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Pythagoras Theorem (Formula, Proof And Examples) PDF Free Download, Overview, Formula, Examples, Practice Problems, Videos, Calculators, Worksheets.
When A Finding Is As Ancient As The Pythagorean Theorem, Which Has Been Around For At Least 4,000 Years, You Would Anticipate That There Wouldn’t Be Anything New To Say About It. There Is Always More To Learn, And In This Instance, It’s Something That Most Mathematicians Had Written Off As Impossibly Difficult For Decades, As Two High School Kids From New Orleans May Now Have Shown.
The Abstract Of The Talk Given By Ne’kiya Jackson And Calcea Rujean Johnson At The American Mathematical Society Spring Southeastern Sectional Meeting On March 18, 2023, States, “In The 2,000 Years Since Trigonometry Was Discovered It’s Always Been Assumed That Any Alleged Proof Of Pythagoras’s Theorem Based On Trigonometry Must Be Circular.”
But Some Say, “That’s Not Quite Right. In This Lecture, We Give A Novel Demonstration For Pythagoras’ Theorem That Is Based On A Key Trigonometric Finding, The Law Of Sines, And We Demonstrate That The Argument Stands Alone Without The Help Of The Pythagorean Trig Equation Sin2x + Cos2x = 1.
This Finding Has Generated A Lot Of Discussion Among Mathematicians. Two High School Students Were Able To Come Up With This, Claims Youtuber Mathtrain In A Recent Video On The Evidence. “I For One Am So Excited That Two High Schoolers Were Able To Come Up With This,” He Adds. In An Area As Established As This Theorem, New Ideas Are Rare, So I’m Eager To Read The Complete Work When It’s Published.
What What Is It About This New Proof That Has Mathematicians So Enthused? Let’s First Review What The Pythagorean Theorem Really States: Theorem States That Given A Right-angled Triangle With The Longest Side Labelled C And Two Sides Labelled A And B,
This Tiny Truth Has Been Shown Literally Hundreds Of Times Throughout The Years, From Simple Geometrical Examples To Computations Involving Differential Equations Or Complex Numbers.
However, Very Few Of Those Arguments Depend On Trigonometry, Which Is The Study Of Angles And Length-to-length Ratios In Geometric Forms. There Is A Valid Explanation For It: Because The Pythagorean Theorem Serves As The Foundation For The Majority Of Trigonometrical Laws, Any Demonstration Of That Theorem Would Logically Ask The Issue Rather Than Truly Answering It.
However, As Johnson And Jackson Note, There Are A Few Exceptions To That Rule, One Of Which Being The Phenomenon Known To Mathematicians As The Law Of Sines. The Law Of Sines States That If We Refer To The Angle Opposite Side An As Angle A, Angle Opposite Side B As Angle B, And Angle Opposite Side C As Angle C
You Don’t Need Anything More Complex Than Elementary Geometry To Establish This Equation, Despite The Fact That It Is Fundamentally A Trigonometrical Connection (The Sine Function, After All, Is Defined As The Ratio Between Two Sides Of A Right-angled Triangle). Thus, It May Be Utilised As The Foundation For Argument Without Posing Any Logical Conundrums Since It Is True Regardless Of The Pythagorean Theorem.
So Far, So Good, But How Precisely Did The Two Aspiring Mathematicians Go On From This Point? Unfortunately, The Proof Has Not Yet Been Published Because It Was Presented As A Presentation Rather Than A Paper. Fortunately, However, A Local News Station That Covered The Girls’ Accomplishment Aired Some Of Their Slides, And Eager Online Mathematicians Jumped At The Chance To Figure Out The Potential New Proof For Themselves.
You Can Really Reassemble The Proof Rather Quickly By Zooming In On The Slides That You Can See In The Movie, According To Mathtrain. It Just Took Me An Hour, And It Was Enjoyable.
The Evidence Mathtrain Goes On To Provide Is Both Clear And Lovely. In Order To Begin Proving The Pythagorean Theorem, We Must First Create A Right-angled Triangle And Identify Its Sides And Angles. So Far, So Commonplace.
The Most Fascinating Part Comes After That. We Increase The Number Of Triangles In The Figure By First Adding The Mirror Image Of The Original Triangle To One Side And Then Extending The Hypotenuse Of This Mirror Triangle Until It Intersects The Original Hypotenuse With A Line.
A New Right-angled Triangle With Sides Measuring C, X, And Z Is The End Product. But The Little Figures That Make Up The Larger One Are What Hold The Key To The Evidence.
The Next Step In The Construction Is To Split The Large Right Triangle Into An Endless Number Of Smaller Ones That Are All The Same As The Original One We Drew, According To Mathtrain. Johnson And Jackson Observe That As You Go Down This Endless Fall, The Ratio Between Each Triangle Increases Is A/b.
From This, We Can See That The Lengths X And Z Are Just The Sums Of These Subsequent Triangles’ Hypotenuses. And Although It Would Be Fantastic, There Are Obviously An Endless Number Of Them To Add Up To.
So, We Immediately Enter The Realm Of Infinite Series, Specifically Infinite Geometric Series. Fortunately, They Are Well-researched And Very Straightforward To Manage, And A Simple Formula Informs Us The Lengths We Need: We Discover That
It’s Clear That Both Equations Are Very Close To One Another – “So Similar, In Fact, That If We Take Their Quotient, These Two Factors Will Cancel Out, And We’ll Be Left With Just 2ab/a2 + B2,” According To Mathtrain.
Finally, The Law Of Sines Is Put Into Action. By Applying The Law Of Sines To The Triangle Formed By Our Initial Right-angled Triangle And Its Mirror Image, We Can See That Sin(2a) Is Also Equal To 2ab/c2. This Is Because The Sin Function’s Definition States That X/z Is Equivalent To Sin(2a).
The Result Is Something That Already Has A Disturbing Resemblance To The Pythagorean Theorem:
According To The 2,000-year-old Pythagorean Theorem, The Square Of The Hypotenuse, The Third Side Opposite The Right Angle, Is Equal To The Sum Of The Squares Of The Two Shorter Sides Of A Right Triangle. The Theorem’s Accompanying Notation, A2+b2=c2, Is Something That Is Covered In Many Geometry Classes.
Mathematicians Have Long Held That Any Purported Trigonometric Pythagorean Theorem Proof Would Fall Victim To The Circular Reasoning Logical Error, Which Is When One Tries To Prove One’s Own Argument.
The Pythagorean Proposition By Elisha Loomis, Thought To Contain The Largest Known Collection Of Proofs For The Theorem, “Flatly States That “There Are No Trigonometric Proofs Because All The Fundamental Formulae Of Trigonometry Are Themselves Based Upon The Truth Of The Pythagorean Theorem,” According To Johnson And Jackson’s Abstract For Their Talk On March 18 In Atlanta, At An Event That Drew Presenters From Prestigious Universities.”
However, Johnson And Jackson Claimed That They Were Able To Demonstrate Pythagoras’ Theorem Using The Trigonometry Rule Of Sines In A Method That Was “Independent Of The Pythagorean Trig Identity Sin2x+cos2x=1” – Without Using Circular Reasoning.
A Local Television Station, Wwl, Covered The Results In A News Release From St. Mary’s. Other Media Sites Ran With The News, But Detractors Rushed To Question If Johnson And Jackson Had Indeed Accomplished What Many Accomplished Mathematicians Had Failed To Achieve.
Johnson And Jackson Did Not Seem To Have Broadly Disseminated Their Evidence As Of Friday. Only That The Two Should Submit Their Findings To A Peer-reviewed Publication Was Made Public By The American Mathematical Society. However, A Youtube User Named Mathtrain Was Able To Recreate The Proof Using The Johnson And Jackson Presentation’s Slides, Which Were Made Available In The Wwl Report.
In His Own Video, Lozano-robledo Dissected Mathtrain’s Reconstruction And Came To The Conclusion That The Kids Had Indeed Done What They Claimed.
He Summarised The Proof’s Use Of “A Fractal Of Similar Triangles” And “Infinite Series” To Calculate The Forms’ Sides In A Subsequent Video.
It’s Very Clever, Remarked Lozano-robledo. “The Proof Is Just So Lovely And Elegant In Itself.”
However, Lozano-robledo Added That It Was Permissible For Individuals To Provide At Least One More Noncircular, Trigonometric Demonstration Of Pythagoras’ Thesis.
In 2009, Jason Zimba Proved That Sin2x+cos2x=1 Could Be Obtained Independently Of The Pythagorean Theorem, However He Adopted A Different Approach. Zimba Was Then At Bennington College In Vermont.
Lozano-robledo Said In The Text That Accompanied His Film That It Was Not Johnson And Jackson’s Fault That People Believed They Were Claiming To Have Accomplished Something That Had Not Been Done In More Than 2,000 Years. He Said That The Pupils’ Abstracts Did Not Include It.
In The 2,000 Years Since Trigonometry Was Created, It Has Been Accepted That Any Purported Demonstration Of Pythagoras’ Theorem Based On Trigonometry Must Be Circular, According To Johnson And Jackson’s Abstract.
After That, St. Mary’s Academy Released A Press Release That Began With That Phrase And Included The Phrase “Unprecedented Research” In The Title.
However, Lozano-robledo Made It Obvious That Based On All Available Evidence Johnson And Jackson Had Produced A Reliable New Pythagorean Theorem Proof, An Achievement That Merited Praise. Two Toddlers In The Eastern Part Of New Orleans Have Reportedly Uncovered A Finding That Many Researchers Have Long Hoped To Make. The Eastern Part Of The City Often Makes The Headlines For Reasons Like Lethal Violence Or Political Corruption.
According To Lozano-robledo, Sometimes New Proofs Provide Mathematicians With Insights That May Be Used In Other Contexts Or Provide Researchers A Tool To Prove Other Theorems That Were Previously Assumed To Be Unprovable. He Said That Encouraging Creative Thinking Is One Of The Key Goals Of Imparting Ingrained Information.
Lozano-robledo Remarked, “It’s Just So Energising — It Makes Me So Happy To See Teenagers Thinking About New Maths And Coming Up With Something Really Important And Fantastic.” It’s Just Wonderful.
According To Jackson, Johnson’s Argument Was Conceived Owing To A Maths Competition Scheduled For The Christmas Break, According To An Interview With Her That Was Published On Thursday By The Times-picayune.
Jackson Said, “We Were The Only Kids That Responded To The Bonus Question. It Posed A Lot Of Questions About How You Would Go From One Step To The Next And Required You To Provide A Proof For The Theorem.
Jackson Said That She Intended To Pursue A Career In Anaesthesia Or Pharmacy In College. Johnson Declared Her Desire To Be An Environmental Engineer. She Has Received More Than 25 Acceptance Letters And Has Received Over $1.2 Million In Scholarship Offers.
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