Rational Trigonometry and a new Foundation For Math
Posted by pwl on August 11, 2009
Two excellent video series by Norman J Wildberger. Very well done and designed to be easy to comprehend. While they start out very simple and basic they get quite deep yet remain as simple as possible. (The bonus series even gets into Relativity, very cool). Wildberger is the math teacher I wish I had all through high school and since, and now through his videos he can be our math teacher! Gotta love online education.
The first series is on the amazing new (and very old) way of doing Rational Trigonometry without using sin and cosine of Classical Trigonometry. It’s Fantastic.
“The new form of trigonometry developed here is called rational trigonometry, to distinguish it from classical trigonometry, the latter involving [cosine, sin and their fellow] functions and the many trigonometric relations currently taught to students. An essential point of rational trigonometry is that quadrance and spread, not distance and angle, are the right concepts for metrical geometry (i.e. a geometry in which measurement is involved).
Quadrance and spread are quadratic quantities, while distance and angle are almost, but not quite, linear ones. The quadratic view is the more general and powerful one. At some level, this is known by many mathematicians. When this insight is put into practice, as it is here, a new foundation for mathematics and mathematics education arises which simplifies Euclidean and non-Euclidean geometries, changes our understanding of algebraic geometry, and often simplifies difficult practical problems.
…
Rational trigonometry deals with many practical problems in an easier and more natural way than classical trigonometry, and often ends up with answers that are demonstrably more accurate. In fact rational trigonometry is so elementary that almost all calculations may be done by hand. Tables or calculators are not necessary, although the latter certainly speed up computations. It is a shame that this theory was not discovered earlier, since accurate tables were for many centuries not widely available.”
Exerpt from the Introduction to “Divine Proportions: Rational Trigonometry to Universal Geometry” by Associate Professor Norman J Wildberger
Here are a few videos from the Rational Trigonometry series with the full series of 46+ episodes here. These first episodes lay the corner stone of Rational Trig, and it only gets better from there!!! Watch them all a number of times and do the examples. Amazing!
“Why classical trigonometry is hard:
The problem is that defining an angle correctly [in Classical Trigonometry] requires calculus [whereas Rational Trigonometry doesn’t]. This is a point implicit in Archimedes’ derivation of the length of the circumference of a circle, using an infinite sequence of successively refined approximations with regular polygons. It is also supported by the fact that The Elements [Euclid] does not try to measure angles, with the exception of right angles and some related special cases. Further evidence can be found in the universal reluctance of traditional texts to spell out a clear definition of this supposedly ‘basic’ concept.” – NJ Wildberger
“Quadrance measures the separation of two points. The easiest definition is that quadrance is distance squared.
… quadrance is the more fundamental quantity, since it does not involve the square root function. The relationship between the two notions is perhaps more accurately described by the statement that distance is the square root of quadrance.
In diagrams, small rectangles along the sides of a triangle indicate that quadrance, not distance, is being measured … .” – NJ Wildberger
“Spread measures the separation of two lines. This turns out to be a much more subtle issue than the separation of two points.” – NJ WildbergerThe spread s between the two lines is the ratio of quadrances. The spread also works out to “square of sine of angle”.
“The spread between two lines is a dimensionless quantity, and in the rational or decimal number fields takes on values between 0 and 1, with 0 occurring when lines are parallel and 1 occurring when lines are perpendicular. Forty-five degrees becomes a spread of 1/2, while thirty and sixty degrees become respectively spreads of 1/4 and 3/4. What could be simpler than that? Another advantage with spreads is that the measurement is taken between lines, not rays. As a consequence, the two range of angles from 0◦ to 90◦ and from 90◦ to 180◦ are treated symmetrically.” – NJ Wildberger
You can find a couple freely available pdf chapters of the book “Divine Proportions: Rational Trigonometry to Universal Geometry” here. Make sure to check out Divine Proportions Overview Chapter 1 as it covers the basics with examples.
You will find some additional pdf papers by the professor here.
The second series is on the Foundations of Math. An excellent review and introduction for everyone at any age.
Here are a few videos from the Foundations of Math series with the full series of 35+ episodes here.
There is also a series of advanced Math Seminars on the topic of Hyperbolic Geometry is Projective Relativistic Geometry.
Rational Trigonometry | Methodikon said
[…] Rational Trigonometry: The new form of trigonometry developed here is called rational trigonometry, to distinguish it from classical trigonometry, the latter involving [cosine, sin and their fellow] functions and the many trigonometric relations currently taught to students. An essential point of rational trigonometry is that quadrance and spread, not distance and angle, are the right concepts for metrical geometry (i.e. a geometry in which measurement is involved). […]
Word Press backlinks said
Word Press backlinks…
[…]Rational Trigonometry and a new Foundation For Math « Paths To Knowledge (dot NET)[…]…
マークバイマークジェイコブス ショルダーバッグ said
http://www.chbigstock.com/%E3%82%AF%E3%83%AD%E3%83%A0%E3%83%8F%E3%83%BC%E3%83%84%E3%81%9D%E3%81%AE%E4%BB%96-category-53.
html
Excellent post. I was checking continuously this blog and
I am impressed! Very useful info specifically the last part 🙂 I care for such info a
lot. I was seeking this particular information
for a long time. Thank you and best of luck.
louis vuitton outlet said
Fabulous, what a web site it is! This web site provides useful information to us,
keep it up.
produce music online said
A motivating discussion is definitely worth
comment. I believe that you need to write more on this subject, it might not be a taboo subject but typically folks don’t discuss these subjects.
To the next! Best wishes!!
phen375 fat burner said
phen375 fat burner
Rational Trigonometry and a new Foundation For Math « Paths To Knowledge (dot NET)
the linden method the anxiety disorder said
It reduces fats and carb absorption, reduces
urge for food, stimulates the metabolism promoting
fats burning and offers you extra vitality.
Individuals who have underlying medical condition should
also consult a physician before taking this
diet pill. With more time now passed, those
rates would be even higher as obesity is still on the rise.
anxiety medication said
Such are the ingredients of Adiphene that its one finest selling level is the shortage of
dangerous uncomfortable side effects. There are
various surgical methods to combat obesity. Different
studies have demonstrated the profits of Ginseng Panax Root Extract.
how to treat panic attacks said
Such are the ingredients of Adiphene that its one finest selling
level is the shortage of dangerous uncomfortable side effects.
There are hundreds to choose from, allowing you
the choice of matching the type with your own health and fitness goals.
Different studies have demonstrated the profits of Ginseng
Panax Root Extract.
ringing in ears said
If you want to know the crystal clear details it is better to view just about any online store web sites
explaining the actual product or service information using elements.
Individuals who have underlying medical condition should also consult
a physician before taking this diet pill. With more time now passed, those rates would
be even higher as obesity is still on the rise.
adjustable dumbbells cheap said
You wont find Adiphene in your native pharmacy or
some huge store like Walmart or Walgreens. Safety is also assured when taking Adiphene
because of its natural ingredients, which means that serious side effects
are unlikely to happen if taken as directed.
There is a section to discuss the biggest loser show, specific diets, and much more.
best product for acne said
If you have been feeling tired and rundown lately, and seem to be gaining weight for no apparent
reason, then it is time for a change. Perhaps the main reason behind this though is that you
have developed a dependence on food. Sonnie Mc – Lemore is a health and fitness blog owner.