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!
