Triangle Inequality

Triangle Inequality
In this video, I define the concept of an absolute value and use it to prove the triangle inequality in R, which is the most important inequality that we'll use in analysis. Enjoy!
Check out my Real Numbers Playlist: • Real Numbers

[@MrCigarro50]

I remember my Math Professor saying..." Learn as much as you can about inequalities, you will need them very very often"

[@lordandprotector]

More often than not. :P

[@myoung1445]

If you're in first year of university or last year of school then take this advice seriously . From a panic cramming 2nd year maths student

[@LegendOfMurray]

I wasn't expecting a dab in a math video. Excellent!

[@tubalnavarro8962]

That triforce in the thumbnail made me click as fast as I could

[@Jim-be8sj]

Good one. If I remember correctly, once you get up to working in function spaces it's time for the Minkowski inequality for many of the same applications that the triangle inequality enables.

[@cerwe8861]

The "dab" you have to train... xD

[@joudy5767]

what a coincidence that i have to prove this for my teacher and you posted this 1 day ago🤩🤩 thank uu

[@fromblonmenchaves6161]

Why in proof 1. case 1 when x ≥ 0 then x = |x| ≤ -|x| ; and not
x = |x| ≥ -|x| ?

[@drpeyam]

It’s a typo I think

[@fromblonmenchaves6161]

@@drpeyam I see. Thankss

[@ragokyo]

11:34 *"the legthgh"*

[@Apollorion]

Is this triangle inequality also true for triangles in non euclidean spaces?

[@EpicMathTime]

Yes, it is. Non-Euclidean geometries take place in metric spaces, and the triangle inequality is required of a metric.
The triangle inequality is foundational for the notion of "distance" or "length" to make sense. Its essence is that the distance from a to b reports the length of the shortest possible route from a to b. That's what we want distance to mean. We don't want a to be 15 miles away from b, but somehow if I go from a, to c, and then to b, I ended up traveling only 12 miles in total.

[@MrQwefty]

The Cauchy Schwarz inequality is driving me mad! Will you do a video on that too?

[@drpeyam]

Already done ✅

[@MrQwefty]

@@drpeyam Whoops! Thank you Peyam :D

[@putin_navsegda6487]

but actually it's Cauchy-Buniakovsky inequality

[@weimeuret7689]

I was hoping you could prove the Reverse Triangle Inequality. I'm also wondering why we need to establish |a-b| is greater than the difference of their absolute values...

[@drpeyam]

There’s a video on that

[@jimmykornelijegunnarsson4265]

I'm wondering why you don't consider the proof by using the dot product (a+bIa+b), seeing as the proof comes out of it naturally.

[@drpeyam]

That’s the way I would have done it as well, but in that class they don’t really mention dot products or R^n

[@jimmykornelijegunnarsson4265]

@@drpeyam Oh cool.

[@sanjeevkumardhiman6783]

It is true coz if both a and b are positive then abs(a+b)=abs(a)+abs(b) but if one of then is negative; let’s say b is negative and let’s say that a=32 and b is equal to -31, then abs(a+b)=abs(32-31)=1; but it is not equal to abs(32)+abs(-31)=32+31=63;in fact the abs(a)+abs(b) is greater than or equal to abs(a+b). :))))

[@LarryRiedel]

I'm looking forward to Jensen's Inequality. :-)

[@drpeyam]

That’s a good one 😄

[@minimurali9700]

Sir , but is there any example or case where one side will be equal to to sum of other side in a triangle ? The inequality says lesser or equal to

[@xibbit6322]

when the triangle is degenerate(smushed down into a line) this is true

[@punditgi]

Awesome explanation!

[@sirjackio_zw]

Did he just do the dab 9:10😂😂

[@foreachepsilon]

My linear algebra professor used to call it the donkey inequality because it’s so simple, even a donkey gets it! (The donkey of course will walk a straight line to get to water, not go in a jagged path.)

[@drpeyam]

LOL, I love this 😂

[@foreachepsilon]

Dr Peyam he still teaches at CSUF, though I guess you’re leaving UCI in the fall.

[@ImNostalgic]

was that the triforce??

[@drpeyam]

Yeah

[@ImNostalgic]

@@drpeyam pogggggg

[@tymothylim6550]

Great video! Thanks a lot for this :)

[@computerguy678]

the triforce :skull zelda refrence

[@DidierDegonon]

you could define abs(x) = Max (x,-x), then your lemma is evident

[@DidierDegonon]

dam you're fast answerer, hello from France!

[@thomasjefferson6225]

Lmao I love the dab!!!! 🤣🤣🤣

[@the_laughingbabe]

You make it more harder for me😢🙆🏽‍♀️

[@Sahanie]

I trust my Toyota Corollary.

[@lesprivatrizal]

Nice, thanks

[@yoannroch6477]

good video!

[@user-qw4vg9lf5v]

Wait what? 🤯🤯, I.. I'm new to this

[@kamaljitkaurdhiman7696]

Analysis is adult version of calculus. Isn't it??

[@drpeyam]

It really is!

[@kelvinmasasire6737]

The explanation's a bit hard to grasp Doc. Its a bit too complex

[@drpeyam]

Good for you

[@mohamedtrfnx6632]

@@drpeyam i dont think its good for him that he didn't grasp it :(

[@visualgebra]

Professor ! Why It is called Triangle Inequality ?

[@drpeyam]

Look at the picture

[@jamesbra4410]

Next measure theory

[@romaindec1717]

i prefer algebra.

[@dm_saj1119]

Hi dr 💚

[@AditYa-sv1nz]

👍