lack of motivation and frustration, particularly with algebra. An algebra maths tutor in Kolkata not only assists and helps with subjects like algebra but also helps in inspiring the student. Students can see immediate improvement and with online classes developed around the schedule of students, there will be no problem in planning to drop or pick up kids from after school programs.

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Maths tutors can identify the needs of a student including any concepts or shortcoming that they do not understand from the previous years. Even algebra and mathematics in particular, forms the base of future mathematics that a student will used throughout their education journey, so it is additionally essential for students to have a solid understanding of algebra and mathematical concepts. But, if a student cannot form a solid foundation, teachers will work on solidifying the previous knowledge first.

The schedule of online tutors are structured around the schedule of students. It indicates that there will be no hassle for students to be somewhere physically and they can choose any time or day that works the best for them.

Maths tutor in Kolkata can make learning fun and help students combat mathematics anxiety and start things in an interesting way. Rather than memorising linear algebra problems and equations that may have no meaning to the student and seem abstract, the best maths tuition professional will talk in terms that the students can easily understand.

There is no need to worry, at the end of the day, if a student is having problems with mathematics all linear algebra. Mathematics tutors can work alongside common standards and can identify the strengths and weaknesses of students and develop confidence so that they can solve linear algebra in an effortless manner. A student will be full grade ahead of where they are now in only 6 months with a professional maths tutor in Kolkata.

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Traits of a Successful Maths Tutor

Finding a Maths tutor in Sydney can be difficult due to the wide range of math studies available, from algebra to trigonometry to calculus. All of these math fields necessitate specific skills that not all tutors may possess. This is where looking for a tutor online can help. By searching for a tutor online, one can ensure that the tutor is qualified to teach the subject matter.

Online tuition makes it possible to teach students about a variety of subjects using e-learning products and materials. Academic assistance to school students, test preparation, and application assistance are the main services provided. Maths tutoring northern beaches aids students' learning by assisting them in developing conceptual clarity, applying these concepts in problem-solving by extensively practicing tests, and monitoring performance through regular assessments administered by the pedagogy.

Find the best online tutor for your math problem only at Contemporary Coaching College

Technology is extensively used to provide the best online tuition services to its students. Technology is just one example of how online tuition can help students improve their learning, performance, and overall development in a short time. Every such online tuition provider optimizes its technology and resources to become the country's leading online tuition service provider.

To use online tuition effectively for a student's overall learning needs, a holistic approach is required. To enable a student to achieve excellent academic results, various forms of teaching are used. These teaching styles can range from collaborative learning to fun activities that encourage active participation, as well as individual mentoring and guidance for customized instruction.

Maths tutoring Roseville made tuition more effective by providing its tutors with the tools and materials they need to provide the best education to their students. To assist tutors in instructing and enabling effective problem-solving, technology-led solutions are used. It assists tutors in addressing some of the following challenges in their daily teaching practices:

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Education is really important in today's society, but how do we understand the amount of education to which we could say it is of the maximum quality?

It can be hard for us to determine, especially if we are talking about our kids and the kind of education we need them to have. When someone graduates with a diploma, it's essential to finish the post graduate level to pursue"higher" learning? Or is it enough that they have the ability to leave college with a diploma in hand?

These are simply a few of the questions which we would love to ask ourselves. There are situations wherein a man or woman isn't even a graduate of a respectable university, and he receives all of the accolades, then you've got a post graduate degree holder who is having trouble finding the ideal career for him. There are no guarantees that the amount of education we achieve defines who we are as a person; in actuality, it's not the amount of education that's important. What matters is the quality of the learning that we could achieve.

Irrespective of the amounts that our kids have achieved from his youth up until the time he leaves the universities, we place lots of worth not on what was taught, but on what he learned.

In certain areas, degree matters. And it matters a lot. Not that we blame themin specific sectors, it's important for us to have the ability to learn the rudiments and jargon, particularly for very technical positions. But in the general international situation, the perception of getting post graduate studies as a qualification might have to be modified.

The actual test of a fantastic employee wouldn't be how he passed a particular certification examination in the classroom, however he can apply what he learned in college to the job in hand. To be the best man for the job means understanding how to do the job, not always possess the degree for this.

If you're parents and are considering the future of your kids, you want to be certain that you concentrate on quality as opposed to simply a degree. Investing in your children's education will let you reap its benefits in the long term.

Though we understand that an uneducated child may struggle in finding the ideal job for them, someone who was raised well by their parents with a fantastic quality education will help them achieve a successful career.

Where to Get the Best Free Education Online

Going out to school is cool but it can also make you flat broken. Rent, books, meal - the faculty expenses list can go endless.

But if you are not that level freak, check out some of the greatest MOOCs (Massively Open Online Course) found at no cost. Bear in mind that they provide free courses online in the kind of video podcasts, audio lectures, and written transcripts, but NOT school credits or degrees.

The Harvard Extension School's Open Learning Initiative made accessible the eight complimentary Harvard courses on video format. You don't need to register simply to see the lecture videos. Sets, Counting and Probability, Abstract Algebra, Intensive Introduction to Computer Science are a few of noncredit courses that are open to the public at no cost and are taught by distinguished Harvard faculty. Read their over 600 evening and online courses, many offered for credit and at an affordable tuition rate, if you're interested in researching what else Harvard Extension School offers.

MIT OpenCourseWare (OCW) online free open publication of MIT course content and materials that mirror almost all undergraduate and graduate courses taught at MIT. Because OCW isn't a credit-bearing or degree-granting initiative, there's absolutely no registration or enrollment procedure. You may work through the materials at your own pace, and in whatever manner you desire.

Supported by a grant from William and Flora Hewlett Foundation (yeah, those supporting HP computers), Open Yale Courses (OYC) provide their notable lectures and other educational materials from some of their collegiate Yale courses free through internet. OYC covers the wide assortment of liberal arts areas, such as social sciences, humanities, and biological and physical sciences.

Each course has a set of class lectures created in top notch video in addition to other course materials like syllabi, problem sets, and suggested readings. The lectures can be found in downloadable movies, audio-only edition, and written transcripts. 

Taught by routine Stanford faculty, Stanford Free Online Courses are especially interactive and envisioned to keep Stanford's leadership in providing excellent education to its students and to the world. Like other MOOC, you do not get credit for your work, but with Stanford, you will be given a statement of accomplishment in case you successfully completed a course. The courses are offered on a lot of different delivery platforms such as Coursera, Class2Go, and Venture Lab.

Free online courses solve lots of other needs which can't be managed by regular schools and aren't meant to replace formal amounts at a respectable degree-granting school.

Jules Mariano is a fulltime freelance search engine and social networking marketing specialist and also a web developer specializing in web content creation. He's the co-founder of VPRO Digital Marketing. He currently writes for BookGator.

[academyofengineers]

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