Terence Tao 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 recognised 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 Terrance 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', 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, Clements says 'you shouldn't laugh at the questions because they get harder towards
the end of the test.' And Terrance 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. Clements 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. Clements asked 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 a seven year old Terence not to know that
either. He does, however, know about the distributive
law and gives the example multiplication over addition, Clements 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 is one of Terence's actual written solutions
to a question that says, 'The length of each side of a square is increased by three metres. The area of the new square is 39 metres 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 metres. The next set of questions given to Terrance
include 'Suppose you decided to write down all the numbers from one to 99,999. How many times would you have to write the
number one?' This one Terrance 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 50,000. He struggled a little bit with a few more
of these questions. And Clements 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 Tao 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, Clements
was shown one of Terence's programmes, it was called Fibonacci and actually contains
plenty of humour. Apparently, Terence wrote many of his programmes
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 programme,
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 programme is. Five weeks later, Clements came back for a
second visit. And he asked Terence if something was a field. He was Terence'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 Terence was able
to learn concepts. They encountered another thing, though, that
he didn't know yet. He was asked for the antiderivative 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 antiderivative of one over x, but was doing things like this
with partial fractions. He was given a space visualisation 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 16 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. 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 programme that
now eight year old Terence Tao had submitted for publication to a student mathematics journal
called Trigon. It is all about perfect numbers and is labelled
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 six'. Euclid proved that a number of this form is
a perfect number, if two to the p minus one is a prime number. So Terence says that he has used this fact
to write a programme in BASIC to find perfect numbers, which starts with a programme for
checking if two to the p minus one 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 Tao 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.
her entire channel is a gold mine for unintentional asmr
TIL I am no where close to a mathematician.
Ends with a cat of the day?
I'm in.
I'm not familiar with using "delta" as a variable - any reason they didn't use a fourth letter in the first example?