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Introduction: Hardy Ramanujan Number Video Lecture - Class 8
FAQs on Introduction: Hardy Ramanujan Number Video Lecture - Class 8
| 1. What is a Hardy Ramanujan number? |  |
Ans. A Hardy Ramanujan number is a positive integer that can be expressed as the sum of two cubes in two different ways. For example, 1729 is a Hardy Ramanujan number because it can be written as 1^3 + 12^3 and also as 9^3 + 10^3.
| 2. How are Hardy Ramanujan numbers discovered? | |
Ans. Hardy Ramanujan numbers were discovered by the famous mathematicians G.H. Hardy and Srinivasa Ramanujan. They were studying the properties of numbers that can be expressed as the sum of two cubes and found that some numbers have multiple representations. These numbers were later named after them.
| 3. Are there infinitely many Hardy Ramanujan numbers? | |
Ans. It is currently unknown whether there are infinitely many Hardy Ramanujan numbers. While many Hardy Ramanujan numbers have been discovered, it is still an open question in mathematics. Researchers continue to search for new examples and investigate the properties of these numbers.
| 4. Can any positive integer be a Hardy Ramanujan number? | |
Ans. No, not every positive integer can be a Hardy Ramanujan number. There are certain criteria that a number must meet to be considered a Hardy Ramanujan number. It must be expressible as the sum of two cubes in two different ways. Numbers that do not meet this criteria are not Hardy Ramanujan numbers.
| 5. What is the significance of Hardy Ramanujan numbers? | |
Ans. Hardy Ramanujan numbers are significant in the field of number theory. They provide interesting examples of numbers that can be expressed as the sum of two cubes in multiple ways. The study of Hardy Ramanujan numbers helps mathematicians understand the properties of numbers and explore new mathematical concepts.
Text Transcript from Video
ROGER BOWLEY: The number is
1,729, which is known as a
Hardy-Ramanujan number.
JAMES GRIME: In mathematical
circles,
this is a famous number.
So there was a famous
mathematician, an Indian
mathematician called
Ramanujan.
Now, he started life
quite humbly.
He was a clerk in India, and he
had no formal training in
mathematics.
However, he worked as
a mathematician
in his spare time.
He had proven some things.
He sent these proofs
to Cambridge.
A lot of the Cambridge
mathematicians just ignored
it, but one Cambridge
mathematician, Geoffrey Hardy,
recognized something in this.
Now, the proofs that he did were
actually things that were
already well known.
But he invited Ramanujan over
to England and started
to work with him.
And Ramanujan is now famous
for being one of the great
number theorists.
Unfortunately, the English
weather and the English food
did not agree with Ramanujan.
This is round about the
First World War.
England is on rationing,
it's cold, it's wet.
He got quite ill.
Geoffrey Hardy visited Ramanujan
in hospital.
To make small talk, he said, the
taxi number that brought
me here was quite
a boring number.
It's 1,729.
ROGER BOWLEY: He couldn't think
of anything to say about
this number.
And he apologized to Ramanujan,
saying I can't--
this is not a very interesting
number.
I'm sorry about this number.
It might be an ill portent.
JAMES GRIME: There's nothing
interesting about that.
Ramanujan said no, there is.
ROGER BOWLEY: It's the smallest
number that can be
represented as a sum of two
cubes in two different ways.
JAMES GRIME: 1,729--
it is 12 cubed plus 1 cubed.
ROGER BOWLEY: Or you could write
it as 10 cubed plus 9
cubed equals 1,729.
JAMES GRIME: Now, these sort
of numbers are now known as
taxicab numbers for
this reason.
ROGER BOWLEY: Now, there was one
thing in the story which
is slightly wrong.
He should've said the sum of two
cubes of positive numbers,
positive integers.
Because you can find another
solution to this if you allow
negative numbers in there.
JAMES GRIME: So Geoffrey
Hardy told this story.
And I think people tell this
story to show how brilliant
Ramanujan was.
He could just come up with that
off the top of his head.
I think the point Geoffrey Hardy
was trying to make is
no, he wasn't a genius.
Ramanujan had come up with
this fact already in his
previous research.
This is something he'd studied,
something he knew.
So he wasn't a genius.
He hadn't just come
up with that.
And I think that's the point
he was trying to make.
ROGER BOWLEY: If you allow
negative numbers, then I could
write down 6 cubed plus minus 5
cubed is the same as 4 cubed
plus 3 cubed, which is 91.
So this is the smallest number
which is the sum of two cubes
in two different ways, if you
allow negative numbers.
And Ramanujan thought it was
that because he only allowed
for positive numbers.
And you'll forgive him, because
he was at death's door
at the time.
1,729, which is known as a
Hardy-Ramanujan number.
JAMES GRIME: In mathematical
circles,
this is a famous number.
So there was a famous
mathematician, an Indian
mathematician called
Ramanujan.
Now, he started life
quite humbly.
He was a clerk in India, and he
had no formal training in
mathematics.
However, he worked as
a mathematician
in his spare time.
He had proven some things.
He sent these proofs
to Cambridge.
A lot of the Cambridge
mathematicians just ignored
it, but one Cambridge
mathematician, Geoffrey Hardy,
recognized something in this.
Now, the proofs that he did were
actually things that were
already well known.
But he invited Ramanujan over
to England and started
to work with him.
And Ramanujan is now famous
for being one of the great
number theorists.
Unfortunately, the English
weather and the English food
did not agree with Ramanujan.
This is round about the
First World War.
England is on rationing,
it's cold, it's wet.
He got quite ill.
Geoffrey Hardy visited Ramanujan
in hospital.
To make small talk, he said, the
taxi number that brought
me here was quite
a boring number.
It's 1,729.
ROGER BOWLEY: He couldn't think
of anything to say about
this number.
And he apologized to Ramanujan,
saying I can't--
this is not a very interesting
number.
I'm sorry about this number.
It might be an ill portent.
JAMES GRIME: There's nothing
interesting about that.
Ramanujan said no, there is.
ROGER BOWLEY: It's the smallest
number that can be
represented as a sum of two
cubes in two different ways.
JAMES GRIME: 1,729--
it is 12 cubed plus 1 cubed.
ROGER BOWLEY: Or you could write
it as 10 cubed plus 9
cubed equals 1,729.
JAMES GRIME: Now, these sort
of numbers are now known as
taxicab numbers for
this reason.
ROGER BOWLEY: Now, there was one
thing in the story which
is slightly wrong.
He should've said the sum of two
cubes of positive numbers,
positive integers.
Because you can find another
solution to this if you allow
negative numbers in there.
JAMES GRIME: So Geoffrey
Hardy told this story.
And I think people tell this
story to show how brilliant
Ramanujan was.
He could just come up with that
off the top of his head.
I think the point Geoffrey Hardy
was trying to make is
no, he wasn't a genius.
Ramanujan had come up with
this fact already in his
previous research.
This is something he'd studied,
something he knew.
So he wasn't a genius.
He hadn't just come
up with that.
And I think that's the point
he was trying to make.
ROGER BOWLEY: If you allow
negative numbers, then I could
write down 6 cubed plus minus 5
cubed is the same as 4 cubed
plus 3 cubed, which is 91.
So this is the smallest number
which is the sum of two cubes
in two different ways, if you
allow negative numbers.
And Ramanujan thought it was
that because he only allowed
for positive numbers.
And you'll forgive him, because
he was at death's door
at the time.
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