Proof: Triangle Inequality Theorem

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Proof: Triangle Inequality Theorem | Real Analysis

Рет қаралды 83,466

Күн бұрын

The absolute value of a sum is less than or equal to the sum of the absolute values for any two real numbers. That is: |a+b| is less than or equal to |a|+|b|. This is called the triangle inequality. It's very useful in real analysis and we'll prove it in today's lesson! The name of the theorem is explained a little below.
If we considered a triangle made of vectors x, y, and z, with x and z starting at the origin, then the vector z is the vector x+y, and thus the theorem says this third side (z) of the triangle must be no greater than the sum of the other two sides: |z| is less than or equal to |x| + |y| and since the vector z is the vector x+y we have |x+y| is less than or equal to |x| + |y|. Speaking strictly in terms of geometry, we'd often use a strictly less than inequality for this theorem because the equality case describes a "degenerate" triangle, where the "triangle" is just a line. However, the "less than or equal to" inequality is perfectly valid and reasonable for real numbers.
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Пікірлер: 94

@WrathofMath 3 жыл бұрын

Check out my growing Real Analysis playlist! kzfaq.info/sun/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

@thedeathofbirth0763 Жыл бұрын

This is how a person who truly understands logic, can beautifully and precisely communicate it . Thank you.

@WrathofMath Жыл бұрын

Thank you!

@punditgi 3 жыл бұрын

Very clear proof. Well done!

@WrathofMath 3 жыл бұрын

Thank you, glad it was clear!

@ryantrynda3029 3 жыл бұрын

why is it called triangle inequality thou

@edisonnogalesantezana4761 Жыл бұрын

The triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.

@androidandroid9591 Жыл бұрын

An excellent AND simple way to grasp a crucial theorem for the understanding of analysis. Txs.

@WrathofMath Жыл бұрын

Glad to help!

@red3987 2 жыл бұрын

Beautiful and concise!! thank you for helping me understand this proof better!

@WrathofMath 2 жыл бұрын

Thanks so much, glad it helped! Thanks for watching and let me know if you ever have any questions!

@red3987 2 жыл бұрын

@@WrathofMath WIll do!!!

@Alan-sv6ym 2 жыл бұрын

We really love you here in south Africa 🇿🇦😭💜💜💜

@WrathofMath 2 жыл бұрын

Thank you so much! Love back from the USA's east coast!

@benjtheo414 2 күн бұрын

if 2:44 wasn't directly obvious to you as it was for me you can use this in the proof: a1

@shhaque9448 10 ай бұрын

This guy really did a 2 hour lecture in 5 minutes

@punditgi 3 жыл бұрын

Can you do an entire video on absolute value identities as building blocks for other theorems? I would also illustrate these both on a 1D number line and on a 2D plane as vectors.

@davaariantara3704 2 жыл бұрын

like |xy|=|x|*|y|? i think you will find it obvious if you proof it by case-by-case like above in the above video or you can even proof it by |x|=squareroot(x^2) ^ just trying to give you an idea that its not hard

@harjassingh7758 7 ай бұрын

thank you bro / sir appreciated !!

@mahmoudalbahar1641 3 жыл бұрын

Many thanks for this good video.

@johnsally8320 Жыл бұрын

My man, you're awesome! Much love from Toronto, Canada.

@WrathofMath Жыл бұрын

Thanks so much!

@valeriereid2337 Жыл бұрын

Thanks for explaining this clearly. If you are able to, please do some work from Abbott's "Understanding Analysis." It would be awesome.

@Bhaskar3D 2 жыл бұрын

It's very good sir. I am from Assam, Indian.

@silvo9460 2 ай бұрын

wow thank you so much sir!!!!!

@H3XED_OwO 7 ай бұрын

beautiful and simple proof!

@WrathofMath 7 ай бұрын

Thanks for watching!

@Divine_wsk21 6 ай бұрын

Can you please do the same proof with a,b and c

@ProgressiveVoice 4 ай бұрын

Thanks!

@husseinsslehi4428 2 жыл бұрын

Very clear explanation thank you

@WrathofMath 2 жыл бұрын

Thanks for watching! If you're looking for more analysis, check out my playlist: kzfaq.info/sun/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

@TahaCodes 11 ай бұрын

Very nice and neat. Thank you. Got yourself a new sub!!!

@jamesjones6359 3 жыл бұрын

Great explanation. Thank you.

@WrathofMath 3 жыл бұрын

My pleasure, thanks for watching and check out my Real Analysis playlist if you're looking for more! kzfaq.info/sun/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Many new real analysis lessons are coming this year, let me know if you ever have any requests!

@samuelbassey6806 9 ай бұрын

Thanks for sharing

@putin_navsegda6487 2 жыл бұрын

great proof, very understandable! Thank you!

@WrathofMath 2 жыл бұрын

Thank you, glad it helped! Check out my analysis playlist for more! kzfaq.info/sun/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

@yibanathimnyanda725 3 жыл бұрын

Thank this really helped.

@WrathofMath 3 жыл бұрын

So glad to hear it, check out my Real Analysis playlist if you're looking for more, many more analysis lessons still to come! kzfaq.info/sun/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

@anasshuaibu2530 2 жыл бұрын

Very good ' but plz prove it again when â and b are real numbers.

@Umar-lj5jx Жыл бұрын

Thanks a bunch

@bekhaddaderrar2111 3 жыл бұрын

Thank you so much for this video

@WrathofMath 3 жыл бұрын

My pleasure, thanks for watching! If you're looking for more real analysis, check out my playlist! kzfaq.info/sun/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

@bonniem1865 6 ай бұрын

Thank you thank you thank you

@WrathofMath 6 ай бұрын

You are very welcome!

@augustus2834 8 ай бұрын

great! the only thing that bothered me a little bit is that we didnt take into account when a number is negative and the other is positive. but that case is trivial so i guess doesnt matter that much ahahaha

@rosepierce9382 Ай бұрын

I think it doesn't really matter because he is only taking the sum of a + b to be negative or positive.

@navjyotsinghvishen3287 2 жыл бұрын

Group theory from basic 🙏

@jeswintitusjv5927 Жыл бұрын

Sir , could I know why this is called as "triangular inequality" ? Is this related to triangle when viewed geometrically ?

@WrathofMath Жыл бұрын

Thanks for watching and the question! Yes, particularly with vectors involved. Suppose we have points A, B, and C, and then three vectors AB, CB, and AC, that make a triangle. The vector AB is the same as AC + CB if you know vector addition. And this side of the triangle, AB = AC + CB has length |AB| = |AC + CB|, and this length must be no greater than the sum of the other two sides, that is |AC| + |CB|.

@ali-hg103 10 ай бұрын

That outro thooo🔥

@WrathofMath 10 ай бұрын

Artist goes by Crayon Angel now - love his stuff!

@TVSuchty 3 жыл бұрын

Can you show that over R^n and all possible dot products, where ||a||^2 = a dot a ?

@qedmath1729 3 жыл бұрын

let a = . By definition of dot product we have a dot a = x^2+y^2. Similarly for a = , ||a|| = sqrt(x^2 + y^2) or ||a||^2 = x^2 + y^2. Hence we see that the 2 are equal.

@TVSuchty 3 жыл бұрын

@@qedmath1729 I mean that u show that the theorem holds over all dot products, where by definition ||a||^2 = a dot a

@gianlucalepiscopia3123 Жыл бұрын

Beautiful

@WrathofMath Жыл бұрын

Thanks for watching!

@user-sw4ih9xg5d 2 жыл бұрын

super good

@WrathofMath 2 жыл бұрын

Thank you! If you're looking for more analysis, check out my playlist! kzfaq.info/sun/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

@samborn3515 Жыл бұрын

Please I stil find this difficult

@existentialrap521 11 ай бұрын

Wrath, once i get this mfing masters degree ima invite you to a carne asada thx G

@daniellin188 Жыл бұрын

-(a+b) does not equal (-a) + (-b) by the way

@Zinani-z7b 26 күн бұрын

It is Darling! Look at it again.

@muhammadasghar7153 4 жыл бұрын

I Asked question but you didn't answer me

@pakistaninstituteofmedical7185 4 жыл бұрын

Yes I also ask question but didn't reply

@ahh61 4 жыл бұрын

I asked a question and he answered me so be happy. Maybe next time ask a better question.

@muhammadasghar7153 4 жыл бұрын

Shut up loser baby hahhahah🤑🤑

@ahh61 4 жыл бұрын

@@muhammadasghar7153 At least I'm not dumb like you. Your question was probably what's 2+2. You stupid. My questions are always answered. He even answered a bunch of my questions on the last live episode.

@WrathofMath 4 жыл бұрын

I'm working on it folks, I get a lot of comments and don't have much time. If you ever come by a live stream, that is time I set aside specifically to answer your questions so you can be sure I'll respond if you come by one of those! Thanks for watching and I'm trying to catch up on the comments!