terence tau is one of the world's best
mathematicians
he has won maths top prize the fields
medal
and many would consider him a genius
his talents were recognized early and he
was a
child math prodigy he was dabbling in
university courses when he was nine
and by 16 he had graduated from flinders
university in adelaide
by 21 he had his phd from princeton
when terence was seven he was taking
some math and physics classes at his
local high school
and it was around this time that he was
visited at home
by ken clements a researcher of
mathematically gifted children
this report is clements's thoughts and
findings on tao
including the results of a math test
given to him so let's take a look at it
it was the day before terence's eighth
birthday
the 16th of july 1983
exactly 38 years ago and it says here
that
terence was given an operations test
knowing that he might find the first few
questions too easy
clement says you shouldn't laugh at the
questions
because they get harder towards the end
of the test
and terence has an interesting reply he
says
the questions won't know if i laugh at
them because they haven't got ears
he went on to score 60 out of 60 on this
test
on average a year 12 student taking the
same test would be expected to get a
score of only 53 out of 60
and here's one of those example
questions question 58
asked if p divided by q divided by
r is equal to delta divided by q divided
by r
then find delta and here is
seven-year-old terence's answer
he writes it out as a fraction and then
just does a bit of algebra
to rearrange for delta getting it to be
p over r squared although it was clear
to clements
that this test was too easy for terrence
and they'd need to move on to some more
difficult things
this next set of questions were
presented to him and they had to be
worked out mentally
they include things like what angle does
an hour hand describe in 20 minutes
terence got all eight of these questions
correct as well
in a total time of nine minutes clemens
noticed that
while terence had been solving the
questions he often justified an
algebraic step
by writing the appropriate algebraic law
for example the associative law next to
it
and so that prompted this following very
interesting conversation
clemens asks what is the associative law
for addition of real numbers
terence answers that it doesn't matter
where you put the brackets
he also gives a correct definition of
the commutative laws
and he also gives a correct definition
of a group something that most math
students don't encounter until
university however it seems that the
first thing to stump him
was being asked what is a field terence
replies
i don't know given that i barely have a
grasp on what a field is now
i think it's forgivable for
seven-year-old terrence not to know that
either
he does however know about the
distributive law and gives the example
multiplication over addition
clemens asks if addition over
multiplication is an example
but terence replies only for boolean
algebras
clements was quite impressed by all of
this not only did he have an astounding
grasp of algebraic definitions
for someone who was still seven years
old but i was amazed at how he used
sophisticated mathematical language
freely i was beginning to form the
impression
that terence preferred to use analytic
non-visual methods
in preference to making extensive use of
visual imagery
here's one of terence's actual written
solutions
to a question that says the length of
each side of a square
is increased by 3 meters the area of the
new square
is 39 meters squared more than that of
the original square
how long are the sides of the new square
terence correctly finds that the
length of the new square is eight meters
the next set of questions given to
terrence include
suppose you decided to write down all
the numbers from 1
to 99 900
how many times would you have to write
the number one
this one terence actually gave an
incorrect solution to
although still an answer that shows
plenty of critical thought about the
question
apparently the correct solution here is
fifty thousand
he struggled a little bit with a few
more of these questions
and clement says that at this stage
terence was
showing slight signs of fatigue although
his interest was still high so he asked
him just
two more relatively simple questions
first he asked him to sketch the graph
of y equals x squared plus x which he
did immediately
i asked him to find the coordinates of
the turning point
and he wrote them down this response
took 20 seconds
i then asked him to sketch y is equal to
x cubed minus 2x squared plus x
and in about one minute he produced this
it is interesting to note that
terence hadn't begun to study any
calculus at school
but when clements had arrived at the tau
household
and was speaking with terence's parents
billy
and grace he had spotted terence sitting
in the far
corner of the room reading a book with
the title
calculus terence's dad was a doctor
and terence's mum was a graduate of
physics and maths
who had worked as a high school teacher
she said that
terence liked to read mathematics by
himself and often spent three or four
hours after school
reading mathematics textbooks terence
also had an interest
in computer programming and had taught
himself basic language by reading a book
and at the end of this first visit
clemens was shown
one of terence's programs it was called
fibonacci
and actually contains plenty of humor
apparently terence wrote many of his
programs
when he was just six years old so let's
take a look at
fibonacci it doesn't just calculate the
sequence
it starts with here comes mr fibonacci
can you guess which year was mr
fibonacci born
write down a number please then if the
user inputs the correct answer they can
start
however if their guess is too far off
they'll be told
no he is already in heaven try again
or they might be told sorry he wasn't
born yet
try again then if they were close it
would print the difference between the
guess
and the actual answer meaning that even
to start this program
the user is guided by a bit of
mathematics
after printing some of the sequence it
ends with
mr fibonacci is leaving now and here
goes his car
this line is perhaps the only reminder
of just how young the author of this
program is
five weeks later clements came back for
a second visit
and he asked terrence if something was a
field
here was terrence's answer clearly
demonstrating that
since the last visit he now knew exactly
what a field was
this kind of answer is something that a
university student would be proud of
and it shows just how fast terrence was
able to learn concepts
they encountered another thing though
that he didn't know yet
he was asked for the anti-derivative of
one over x
and responded that he had not got up to
that yet in his reading
but on a third visit not long after the
second
terence had not only mastered the
anti-derivative of one over x
but was doing things like this with
partial fractions
he was given a space visualization test
which again made clements come to the
conclusion
that terence preferred to use non-visual
analytic methods even if they required
more complicated thinking
than more visual methods for example
with this question here
terence said that he checked each shape
by the reflection law
which was that each point had an image
on the other side of the mediator
and he did not imagine each shape being
folded along the dotted line
included in the report was terence's
school timetable
the subjects marked with the asterix
took place at the primary school
whilst all the others took place at the
high school
his teachers said that whilst
seven-year-old terence did have the
academic ability of a sixteen-year-old
his maturity was still that of a
seven-year-old
and terence's parents had to balance
giving him a childhood
that challenged him intellectually but
also allowed him to fit in
socially and emotionally the report
makes it clear that
rather than being pushed along one rigid
track of mathematical education
terence was allowed to pursue the topics
that interested
and challenged him the most he loved
mathematics
and the efforts of those around him were
to ensure that he did not become
bored or frustrated with non-challenging
work
let's finish with a look at a program
that now
eight-year-old terence tau had submitted
for publication
to a student mathematics journal called
trigon
it is all about perfect numbers and is
labeled here as
terence's first published paper a
perfect number is one such that
all its factors including one but
excluding itself
add up to itself for example six is a
perfect number
because six has factors one two three
and six
and one plus two plus three is equal to
6.
you could proved that a number of this
form
is a perfect number if 2 to the p minus
1 is a prime number
so terence says that he has used this
fact to write a program in basic
to find perfect numbers which starts
with a program for checking if
2 to the p minus 1 is prime
then he computes perfect numbers up to
10 to the 13. yet again another
example of just how outstanding he is
it's been humbling for me to read
through these notes i hope that it
doesn't seem discouraging to any
aspiring mathematicians out there
as i'm sure terence tau would even agree
that many of the great mathematicians
followed a more conventional path
through education
thanks for watching i hope you've
enjoyed this video and
thanks as always to my patreon
supporters for making these videos
possible
a special shout out to today's patreon
cat of the day leon