Shakuntala Devi’s Biopic
Making of the Genius, On Visualising Numbers, A look at one Show.. Some Random musings..
It was a revelation kinds seeing her personal story. She was that Math Goddess in school and college days. Campus Interview time Infosys aspirants would crack her book top to bottom.
From what the movie depicts.. Her super natural gift with numbers, difficult childhood, fearless courage and chase of fame and name like a typical actress celebrity who staged the theatre of numbers.
All that is her personal struggle. What really intrigued me very much is a 5 year old with no formal training doing complex cube roots. I kept thinking about it. There can be 4 possibilities. She saw her neibhour doing it and got fascinated in the subject silently. Problem Statement is exposed to her. And she kept practicing it from then with single minded focus like a singer practising music. The money and fame perhaps fueled her cause. At a later part of her life she could have honed her skills and that part does not excite me.
a) perhaps she mimicked what she saw and she had a mind of a computer.. multiple registers, accumulators and high speed computation.. Is this possible?
b) she was trained to use a Vedic math like dimension reduction algorithm and combined with her intellect and memory.. But does not seem like from the movie.
c) probably the dimension reduction algorithm was hardwired in her architecture and she used it without being aware..
d) perhaps she had a different cognition of numbers. Perhaps each number had a spatial representation, and she could glance through the cube roots through it.
May the the movie just made up the uninitiaited cube root thing May be she was just good with numbers to begin with. And her father being a performer and lion trainer kinds trained her to practise and practise and she aced at this. May be the cube roots was a later venture. Hmm.. Needs validation from the movie.
A few videos pop us in my youtube reccommendation. This one seems to remove the doubts created by the movie. A brilliant ask of very complex questions from HoD of an American University and few other Top Notch math and economics students from UC Toronto. Apparently her mother’s side were from priestly families and good at Astrology she says. Astrologers were very good at math and priestly families with the chants etc have a broadened memory. And the Vedic Math must have come through this path.
There are child prodigies with no classical music training or knowledge who can sing a Raaga by just listening. If that is possible well this is in similar lines. Do we see numbers in some boring domain and they see in spatial domain? Like circle in a circle? They see the seed within?
In a TeD show that I had watched earlier.. The colour and number sensing part are next to each other in the posterior part of the brain. Was discussing with another friend and her words.. “Dr V S has researched this extensively. He calls it synesthesia. It is heightened among creative people almost like a cross wiring of sense. A sculptor sees the work in the block of wood he carves.”
“Maybe in some sense, numbers not just came in colour but they had ‘personality’ and came ‘alive’ to her. When we use cross referencing of senses- fragrance of a song, hot summer, a melody that is sepia toned, etc… perhaps it takes on a 3D character in our minds. Remember that scene from Ramanujan’s life- he’s in the hospital and Hardy comes visiting him. R asks him how he came, Hardy says by cab. R asks for the cab number and instantly recognises it as the smallest sum of two perfect cubes or something. Numbers had a personality in their heads.”
On the topic of visualizations, simple dimensionality reduction and order and beauty in numbers, let us see some examples.
a) Integer Cube Roots of 6 digit numbers
b) N * N multiplication
Integer Cube Roots of 6 digit numbers
Cube of 100 is 1000,000, a 7 digit number.
Cube of 10 is 1000.
Hmm.. Perfect Cube-root of a 6 digit number is ‘not an out of range complicated number that the mind cannot hold.. It has to lie between 99 and 10.
a) We need to crack the 2 digit number AB. The unit’s place B and the ten’s place A. Cube root of 132651, let us call it AB
b) Let us divide the 6 digit number into 2 chunks of 3 digits 3 digits. The left 3 numbers contribute to A and the right 3 numbers contribute to B.
In 132651, 132 contributes to A, 651 contributes to B. Hope you are with me till here.
c) There is this cool thing about cubes of numbers till 10. There is a one to one relation between the number and its cube. The ending digit is either the number itself or its ten’s complement. Observe this table.
For {1, 4, 5, 6, 9, 0}, their cubes end in {1, 4, 5, 6 , 9, 0} which is the number itself..
For the remaining digits {2,3,7,8}, it ends in the 10’s complement [10–x]..
which is {8, 7, 3, 2}
This is a simple and remarkable, 1 to 1 unique association between the number and its cube. Some what like a wave.. 4, 5, 6 at the center, 1, 9, 0 at the edge.. 8,7, and 3,2 in between.
1 001
2 008
3 027
4 064
5 125
6 216
7 343
8 512
9 729
10 1000d) Hence, by looking at just the last digit of the 6 digit number, one can instantly deduce the unit digit B.
In 132,651, B is 1
In 13,824, B is 2
In 166,375, B is 5
e) And by looking at the first 3 digits one can judge the Ten’s digit A.
In 132,651, A is 5, as 5³ < 132 < 6³
In 13,824, A is 2, as 2³ < 13 < 3³
In 166,375, A is 5, as 6³ < 166 < 5³
Is n’t this algorithm not brilliant? It needs memorisation or sense of cubes till 10 though. I have taught it to so many 5th graders and little kids have a way of memorizing things, even if they cannot appreciate the beauty in it.
Need help memorising cubes 6 to 10? Notice how it grows exponentially
6 => 216. 216 adds to 9
7 => 343 is apalindrome
8 => 512 is easy peasy, 1K or 1024/2
9 => 72-9 adds to 9-9
Vedic Math uses so much of visualisation. Fractions have a circular graph. Numbers are visualised as 9 point circles. It is not a coincidence that we have clock with 60 minutes and 360 degrees around a point from the Babylonian Civilisation.
There are handy methods for all kind of specialised operations.. prime numbers detection, 9 and 11 operations.. number ending with 5, numbers clustered around multiple of tens . I just have a taste of these and have not trained myself on these aspects.. I have a feeling the number sense and appreciation would be so different in this kind of environment.
In a regular N * N multiplication.. The regular math would need around N*N multiplications. If we used divide and conquer technique and kept dividing into smaller units till it reduced into a 2*2 multiplication.. the complexity would be reduced logarithmically. It is called karatsuba multiplication in Algorithm Course. In vedic Math, it is called Nikhilam.. It is a kind of inplace multiplication without need for multisteps and accumulators.. And having a flower like design in the mind aids 3 * 3 multiplication so easily..
She had a gypsy kind of a travelling life perhaps, and did not get a chance to settle and use her intelligence like Ramanujan. Nevertheless we have witnessed a genius in our era and keeps us wondering how less we know about the human cognition or perhaps people in the past who were like her. Sha-kuntala.. It literally means curled up and convoluted and she was..
YouTube popped up this in my feed without any search and it was brilliant watching it. A doordarshan broadcast in the year of my birth. She tells at the end. Human beings should not be dehumanized by the machines.
Let’s try to guess and attempt a few integer root questions as it is more on deduction and not on create space like in multiplication.
8th root of 5764801. Taking cues from the cube root solving., This number has to be a single digit as it is less than 8 digits. {1, 3, 7, 9} have 1 in their 4th power hence 8th power. Answer seems 7 as 1⁸ is 1, and 3⁸ is 81*81 which is about 10000,and 9⁸ should be a big number.
4th root of 3895,0081
It has be a 2 digit. Again {1,3,7,9} are candidates for units place. 7⁴ ~ 2500, power of 8 ~ 65*65 ~5625
77 or 79 it boils to..
Second root of third root of fourth root of 1677716
Ah she does the reverse math and detects one digit is missing and there is an Anamoly in the question. The number is 16777216. My guess is it has to be very simple.. 4th root of 8 digit becomes 2 digit. Cube root of 2 digit number is one digit. And a one digit number having a perfect square root. That makes it 2 or 3. And since the 4th power is 6, it has to be 2.. And I am bang correct :)
How to get the missing number.. 2 power 48. 1024 power 3 times 8. Needs little flexing. Will leave at this :)
An old Oscar best song winner..
Like the circles that you find in the windmills of your mind!
Round like a circle in a spiral, like a wheel within a wheel
Never ending or beginning on an ever spinning reel
Like a snowball down a mountain, or a carnival balloon
Like a carousel that’s turning running rings around the moon
Like a clock whose hands are sweeping past the minutes of its face
And the world is like an apple whirling silently in space
Like the circles that you find in the windmills of your mind!
