This is clay, mine came from the store but you could even dig it up yourself from the
earth where the soil contains clay minerals. When you bake it, it becomes hard so you can
make an assortment of random objects but people much more talented than me have been using it
for thousands of years for things like pottery and buildings, and clay was even the first known
writing medium. I find it delightful to be able to look at ancient clay tablets from the Babylonians
4000 years ago, many of them are mathematical and show an amazing level of knowledge and accuracy.
Scribes were taught how to record the information onto the clay at a kind of scribe school and it
is within the relics of these schools that we can find the first known homework assignments.
What's great is that it's actually not too hard to read these tablets for ourselves,
the numbers are marked in the clay with a wedged tool.
This is one,
two, three,
four, five,
six, seven,
eight, nine,
and this is ten. So, twenty,
thirty, forty,
fifty, and since they're using base sixty,
this symbol is again 60. So it could be 60 or 1 depending on its place within the number.
To get the hang of it let's start with the work of a first-year student at scribe school where
the writing is a lot larger and clumsier than the more accomplished students. These are some
relatively easy multiplication exercises. In the first row we have 50 multiplied by 45. We would
normally work that out to be 2250 in our number system but the answer written here looks like 37
30. we can go ahead and see that this is indeed the same as what we would get, since there is no
symbol for zero the place values are up for some interpretation but for this one the 30s
is essentially the ones position and 37 is the 60s so 37 times 60 plus 30 is equal to 2250.
At school we often have to memorize the multiplication tables but something that
all scribe students would have had to memorize is a table of reciprocals which would make division
a lot easier. This is an example of one such table written by a student but apparently there are
so many errors in it that an angry teacher has crossed it out completely on both sides. These
tables would list that one-third of 60 is 20, one-fourth of 60 is 15,
a fifth is 12, a sixth is 10, and so on.
60 is a great base to use because it divides into lots of numbers exactly. To divide by any number
on this table you just have to multiply by the reciprocal but if your teacher is a big meanie
they might want you to divide by a weird number like 7 or 13 which 60 doesn't divide exactly into.
These kind of problems are a lot harder and you see an example of one of these questions
on this tiny little tablet here. It's small and can fit easily in the palm of your hand
and it was likely intended for a student of scribe school to take it home with them to complete and
return to the teacher with the solution. It is one of our earliest ever examples of homework. It asks
the student to show that 1 1 1 1 divided by 13 is 4 41 37. For us that number is 1 plus 60, plus 60
squared, plus 60 cubed which is two hundred and nineteen thousand six hundred and sixty one.
That we will want to divide by thirteen and for us doing it on a calculator that gives 16 897.
The answer shown on the tablet converted to our number system is 37 plus 41 times 60
plus four times 60 squared which is the same sixteen thousand eight hundred and ninety seven.
The babylonian students would likely have had to solve this by doing 60 cubed, 60 squared,
60, and 1 all individually divided by 13 and adding up the results and the remainders.
I just find it amazing that not only were they doing pretty hard math problems but
the way they recorded their working and answers
are so simple that we can actually understand and replicate them thousands of years later.
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What got me interested in babylonian math again recently was a paper that was published a few
weeks ago by an Australian mathematician who had discovered the oldest known example of
applied geometry. The tablet Si.427 shows land boundaries and was drawn by a scribe who would
have graduated school and gone out into the field. Practical issues of land ownership,
sales and taxes is after all exactly what a scribe would have trained for. The land in question is
split into very precise shapes but to make truly perpendicular lines was not trivial and something
that is pretty special about this tablet is that it shows the Babylonians using Pythagorean triples
in order to ensure that the perpendicular lines do indeed meet at right angles.
It is Pythagoras's theorem that says in a right angle triangle a squared plus b squared is equal
to c squared and a Pythagorean triple is three positive integers that meet this condition
such as three four and five. This tablet was made at least a thousand years before
Pythagoras was even born but the concept was also used in India and China before then so Pythagoras
was hardly the first person to invent the idea anyway. Here you can see a triangle with sides
seven-thirty, eighteen, and nineteen-thirty, for us that's seven and a half, eighteen, and
nineteen and a half which is a scaled Pythagorean triple. It's just 1.5 times the triple 5, 12,
13. This rectangle here has the same dimensions and was probably used to keep these sides in equal
length which could then be extended to make the rest of the boundaries. This discovery of how they
used Pythagorean triples was only published a few weeks ago but we had already known that they knew
trigonometry from another Babylonian tablet Plimpton 322, a list of Pythagorean triples
together with the information needed to resize them. This list likely belonged to a teacher
maybe at one of the scribe schools and served as a powerful reference. The fact we are still
learning so much about this ancient civilization and how they did mathematics is really exciting.
Here are a few more fascinating homework problems; this palm-sized tablet shows a circle drawn inside
a square and we believe the question for the student was to find the side length of the square
given the shortest distance from the circle to the square and the area of the region between the
circle and the square. If we label that area B, the distance d, which is written on there as 15,
the side length of the square as S, and r as the radius of the circle, then we have these
two equations. That the area of the square minus the area of the circle is B and that
two times the distance plus two times the radius of the circle is the side length of the square.
We can eliminate the variable r and then doing a bit of work to get a quadratic
that can be solved for S. For this problem the Babylonian approximation of pi was 3 and we've
been able to figure out that the value of B was likely given as what we would call 2376.5625
The quadratic was one that seemed fairly tricky so I used WolframAlpha to solve it
and that gave me the answer of 52 and a half which matches up with what you'll see written on the top
left of the tablet it says 52 30 and on the tablet you'll also find that value squared
which is the total area of the square. Isn't it amazing that this slab of such ancient clay
can help us here in 2021 to sharpen our math skills even without understanding
any of the historical or cultural context of when it was written.
Another amazing little tablet is this one here, YBC 7289 contains an approximation for the square
root of two which is accurate to six decimal places and that's the greatest known computational
accuracy found anywhere in the ancient world. The square root of two is an infamous number,
it's irrational, it goes on forever and can't be written as the ratio of two integers. The
rumour goes that Pythagoras and his squad were so shocked by this concept that they drowned a guy
at sea for so much as suggesting it but here are the Babylonians a thousand years earlier
assigning it for homework. They have a square with side length one and that makes a triangle where we
can use what we call Pythagoras's theorem to find the hypotenuse squared by adding up the squares of
both the sides in this case the hypotenuse squared is one plus one so the hypotenuse is the square
root of two. The number given here on the tablet is 1 24 51 10 which as a decimal is 1.414213
So now we know the ratio of the diagonal to the side of any square and can scale it.
On the tablet one side is labelled 30 and say for example we wanted to find the diagonal of
a square with side length 30 we would just do 30 times the square root of 2
and doing that gives us the last number here on the tablet 42 25 35 which is 42.426 for us.
So I guess maybe it's worth taking some care with your homework assignments just in case they get
studied by historians in the future and used to make judgments on how advanced your entire society
was. Apparently only half the tablets still in existence have been translated so I'm sure we
will find plenty more interesting stories within them and if you want to discover something new for
yourself then perhaps it is a topic worth pursuing further, I've listed all of my sources down in the
description. Thanks for watching this video and thanks to my patreons for supporting my channel
and a special shout out to today's patreon cat of the day who is looking awfully dog like, Mo.