What Happens If We Add Fractions Incorrectly? #SoME3

What happens if we add fractions incorrectly? Can we do something with that? This video covers mediants, Simpson's paradox, Farey sequences, the Stern-Brocot tree, Ford circles, rational approximations, and Hurwitz's theorem.

Additional sources:

0:00 Definition of a mediant
1:52 Simpson's Paradox
6:03 List of all rational numbers
15:41 Visualizations
17:08 Ford circles
18:41 Rational approximations and Hurwitz's Theorem
25:20 Outro: Why did we do all of this?

Intro riff taken from: Nikolai Kapustin - 8 Concert Etudes, Op. 40: III. Toccatina

Music Credit:
Elegance / Megan Wofford
Technicolor Dreamscape / Franz Gordon
Joy in the Little Things / Sayuri Hayashi Egnell
Reve d'enfant / Magnus Ludvigsson
Lovely Dinner / Franz Gordon
When Sun Meets Moon / Gavin Luke

[kajacx]

Imagine teaching fractions in elementary school and a kid says "I'm not doing addition wrong, I'm computing the median" and then explains everything from this video.

[humbledbjesus]

i was the apprenticeship instructor for the roofing program in calgary for 13 yrs...

my most fulfilling memory was being able to teach fractions and the metric system to 40 yr old roofers with learning disabilities, addiction problems and 'incomplete' education scenarios...

seeing the look on someone's face when they actually get it and feel good about themselves...i was so blessed to help...

[thatapollo]

i kid you not, THE DAY after i watched this i had a math competition, and one of the problems was to find the fraction will the smallest denominator between 2 fraction, i tried trial and error and then remembered this video and immediately got the answer

[OnkarPawar]

I'm in high school. Clicked with great curiosity to watch and expand my knowledge but it slowly kept getting more and more complex until my brain couldn't understand

[toferg.]

1:17 This function has fascinated me for years. To me the practical application is when you divide something up and divvy it evenly amongst multiple people. Take a wandering band of fishermen living around ancient Sumeria for example. Let's say 13 guys catch 6 lb. of fish. They happen across some old allies they still love, and decide to have supper together. The other wandering band is 8 guys who caught 4 lb. of fish. How can they equally allot the fish to each fisherman, in the combined supper that night? 6/13 (+) 4/8 = 10/21 . Each person gets 10/21 lb of fish to eat.

Now let's say you come along, and tell them, you need to reduce the fractions first. So, the equation becomes 6/13 (+) 1/2 = 7/15 lb. See why we need to do away with this rule?

Now i'm much less familiar with vectors, than making sure i eat enough. But from what you're showing around 5:00 in, it looks like my example should work with vectors also. The demand to reduce the fractions isn't a good idea.

[DeJay]

Even if the other entries in SoME3 are incredible, this deserves at the very least an honourable mention.

[Nolord_]

Finally someone talked about this ! Last year I rediscovered most of this, in a attempt to find an algorithm that converts computer floating point numbers into a ratio, without suffering from the precision loss of floating point arithmetic. I couldn't find anything about this on the internet, until a friend of mine did. I'm happy more people learn about this simple but very interesting maths concept !

[iabervon]

If ad-bc=1, you can use Pick's Theorem to prove a bunch of properties of the mediant. The parallelogram is a lattice polygon with area 1, and we know 4 points on its border. Since the area is the number of interior points plus half the border points minus 1, there must not be any points in the interior and no more points on the border, so there cannot be any rational numbers that would fall in this region or on its border.

[anjanavabiswas]

It is just nice when u start from very simple things and sort of play around with it to discover interesting observations. It feels like going in the reverse direction when u are reading a theorem. Instead of having a very complicated unintuitive statement thrown at you and then having to use every single brain cell to figure out why that even works in the first place, this just feels very satisfying. It feels like the thought process flows naturally, without resistance.

[wilderuhl]

I love it when different bits of math come together so sensibly and beautifully. Excellent video.

[coaster]

It was lovely getting an intuition on where the mystifying square root of 5 comes from in Hurwitz’s theorem!

[amritawasthi]

Damn criminally underrated video. Good work man. Keep em coming

[Jackrabbithero]

Excellent video. I felt like every time you introduced a new concept, I had some questions pop into my head and thought "I'll have to google this after..." but then you answered the question in the video!

[BlackEyedGhost]

School teachers: "You can't divide by 0"
Mediant: "Hold my beer"

[NinjaOfLU]

Frick yeah, new zhuli video just dropped!

[skylark.kraken]

I planned on covering this topic as an interactive website for SoME1 for use as estimations. You came at it from a much better angle and covered more than I would have

[Oscar-vsyw]

Wow, I saw the thumbnail and thought that the video would be something really simple, but then saw it was nearly 30 minutes long! Unexpectedly turned out to be an absolutely banger! Good job man!

[juzbecoz]

This is a banger!let me share it! Eye opener

[isavenewspapers]

I like how my thought upon seeing the thumbnail was, "So like vectors?"

Yep.

[banaverhel]

The intro riff sounded vaguely familiar to me, so I checked the description and it’s Kapustin! Great taste in music and great video