Every now and then I hear stories about people who can perform amazing feats of mathematics,
often in their head.
And through comments on my videos I've learned about one such person named Shakuntala Devi,
also known as The Human Computer.
She holds the world record for the fastest human computation which she gained in 1980
by multiplying two 13 digit numbers together in her head.
And she was able to give the answer in 28 seconds, and that was just the amount of time
it took for her to say the answer out loud.
There are some clips of her on YouTube performing math shows where she answers questions from
the audience in her head much faster than her scribe can even write them down on the
chalkboard.
Devi: "Mr. Sherma, you want the 8th root of this number?
7 is the answer right.
I can even give you the steps.
7 multiplied by - Yogesh you're not writing anything!
Please go ahead.
7 multiplied by 7 is 49, multiplied by 7 is 343, multiplied by 7 is 2401, multiplied by
7 is 16807, multiplied by 7 is - oh you don't want to do anymore?
Ok then well I will do it anyway for my satisfaction - multiplied by 7 is 117649, multiplied by
7 is 823543 and that multiplied by 7 is the answer you wrote on the blackboard."
She loved doing multiplications, finding roots of numbers, or even figuring what day of the
week a certain date fell on.
Devi: "25th April 1960?
That's a Monday"
And these are all amazing but what really amazed me also is just how funny she was.
Devi: "Will you compete with me?
Oh goodness that's no use, I mean you can't get approximate numbers it's got to be exact.
The machines have their limitations whereas humans have no limitations."
She had confidence and made jokes and was really good at putting on a show.
And that's not something that's always so common for someone with these abilities.
Which actually does have a name by the way, hypercalculia.
Her charisma on stage might make a bit more sense when you understand a bit about her
background.
Devi was from India and her.
father was a circus performer.
It was him who discovered her ability in mathematics while he was teaching her a card trick when
she was only 3 years old.
After then he started taking her on math road shows and she went on to impress and entertain
people all over the world.
And she did all of that without having any formal education.
When I see what she was able to do, my question is 'how?'
And luckily she did write a book explaining some of her methods and I have that book her
today.
It is called Figuring and I would like to go through it with you and show you some of
her explanations.
In her introduction Devi says that she has a desire to show those who think mathematics
is boring just how beautiful it can be.
And she says to understand the methods describes you need nothing more than a basic knowledge
of arithmetic.
She then has a chapter on the digits where she explains each number one by one and the
patterns that can be seen with it.
She refers to something called secret steps in multiplication where after doing a normal
multiplication and being left with a 2 digit number, she will add up those two digits to
get left with a single number.
For the 1 times tables this last digit will follow the pattern that it increases by 1
each time.
Something similar can be seen in the 2 times table.
There are likewise patterns in the secret steps of the 3 times tables, 4 times tables,
and all of the digits really.
And there's always some kind of pattern that she manages to find in these remaining digits.
These patterns give rules to working out multiplications and I guess ways to check your answers, and
show how she played with numbers to such an extent that she could find patterns everywhere.
Since multiplication was what she had the world record for, we'll take a little look
at some of her methods for multiplication.
She starts with just multiplying two 2-digit numbers, 63 and 48.
And she gives in order in which it should be done.
We start here on the right and do this column which is 3x8.
It's 24 so you should put a 4 down and carry the 2 over to our next one.
Here we will add up the products of the diagonals so we'll do 4x3 plus 6x8 plus that remaining
2 that we were carrying.
That will give us 62.
So write the 2 down and carry the 6.
For our last one we'll do 4x6 plus the 6 that we carried and that will be 30.
So by the end of that we'll have the answer 3024.
And this method can be extended to 3-digit numbers, 4-digit numbers and all the way up
really.
She gives an example here of multiplying two 6-digit numbers this way.
It takes a few extra steps and she acknowledges that most people would want to do this with
a pen and paper but in the end she does get the correct product.
How she manages to do it so fast is still beyond me even though I've read some of her
methods.
But maybe it's a combination of raw talent, a lot of practice, and spending time to notice
relationships between numbers to speed up each step of the process.
Something else that she did a lot of on stage was finding square roots, so let’s take
a little look at one of her methods there.
She says that first you’ll want to learn a table of squares of numbers from one to
nine off by heart.
And from this you can see that a square ending in a one would have a root ending in either
a one or a nine.
And a square ending in a four would have a root ending in either a two or eight.
Squares ending in a nine or a six will also have two possible root endings, and a square
ending in five will only have one possibility which is that the root also ends in five.
Now to use this table to find square roots, say that we have a four digit number: 6, 2,
4, 1.
We’ll break that down into 62 and 41.
Take the 62 and find where it lies on the table.
It is less than 64 but more than 49, so the highest possible first digit of the root would
be a seven.
Now we move on to using 41 to try and find the units digit of the square root.
We know that since it ends in a one, that the root must end in a nine or a one, and
to decide which one to use, what we do is we take the first number of the root, in this
case seven, and we multiply it by itself plus one.
So multiply it by seven plus one.
That’ll be seven times eight equal to 56.
And since 56 is less than 62, which is what we used to find that first number of the root,
then we will take the highest of the two possible digits figures, so in this case it will be
a nine.
If 56 had been higher than 62, then we would’ve taken the lower of the two possible digit
figures and that would’ve been a one.
So the two digits we worked out are seven and nine, to give the root of 6241 as indeed
79.
There’s a few seemingly random steps there but what we did is we broke our number down
into two, we used the first half of it to find the first digit, and we used the ending
of the second half of it to find out what our root should end in.
To do larger squares, you will need to remember a bit more of the table, and of course this
goes on for even larger numbers still.
And there’s a pretty similar method for how she works out cube roots and roots of
higher powers as well.
And there’s one more fun method that she talks about here and this one’s probably
the easiest to understand.
It is how she works out what day of the week any date will fall on.
She says that you’ll need to memorise four tables.
Table one, which is a few multiples of seven.
The second, which corresponds each month to a digit.
And the third table which takes each fourth year, maybe these are leap years, I’m not
sure, and corresponds them to a digit as well.
And this table repeats, so you only really need to memorise a section of it.
And there’s a fourth table here which will turn the result of our calculation which will
be a number from zero to six, into a corresponding day of the week.
So if we wanted to work out what day of the week was the 9th of April 1932, the first
thing that we do is reduce the day of the month to a number less than seven.
And to reduce a number to something less than seven, we would need to subtract any of these
four numbers from table one, depending on how big it is.
In this case, we would do nine minus seven, and be left with two.
We then use table two and find out that April corresponds to a six.
This is added to the two, so we now have eight.
We always want to remain lower than seven since there are seven days in the week, so
we will subtract seven and be left with a one.
Now we’ll take a look at the year, 1932, and we’ll find from this table three that
it corresponds to a five.
Since 1932 was one of the years on the table, all we need to do is add the digit from previously
which is a one, so five plus one is equal to six, and we can just use table four to
find out what day of the week six corresponds to, and that will be a Saturday.
There are slightly different rules if the year doesn’t fall exactly on that table
of leap years, but it would just involve finding the previous leap year plus the difference
between that leap year and the year in question.
Again, a few seemingly random steps here, especially when I just read them out.
But with a little bit of practise, this method becomes something kind of routine, you just
need to memorise those tables and these few steps, and you can work out any day of the
week you want.
She talks about some of her methods to check her answer such as this one that she refers
to as casting out the nines, because even a human computer will occasionally make mistakes,
but she’s also able to check things in her mind so it means it’s much less likely a
mistake will slip through.
To finish I'd like to take a little look at her conclusion.
She says that mathematical discoveries, like all others, may come from the need to solve
practical problems.
But a more important motive is the pure delight some individuals take in invention and discovery.
I’ve seen a little bit of criticism online in referring to Shakuntala Devi as a mathematician,
since what she does is really just mental arithmetic, some would call it, but I would
say she deserves to be called a mathematician because of her outlook on what mathematics
is.
A lot of these methods in this book she comes up with herself by spending hours in the company
of numbers, thinking about the patterns between them and thinking about ways to group them
together, different ways to solve the problem, and finding various tricks that will work.
But I think all of that does come down to the nature of mathematics which is experimentation
and finding delight in numbers.
She says the intent of this book is that she hopes the readers will come to share this
feeling for the subject, and so I hope by showing you a little bit through it in this
video, you’ve gained a bit of that appreciation.
Thanks for watching to the end of the video, and a big thanks to my Patreon supporters
who are the people who make these videos possible in the first place.
And a shoutout to today’s patron cat of the day, Toth, who doesn’t look too much
like a cat but still gets a few minutes of fame.