WildTrig19: Trigonometry with finite fields (I)

An introduction to finite fields, based on first understanding rational numbers. This will be the basis of extending geometry and trigonometry to the finite setting.

Category:

Science & Technology

Tags:

• rational
• trigonometry
• finite
• fields
• geometry
• fractions
• wildberger

[mathdanielusa]

Wildberger,

Thanks for the video. So 2/3 = 6?! Can you

explain a lttle more about mod7. Is this similar to Base7? When subtracting 7 from 12/21 to get 5/14 and ending up with an integer- is that right? Fractions that become integers-is this linear algebra? I have not had that yet. Thanks.

[njwildberger]

When working in F_7, you set 7 equal to 0. That is the essential rule---otherwise you do arithmetic as usual. So 2/3=9/3 because we have added 7 (actually 0) to the numerator. But then 9/3 is equal to 3/1 as usual, and so we conclude that 2/3=3. As for 12/21, that would be undefined in F_7, because the denominator is zero.

[danieloreto]

How can the problem of applying RT generally be solved if all the problems we might need to solve are given in terms of angles? I also like to thank you for sharing this approach to trigonometry, knowing new methods its always motivating!

[njwildberger]

If you know an angle x, and want to find the corresponding spread s: s is the square of sin x. That means that x is the square root of the arcsin of s.

[danieloreto]

,what about the enormous variety of phenomena this functions describe? what about the role this functions play in complex number representation, calculus development, vectors, etc. Also, i think that an approach to trigonometry based on the ratios of the sides of a triangle is old fashioned and can be taken as a special case of the general definitions of the trig functions in a unit circumference.

[njwildberger]

Circular functions are important in advanced mathematics. They are just not that important in trigonometry---the study of triangles. When circular motion is involved, so will be the circular functions, including harmonic analysis, complex analysis, parts of physics.

But not triangle and elementary geometry.

Try to keep circles and triangles separate in your mind. We do not need a theory that covers both at once.

[supermanifold]

strictly speaking, you should be identifying F_7 as the set of residue classes modulo 7

from here, you can construct its field of fractions

which, of course, is isomorphic to F_7 itself

though I understand your motivation for doing all this, I think this little excursion into finite fields is completely misleading, especially for someone viewing this video with little or no background in abstract algebra

-supermanifold

[danieloreto]

I repeat it again. Its a GREAT ACHIEVEMENT ti solve triangles without trig functions, but i've been thinking about it and the people (i include myself) which want to use RT to solve the many aplications of triangle solving, how can we use this method when angle measures are given? i mean, we are just able to apply RT when we know at least the length of two sides of the triangle, which not always is possible. (we maight just be given a side and an angle).So, I'd like to ask you

[danieloreto]

Hey, excellent series!! I like the rational approach you give to triangle resolution, as you said, it allows us to sove triangles without needing any trascendental function. I agree with you when you say that the 90/45/45 and 90/60/30 are the only triangles most students are familiar with. But in my opinion, your comment about replacing "classical" trigonometry with rational trigonometry is somewhat difficult. Eventhough is a great achievement to sove triangles without trig fuunctions

[njwildberger]

Hi---Thanks. Most of the video is elementary, but my take on finite fields is non-standard, stressing the importance of thinking in terms of FRACTIONS. This allows us to calculate more smoothly, without having to constantly reduce to integers. Hopefully this will become clearer with the connection with rational trig in future videos.