Plus 16 tan to the negative 1 x 8 X minus 3 over root 7 all over 7 root 7 plus
constant. Any questions? When am I ever going to use this? The truth you're
probably not. The number of adults that make use of the integral 1 over 1 minus
3 X plus 4 x squared all squared is small. Most people aren't planning on doing
research in pure mathematics or becoming a rocket scientist. So why should you
bother learning this stuff? Well let me answer that with a story. During the
Second World War the American military were faced with a problem. You don't want
your planes to get shot down by enemy fighters so you put armor on them but
armor makes them heavier and heavier planes are slower and use more fuel. Not
enough armor is a problem, too much armor is a problem. So what do you do? Somewhere
there's a sweet spot, but where? That's the question the Army's best scientists
were trying to answer. Among them was Abraham Wald, grandson of
a rabbi, son of a baker, and a brilliant mathematician. The American military came
to Wald with some data they thought would be helpful. When their planes
returned from battle they were covered in bullet holes but the holes weren't
evenly spread. There were always more bullet holes in the fuselage, the main
body of the plane, and barely any in the engine. The military thought this was a
good opportunity to make the planes more robust while cutting down on armor. Put
more armor where the planes get hit the most, the fuselage, and less armor where
they get hit the least, the engines. But exactly how much more or less to put,
they didn't know. That's why they came to Abraham Wald, but they didn't get the
answer they expected. He told them that the armor doesn't go where the bullet
holes are but where the bullet holes aren't, the engines. After all, where were
the missing bullet holes? Wald's answer? The missing bullet holes were on the
missing planes. The reason planes were coming
with fewer hits to the engines weren't because they weren't getting hit in
the engines, but because the ones that were didn't make it back.
The fact that the majority of returning planes had more bullet holes in the fuselage
meant that hits to the fuselage could be withstood, so if anywhere, armor
could be spared there. It's the same reason there are a lot more patients in
the hospital's recovery Ward with bullets in their legs than in their
chest. It's not because people don't get shot in the chest, it's because when they
do they don't end up in the recovery ward. The chest is the point of greatest
vulnerability so it makes sense that we wear bulletproof vests there. The engine
is the point of greatest vulnerability for a plane so that's where the extra
armor goes. Wald's idea was quickly put into effect and was still being used by
the Air Force during the Korean and Vietnam wars. This seems obvious once
it's spelled out but why did it fool the military officers? They had much more
experience with war, planes, and armor than Abraham Wald did. Why could he see
what they couldn't? It all came down to the way they thought. Without knowing it
the military officers were making the assumption that the planes that came
back were a completely random sample of the planes that left. If that were the case
then it would make sense to assume that some areas were just getting hit more
than others. But if you think about that for just a moment you realize that
that assumption doesn't make sense. There's no reason at all to expect that
all parts of the plane are getting hit except the engines, or that all parts of
the plane are equally sensitive to gunfire. Wald thought "what assumptions am
I making and why am i making them?". That's a nice story but where was the math?
Abraham Wald didn't use any formula, trig identity, or integral to solve the
problem. Wasn't it just common sense? The question still stands, why should you do
page after page of integrals you're never going to use again? Well look at it
this way. Rote learning and formulas are to math as weight-training and repetitive
drills are to soccer. If you want to be a professional soccer player you need to
spend hundreds of hours lifting weights and zigzagging through traffic cones. You
would never actually see a professional athlete doing any of these things on the
field, but you do see the agility and strength they acquire
from doing them. But say you don't want to be a professional soccer player you
just want to play the friendly weekend game. That's fine, you can still play
soccer and you're probably going to be a fitter and healthier person if you do.
It's the same for math, you don't have to get into a career in mathematics but you
can still do math, and you're going to have better reasoning skills, more common
sense, and make better life decisions if you do. In his book The Power of
Mathematical Thinking, Jordan Elenberg describes math as a
bionic prosthesis you can attach to your common sense, allowing you to reach
further than you ever could have without it. Or like Tony Stark's Iron Man suit
which allows him to do things he'd never be able to like punch through brick
walls, but from Tony's perspective he's just punching the wall exactly like he
normally, would just a lot harder. So if anyone ever asks you "When am I ever
going to use this?", you can tell them about Abraham Wald, soccer, and iron man!
And if for any reason it doesn't make sense, you can show them this video.
Thanks for watching guys! This video was largely based on a chapter of the book
The Power of Mathematical Thinking, How to Not Be Wrong by Jordan Elenberg. I've
put a link to it in the description below if any of you are interested in
reading more about that. Oh and this video was part of a giant collaboration
I did with some of my educational youtuber friends, if you click on this
playlist you'll find all kinds of really interesting videos about education from
how dirt helps you learn to why textbooks are so expensive, so you should
definitely check that out next. That's all for me and I'll see you in the next
video. Bye!