Special numbers

Jai Ganesh · 2006-10-16 16:05:26

1. The special quality of the number 64 : it is both a perfect square and a percet cube. The smallest such number parat from zero and one . Let's call it it g3 for convenience.
g4 is 4096
g5, g6 is 1152921504606846976
g10 is approiximately 7x10^758
g20 is 2.125219x10^10^70077344
g1000 is approximately 10^10^433
You can see how these numbers grow as the base value increases.
Interestingly, gx would always lie beteen x^x and x^x^x.

mathsyperson · 2006-10-17 00:34:04

How come g20 is more than g1000? Is g20 meant to have that second ^10 in there?

Jai Ganesh · 2006-10-17 15:54:47

mathsy,
g1000 is the smallest number apart from from zero and one thats a perfedct square, cube, .....1000th power. No wonder its greater than g20. It is of the magnitude of 10^10^433. Much biggerthan a googolplex.

mathsyperson · 2006-10-17 21:48:20

Yes, exactly. g1000 *should* be a lot bigger than g20, but according to the 1st post, it's not.

g20 is 2.125219x10^10^70077344
g1000 is approximately 10^10^433

Jai Ganesh · 2006-10-18 15:54:10

That was a mistake, mathsy, Admitted.
g20 is 10^70077344 and not as given.

Patrick · 2006-10-18 22:02:59

ganesh wrote:

1. The special quality of the number 64 : it is both a perfect square and a percet cube. The smallest such number parat from zero and one . Let's call it it g3 for convenience.
g4 is 4096
g5, g6 is 1152921504606846976
g10 is approiximately 7x10^758
g20 is 2.125219x10^10^70077344
g1000 is approximately 10^10^433
You can see how these numbers grow as the base value increases.
Interestingly, gx would always lie beteen x^x and x^x^x.

Nice post, thanks for providing the numbers

mathsyperson · 2006-10-18 23:26:53

Each of these numbers is basically 2^{lowest common multiple of every number up to x}, which explains why it grows so quickly.

To get from gx to g(x+1), you need to take it to some kind of power. This could be squaring, cubing, or anything up to ^(x+1) if x+1 is prime. Of course, there are other times when the power is one, but overall that explains why it grows so quickly.

How did you work out g1000?

Jai Ganesh · 2006-10-22 20:27:08

With a hand held scientific calulator, 18 years ago. There are nearly 168 prime numbers from 1 to 1000. The highest powers of these numbers upto 1000 have to be multiplied. You get something close to 10^433. Therefore g1000 is roughly 10^10^433.

Zeroface · 2006-12-29 10:28:30

ganesh, you are well meaning and share an interest in the peculiarity possesed by certain numbers. But I take an interest in 0, the Alpha and Omega of all numbers. A number that alone signifies nothing but when merged with other numbers signifies much more. It is odd but true and we all know of these facts. Face it ganesh, 0 is awesome:D

Jai Ganesh · 2007-01-19 23:01:13

Zeroface,
I know the importance of zero.
I know it is an indispensable part of counting in any system.
And I also know India gave zero to the world
Wonder how people counted before!

Patrick · 2007-01-20 06:46:56

ganesh wrote:

Wonder how people counted before!

Counting things without the concept of 0 was never really a problem, I guess. Why on earth would you count nothing? Waste of time

Jai Ganesh · 2007-01-20 17:20:25

Patrick,
Imagine, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11!!!!!!!

mathsyperson · 2007-01-21 06:52:10

Romans managed to do fairly well by using a system where the value of a number was determined by adding all of its digits, rather than using a base system.

That worked alright for them because they presumably kept their numbers smallish, but it wouldn't for us because once we get past 1000, we run out of numerals and have to just put a bunch of M's. So we use bases instead.

At first, people didn't use 0, and just left a gap where one should be.
6, 7, 8, 9, 1 , 11, 12, 13...

But this didn't work very well at all, because if you had a number like 1 2 , then no one has any idea what it's meant to be. 1020? 10020? 12? 10002? 1 and then 2? It could be anything.

And then some clever person had the idea of making up a 0 digit, and that seemed to work so we've used it ever since.

Patrick · 2007-01-21 11:33:50

ganesh wrote:

Patrick,
Imagine, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11!!!!!!!

Haha, true, but why would you count 11!!!!!!! after 9? there's quite a difference between there two. (had to get back on ya )

landof+ · 2007-09-09 00:38:55

Nooo!!!
It's
, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1:), 11, 12, 13, 14, 15....

krassi_holmz · 2007-09-09 06:21:25

Another counting system without null:
1-1
2-11
3-111
4-1111
ect.

Math Is Fun Forum

#1 2006-10-16 16:05:26

Special numbers

#2 2006-10-17 00:34:04

Re: Special numbers

#3 2006-10-17 15:54:47

Re: Special numbers

#4 2006-10-17 21:48:20

Re: Special numbers

#5 2006-10-18 15:54:10

Re: Special numbers

#6 2006-10-18 22:02:59

Re: Special numbers

#7 2006-10-18 23:26:53

Re: Special numbers

#8 2006-10-22 20:27:08

Re: Special numbers

#9 2006-12-29 10:28:30

Re: Special numbers

#10 2007-01-19 23:01:13

Re: Special numbers

#11 2007-01-20 06:46:56

Re: Special numbers

#12 2007-01-20 17:20:25

Re: Special numbers

#13 2007-01-21 06:52:10

Re: Special numbers

#14 2007-01-21 11:33:50

Re: Special numbers

#15 2007-09-09 00:38:55

Re: Special numbers

#16 2007-09-09 06:21:25

Re: Special numbers

Board footer