What this calculation

[Mikefromlondon]

Fair enough. For what it's worth, my calculator gave exactly 13,983,816!

Kind Regards, John

That was when i used my under powered desktop calculator (Citizen CT-600) most likely it rounds off to 6 decimal places, hence a small error, and indeed when I used my phone calculator I got exact. Worth doing these kinds of things where who knows we could perform complex computations using less computation power but different technique, so achieving results without using the most powerful computers if you haven't got one.

 

[JohnW2]

That was when i used my under powered desktop calculator (Citizen CT-600) most likely it rounds off to 6 decimal places, hence a small error, and indeed when I used my phone calculator I got exact.

Your phone calculator was 'rounding to 8 significant figures.

However, there is no need for any rounding in your example - 13,983,816 (not 13,983,815.99) is the correct answer. You can see that the answer must be a whole number by 'partially factorising' as follows:

(49 x 48 x 47 x 46 x 45 x 44) ÷ (6 x 5 x 4 x 3 x 2 x 1)
= 49 x (48 ÷ 6) x 47 x (46 ÷ 2) x (45 ÷5 ÷ 3) x (44 ÷ 4)
= 49 x 8 x 47 x 23 x 93 x 11
which obviously gives an exact 'whole number' answer

Kind Regards, John
Edit: Omission corrected
Edit2: consequential error corrected

 

[EFLImpudence]

Haven't you forgotten the '3' ? Or am I missing a point?

That doesn't in this case, but could result in other than a whole number.

 

[JohnW2]

Haven't you forgotten the '3' ? Or am I missing a point?

Apologies - typo (omission) when copying from my scrap of paper. I've now corrected it. Thanks for noticing.

Quite apart from the mathematical approach (such as I illustrated) it is apparent that the result of any such combination calculation must always be a whole number - since what (in context) we are determining is the number of ways in which one can select X numbers (in any order) from a set of Y unique numbers. The "number of ways" one can do something obviously has to be a whole number.

Kind Regards, John

 

[EFLImpudence]

Ah, yes. Of course.

 

[Mikefromlondon]

= 49 x (48 ÷ 6) x 47 x (46 ÷ 2) x (45 ÷5 ÷ 3) x (44 ÷ 4)
= 49 x 8 x 47 x 23 x 9 x 11
which obviously gives an exact 'whole number' answer

Kind Regards, John
Edit: Omission corrected

Still a slight error where you inserted 3 in red, the product of that bracket is now 3 not 9. Just pointing the error. (i.e. 45/5/3 = 3) How do you get a divide sign?

Edit, I mean instead of saying the Product, I should have perhaps said the Result of that bracket.(45÷9÷3) = 3 (by the way many thanks for symbols)

 

[JohnW2]

Still a slight error where you inserted 3 in red, the product is now 3 not 9. Just pointing the error. (i.e. 45/5/3 = 3)

I don't see why you think that's an error. With my correction, all the 'multiply' numbers (49, 48, 47, 46, 45, 44) and all the 'divide' numbers (6, 5, 4, 3, 2 - and '1') are now accounted for. All I have done is rearranged the 'alternate multiplication and division operations' method I explained at the start.

How do you get a divide sign?

Use the 'symbol' facility. Click on the Omega symbol (next to the camera) at the top of the reply box, and then scroll down.

Kind Regards, John

 

[Mikefromlondon]

I don't see why you think that's an error. With my correction, all the 'multiply' numbers (49, 48, 47, 46, 45, 44) and all the 'divide' numbers (6, 5, 4, 3, 2 - and '1') are now accounted for. All I have done is rearranged the 'alternate multiplication and division operations' method I explained at the start.
Use the 'symbol' facility. Click on the Omega symbol (next to the camera) at the top of the reply box, and then scroll down.

Kind Regards, John

No the procedure you laid is correct, but you corrected one error but did not correct its result in that bracket where you inserted the red 3, since 45 divide by 5 =9, and then 9 divide by 3 = 3.
so this is where the problem is , which I will highlight in blue.
= 49 x 8 x 47 x 23 x 9 x 11

therefore 9 in blue should now be 3.

so your third line should read as this : = 49 x 8 x 47 x 23 x 3 x 11

Repeated edit.
Edit, I mean instead of saying the Product, I should have perhaps said the Result of that bracket.(45÷9÷3) = 3 (by the way many thanks for how to get ÷symbol)

 

[JohnW2]

No the procedure you laid is correct, but you corrected one error but did not correct its result in that bracket where you inserted the red 3 .... therefore 9 in blue should now be 3.

Ah, right - that's now corrected, too!

Kind Regards, John

 

[Mikefromlondon]

using a normal calculator, you can find the value of any number to the power of 2 ( binary) e.g. enter 2 and press X and then press = button, each time you press = will give you the power of previous number in bianry equivalent. I am not sure if I can make myself explain better. (In some calculators you may have to use the x button twice

so you should get 2, 4, 8, 16, 32, 64, 128, 256, and so on. My calculator can only do this to 35 power of 2, or 32 bit long binary number, most calculators can do this but my phone calculator cannot do this even though it is more powerful as it uses software rather than logical function.

A 1 bit binary number can represent only 2 values e.g 0 or 1, so a 2 bit binary number can represent 4 values and so a 3 bit binary number represents 8 values and a 4 bit binary number represents 16 different values, so in order to get a value of any number you press = sign for each number.

so 2 x = should give you 4, and press = again should give you 8, press = again should give you 16.

This is useful if you want to work out what is the value of a binary number 12 bit wide, so it should be 2048, but remember you will need to press = button 11 times and not 12, because you start off with 2, and not with 1, as 1 will always give you 1 if you multiply it by itself.

of course there other methods too, but this is using just an ordinary calculator

 

[Mikefromlondon]

Question for you

If you were to tell an alien race who knows nothing about our interpretation of numbers, how would you tell them what 1 means or what 2 means?
in other words the symbol 1 as we type and as we know, how would they know that the symbol 1 means one something? so how would you teach them what the symbol 2 means? sure you are not going to show then two fingers are you?

I remember our scientist sent a signal with binary code, which i think was wrong, it should have been decimal system.

 

[JohnW2]

using a normal calculator, you can find the value of any number to the power of 2 ( binary) e.g. enter 2 and press X and then press = button, each time you press = will give you the power of previous number in bianry equivalent. I am not sure if I can make myself explain better. (In some calculators you may have to use the x button twice .... so you should get 2, 4, 8, 16, 32, 64, 128, 256, and so on. My calculator can only do this to 35 power of 2, or 32 bit long binary number, most calculators can do this but my phone calculator cannot do this even though it is more powerful as it uses software rather than logical function.

Yes, all but the most basic calculators can do that, but most require one to press the 'x' ('times') button twice. How far they can go depends on the particular calculator.

The Windows calculator can do it, and goodness knows how far it will go - I've just tried and gave up at 100 (i.e. 2 to the power 100). I suspect that it has an incredibly high limit. If one uses it to do a direct power calculation (not by successive multiplication), mine will go up to just over 2^144,200 (i.e. 2 to the power 144,200).

Kind Regards, John

 

[JohnW2]

Question for you .... If you were to tell an alien race who knows nothing about our interpretation of numbers, how would you tell them what 1 means or what 2 means?

As you go on to say (but disparage), one almost certainly wouldn't - one would use binary, which is all but unambiguous (to anyone having the technology and knowledge to receive the signal) - with only two symbols, their meaning would be obvious to any 'intelligent' being.

I remember our scientist sent a signal with binary code, which i think was wrong, it should have been decimal system.

As above, despite your disapproval, binary is the obvious way to go. Our base-10 'decimal' system is essentially arbitrary (and in many senses not ideal). Had it not be for the fact that we have 10 'fingers and thumbs' we would probably be using some other number base - maybe (in decimal) 8 or 16.

Kind Regards, John

 

[ban-all-sheds]

Question for you

If you were to tell an alien race who knows nothing about our interpretation of numbers, how would you tell them what 1 means or what 2 means?
in other words the symbol 1 as we type and as we know, how would they know that the symbol 1 means one something? so how would you teach them what the symbol 2 means? sure you are not going to show then two fingers are you?

Something like this?

1 ■
2 ■■
3 ■■■

https://en.wikipedia.org/wiki/Alex_(parrot)

I remember our scientist sent a signal with binary code, which i think was wrong, it should have been decimal system.

I remember in my first year at Uni sitting through a series of lectures for a maths module which made no sense whatsoever - maybe I missed at the very start the lecturer telling us what the topic was, but if so I wasn't the only one - it wasn't that we couldn't understand what we were being taught, it was that none of us seemed to understand why, or what it was all for, etc.

After a whole term the pieces fell into place and we realised that we could then describe basic mathematical operations (+,-,x,÷) without needing any reference to numbers. Think about that - how would you explain to someone what addition was without any examples involving numbers of things?

The point is that it can be incredibly difficult to communicate concepts to another being from absolute first principles, or indeed from no first principles, i.e. without being able to assume any understanding of any underpinning knowledge or concepts.

We know that at even at a few months of age human babies understand cardinality - they can't "count" and clearly they don't know the words one, two, three etc, or the symbols 1, 2, 3, but they recognise the difference between ■■ and ■■■. How would you communicate even that sort of thing to a hypothetical race who were conscious of more than 3 dimensions, and for whom, for some reason, all numbers just have to have more than one component?

 

[bernardgreen]

How to talk to aliens ( or more precise intructions on Voyager Spacecraft DVD message for aliens who find the craft )

http://goldenrecord.org/#discus-aureus ( click on the various images. )