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Did Issac Newton know about numeral systems?


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7 hours ago, Eise said:

Yes, it is obvious that you are very confused. So why did you do so long as if you understand what 'base' means here?

Let's try to explain it 'my way'. You must distinguish between the designator and the designated. E.g. when I ask you what a chair is, you could say 'it is a piece of furniture, that is designed to sit on'. You would feel pretty fooled if I would answer 'no, it is a word of 5 letters'. (I should have written then 'chair', between single quotes, to make clear that I meant the designator, not the designated. But hey, I wanted to fool you.) On the other side the same designated can be designated in another language, e.g. in Dutch as 'stoel'. Science usually does not change when you change the language. 

It is the same with numeric bases: on one side there are the numbers, at the other side there are their representations. Mathematics, and therefore all sciences using mathematics, are of course independent of the number base you use to express the numbers.

Here a list of 'objects' and some translations:


Chair        Stoel                      translation in Dutch
666          29A                        translation in hexadecimal
666          1232                       translation in octal
666          1010011010                 translation in binary
666/18 = 37  29A/12 = 25                translation in hexadecimal
666/18 = 37  1232/22 = 45               translation in octal
666/18 = 37  1010011010/10010 = 100101  translation in binary

So in the second column, we have only translations. The maths stays exactly the same, just a the physical characteristics of a chair are exactly the same as een stoel. We just have to keep an eye on which language we use, and be consistent. As an example: if we would think that '1232/22 = 45' is written in decimal, it would be wrong: in decimal 1232/22 = 56. But those are just symbols. You should always be aware of what they mean.

You meant tan( #$@ ) = @   ^_^ (assuming the last symbol in your 'language' stands for '0' ).

Numeric Bases was the gap I had..

"Thanks" for clarifying this.

This reminds me of "cryptography" or

The "Ceaser Cypher " That uses "keys" a position and an output "letter by letter" where the key is a constant..

I know this sounds weird of a question.

But what base does algebrea use?

I ask becuase it has a 2 above the x:

x^2

 

I found this for Numeric Base..

In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is ten, because it uses the ten digits from 0 through 9.

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8 hours ago, CuriosOne said:

I'm totally confused on what you mean by "base and powers." I don't understand these sequences either...I see five sets of numbers, but dont understand what they mean..

Endy018 was trying to show you how to count to 5 in different bases.  In base 10 there are 10 single digits (if you include 0) and no single digit for 10.

So to count to 5 in base 10 you can count 1, 2, 3, 4, 5.

In base 5 there are 5 single digits (if you include 0) and no single digit for 5.

So to count to 5 in base 5 you can count 1, 2, 3, 4, 10.

In base 2 there are 2 single digits (if you include 0) and no single digit for 2.

So to count to 5 in base 2 you can count 1, 10, 11, 100, 101.

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18 hours ago, studiot said:

 

Towards the beginning of this thread I asked you a simple question you have yet to answer.

 

 

This question was intended to help you think about your question on bases.

 

In modern times we enjoy a very efficient representation of numbers the means the 'base' is the count (number or integer if you like) of different symbols that are required to represent any number value whatsoever, when used in combination.

So base 2 has two symbols :- 0,1                and  base 10 has 10 symbols:-  0,1,2,3,4,5,6,7,8,9 
The next number in the base 10 system is represented by a combination of two existing symbols   as 10.
Using only two combined symbols will take us up to 99 and then we require to start combining 3 symbols for 100

As soon as we start combining symbols we also require a convention to distinguish between say 32 and 23.
This modern convention is one of the strengths of the modern system.

 

However it was not always so

For instance Roman and earlier civilisations used fewer symbols and as a consequence they representations contained more than one base, which made their arithmetic much more difficult.

 

 

 

 

When I think of 1000 and 1, I see many things..

I see a conversion factor = 1 as in the inverse of this times 2 = 1/2

1/10^3 =0.001 where 1/0.001 "un" does the conversion, I also see 1010 looks like a binary number "who knows." I also see units of kg, or even microns...Its very hard to say what I see..

When I see "our more popular choice of base 10 I see both base 10 and base 2 as x^2 

x= base 10

^2 = base 2

Why would I think this??

For the rule of "like terms." 

By the way....How does minutes, hours and seconds apply to any of this to be something "reasonable" as for as the human perception of time goes??

In other words we do well with 26 = 26 letters of the alphabet, but how do we link numbers to "time?" 

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4 minutes ago, CuriosOne said:

When I see "our more popular choice of base 10 I see both base 10 and base 2 as x^2 

x= base 10

^2 = base 2

Why would I think this??

I don't know. 

X^2 = X * X

^2 has nothing to do with base 2.

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8 hours ago, Sriman Dutta said:

You seem not to understand what is meant by bases.

I will be trying to explain it in simpler terms.

Any quantitative thing is a number. We use 10 characters 1,2,3,4,5,6,7,8,9,0 to represent them. Count how many characters we use? It's ten characters. Therefore our standard base of calculation is ten. 

Now imagine a civilisation living in a far off galaxy (don't ask questions like where are they bla bla, I'm just trying to explain). They evolved just like us. However unlike us, they are familiar to calculate numbers in base 4. They use the characters @,#,$ and & to represent all kinds of numbers. So they have base 4.

That's basis. It has nothing to do with calculus or trigonometry. 

Trigonometry doesn.t require a special base. Why would it? tan 45 =1 in our base and character set. It will be @ in that far off civilisation's base and character set. That doesn't mean the two things are different. 

 

If you find it hard to understand, for a moment forget everything about number system and things taught. Try to see what I'm trying to say. 

 

 0 9 8 7 6 5 4 3 2 1 is understood, but when we add these to "units" from derived math formulas is where I get confused, very confused and "one of the reasons are:" You cannot have 60 seconds in one "completed cycle" ie time doesn't say 1:60 pm, it goes from 1:59 pm to 2:00 pm in a cycle of 59 seconds, not 60...Is this correct??

So what happens when we apply your explanation to time? Do we have 0->59 charectors of time?? 

I think were not including the most fundamental part of numbers and nature..

7 hours ago, Sensei said:

...just in typical usage..

There are existing non-integer numeral systems, e.g. base PI, or base e, too..

https://en.wikipedia.org/wiki/Non-integer_base_of_numeration

https://en.wikipedia.org/wiki/Non-standard_positional_numeral_systems

 

 

Well this just keeps getting more and more interesting!

""""""""""""Thanks for telling me this""""""""""""

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1 hour ago, Bufofrog said:

Endy018 was trying to show you how to count to give in different bases.  In base 10 there are 10 single digits (if you include 0) and no single digit for 10.

So to count to 5 in base 10 you can count 1, 2, 3, 4, 5.

In base 5 there are 5 single digits (if you include 0) and no single digit for 5.

So to count to 5 in base 5 you can count 1, 2, 3, 4, 10.

In base 2 there are 2 single digits (if you include 0) and no single digit for 2.

So to count to 5 in base 2 you can count 1, 10, 11, 100, 101.

^This.

For lower values, I'll also normally say the individual digits to make differentiating easier.

ie. One(1), One-Zero(2), One-One(3), One-Zero-Zero(4), One-Zero-One(5)

 

1 hour ago, CuriosOne said:

But what base does algebrea use?

I ask becuase it has a 2 above the x:

x^2

 

You assume base 10 unless otherwise indicated.

 

For x^2 that '2' just indicates that you are multiplying it by itself that many times.

x^2 = x multiplied by x twice = x raised to the power of 2 = x squared

 

1 hour ago, CuriosOne said:

 0 9 8 7 6 5 4 3 2 1 is understood, but when we add these to "units" from derived math formulas is where I get confused, very confused and "one of the reasons is: You cannot have 60 seconds in one "completed cycle" ie time doesn't say 1:60 pm, it says 2:00 pm

So what happens when we apply your explanation to time? Do we have 0->59 charectors of time?? 

I think were not including the most fundamental part of numbers and nature..

Time is a mixed base system based on the bases 12 and 60(12*5).

image.jpeg.c8142343ad7bf6f31d4eb3d6fe10fb05.jpeg
Originally based on counting each of the 12 colored areas above and everytime you reach twelve lowering one finger of the other hand.
 

 

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38 minutes ago, Bufofrog said:

I don't know. 

X^2 = X * X

^2 has nothing to do with base 2.

what's the capitol x for?

about ^2 makes sense, ...

undoublty "then" its ratio, length and time based all wrapped up in a nice tiny box of x...lol

24 minutes ago, Endy0816 said:

^This.

For lower values, I'll also normally say the individual digits to make differentiating easier.

ie. One(1), One-Zero(2), One-One(3), One-Zero-Zero(4), One-Zero-One(5)

 

 

You assume base 10 unless otherwise indicated.

 

For x^2 that '2' just indicates that you are multiplying it by itself that many times.

x^2 = x multiplied by x twice = x raised to the power of 2 = x squared

 

Time is a mixed base system based on the bases 12 and 60(12*5).

image.jpeg.c8142343ad7bf6f31d4eb3d6fe10fb05.jpeg
Originally based on counting each of the 12 colored areas above and everytime you reach twelve lowering one finger of the other hand.
 

 

Ok, lots of questions here...if you may..

The counting base 10 system is the only one I understood so far...I will work on the others..

For the 60 and minute question.

Is that 5 in 5*12 base 5?? Or is it some "form" of square root "base system" for 25*4 = 10^2 100 cm for the base 12 system...Maybe I should just open a new thread, but this is on topic..

Just to let you know I get this idea from 100 cm in one "meter second" the SI unit for the second, but from real time, not the speed of light...

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2 hours ago, CuriosOne said:

what's the capitol x for?

about ^2 makes sense, ...

undoublty "then" its ratio, length and time based all wrapped up in a nice tiny box of x...lol

Ok, lots of questions here...if you may..

The counting base 10 system is the only one I understood so far...I will work on the others..

For the 60 and minute question.

Is that 5 in 5*12 base 5?? Or is it some "form" of square root "base system" for 25*4 = 10^2 100 cm for the base 12 system...Maybe I should just open a new thread, but this is on topic..

Just to let you know I get this idea from 100 cm in one "meter second" the SI unit for the second, but from real time, not the speed of light...

5 in base ten. Always assume base ten until told otherwise or if it is obvious in context(dealing with base-16 hexadecimal codes with letters as numerals for example).

This is only alongside numbers, but you would normally use subscript to identify the base being used(if it needs to be made obvious). The number indicating the base, should itself be in base ten to avoid confusion.

ie. 5 in base ten = 510

 

An old joke goes that there are 10 kinds of people in this world, those that understand binary and those that don't.

The joke being that 'one-zero' in binary or 102, actually means just two in base ten.

 

You already understand counting in different bases, you just have to adjust your thought process slightly to see that.

If I say 90 minutes from 3 AM, your brain will automatically add the two correctly and give you the correct time, without you even noticing it.

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45 minutes ago, Bufofrog said:

It is a variable, just like x, a, b... Etc.  A variable just represents any number you choose.

Oh, I thought capitol x as in capital X had some relavance....lol

57 minutes ago, Endy0816 said:

5 in base ten. Always assume base ten until told otherwise or if it is obvious in context(dealing with base-16 hexadecimal codes with letters as numerals for example).

This is only alongside numbers, but you would normally use subscript to identify the base being used(if it needs to be made obvious). The number indicating the base, should itself be in base ten to avoid confusion.

ie. 5 in base ten = 510

 

An old joke goes that there are 10 kinds of people in this world, those that understand binary and those that don't.

The joke being that 'one-zero' in binary or 102, actually means just two in base ten.

 

You already understand counting in different bases, you just have to adjust your thought process slightly to see that.

If I say 90 minutes from 3 AM, your brain will automatically add the two correctly and give you the correct time, without you even noticing it.

There is something else, I must add...

Saw this in a conversion base "calculator" as I stumbled on..

 "quotient" of "Base Systems"???

As in "The Difference Quotient?"

Used in calculus??

How Odd!

Where do those sum remainders go??

20201214_130545.jpg

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2 hours ago, CuriosOne said:

Oh, I thought capitol x as in capital X had some relavance....lol

There is something else, I must add...

Saw this in a conversion base "calculator" as I stumbled on..

 "quotient" of "Base Systems"???

As in "The Difference Quotient?"

Used in calculus??

How Odd!

Where do those sum remainders go??

 

That looks off but if you choose something like

6 to base 2

Division Quotient

Remainder

 
6/2 3 0  
3/2 1 1  
1/2 0 1  

Reassembling the remainders from bottom to top, you get 1 1 0

so One-One-Zero in base-2 or 1102 = 610

 

Bases come  into play with logarithms in calculus.

ie.

log base-b, with the typical base you'll use being an unspoken 10.

log10(100) = 2

 

1 hour ago, studiot said:

The story of how we got from there to here (today) is quite interesting.

Indeed. I still wonder if we'll ever switch from predominantly using base-10 again. Interesting to consider.

 

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14 hours ago, Eise said:

Yes, it is obvious that you are very confused. So why did you do so long as if you understand what 'base' means here?

Let's try to explain it 'my way'. You must distinguish between the designator and the designated. E.g. when I ask you what a chair is, you could say 'it is a piece of furniture, that is designed to sit on'. You would feel pretty fooled if I would answer 'no, it is a word of 5 letters'. (I should have written then 'chair', between single quotes, to make clear that I meant the designator, not the designated. But hey, I wanted to fool you.) On the other side the same designated can be designated in another language, e.g. in Dutch as 'stoel'. Science usually does not change when you change the language. 

It is the same with numeric bases: on one side there are the numbers, at the other side there are their representations. Mathematics, and therefore all sciences using mathematics, are of course independent of the number base you use to express the numbers.

Here a list of 'objects' and some translations:


Chair        Stoel                      translation in Dutch
666          29A                        translation in hexadecimal
666          1232                       translation in octal
666          1010011010                 translation in binary
666/18 = 37  29A/12 = 25                translation in hexadecimal
666/18 = 37  1232/22 = 45               translation in octal
666/18 = 37  1010011010/10010 = 100101  translation in binary

So in the second column, we have only translations. The maths stays exactly the same, just a the physical characteristics of a chair are exactly the same as een stoel. We just have to keep an eye on which language we use, and be consistent. As an example: if we would think that '1232/22 = 45' is written in decimal, it would be wrong: in decimal 1232/22 = 56. But those are just symbols. You should always be aware of what they mean.

You meant tan( #$@ ) = @   ^_^ (assuming the last symbol in your 'language' stands for '0' ).

So 360 = 4

3 hours ago, studiot said:

There is good evidence that Man's first number system was base 1.

The story of how we got from there to here (today) is quite interesting.

I'm still trying to figure out if our modern technology inherited the ideas of base systems and if they are "error prone" to human flaws..

3 hours ago, Endy0816 said:

That looks off but if you choose something like

6 to base 2

Division Quotient

Remainder

 
6/2 3 0  
3/2 1 1  
1/2 0 1  

Reassembling the remainders from bottom to top, you get 1 1 0

so One-One-Zero in base-2 or 1102 = 610

 

Bases come  into play with logarithms in calculus.

ie.

log base-b, with the typical base you'll use being an unspoken 10.

log10(100) = 2

 

Indeed. I still wonder if we'll ever switch from predominantly using base-10 again. Interesting to consider.

 

The logarithms bring upon an issue dealing with calculus and that is "nature and machine languages."

I mean to say integration..

Is that why x^2 looks like a parabola when you graph y=x^2

How in the world does anything x^2 get that shape????

 

From Wikipidia..

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x.

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9 hours ago, CuriosOne said:

I'm still trying to figure out if our modern technology inherited the ideas of base systems and if they are "error prone" to human flaws..

I don't know what you mean.

There has been an ongoing cycle of improved theoretical abstraction resulting in improved technology resulting in refined abstraction resulting in.....   since time immemorial.

I think it would help you enormously if you recognised the idea of abstraction and the abstract and its connection to the physical world.
I think it (not) understanding this underlies your (current) difficulties in numbers/bases/units/dimensions/powers  and probably other areas.

 

Edited by studiot
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And, for CuriousOne's  (and also any non-trolls) benefit.

Computer programmers are among the people who (fairly) commonly use other number bases, rather than base 10 which most of us use.
And, because of that, they need to say what base they are using.

In particular they commonly use bases 2, 8, 10 and 16.

Binary , Octal, Decimal and hexadecimal.
And they designate them as BIN, OCT, DEC and HEX.
So, for example the number of miles between London and Slough would normally be written as 22.

If we want to be a bit more specific, we need to say that we ae using base 10- known as decimal.

 The number of miles between London and slough  is 22 DEC

That means that the number is two lots of ten and 2 lots of 1

On the other hand, you could also express it in Binary

The number of miles between London and slough  is 10110 BIN
That's

1 times 16 plus
0 times 8 plus 
1 times 4 plus
1 times 2 plus
0 times 1

 

Or we could write it in Octal
The number of miles between London and slough  is 26OCT

That's 

2 times 8 plus
six times 1

OK, back to the joke.

Now, the number of days you have to wait from the start of October before you get to Halloween is 31.

Again, if we are being careful to say what base we use that's 31DEC

 

And consider the number of days into  December you have to go to reach Christmas:25.

It's a base 10 number so we can clarify that: it's 25 DEC

But we could express the number 25 DEC in octal.
It's 3 times 8 plus
1 times 1

So it's 31 OCT.

25 DEC is 31OCT

 

 

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31 minutes ago, John Cuthber said:

And, for CuriousOne's  (and also any non-trolls) benefit.

Computer programmers are among the people who (fairly) commonly use other number bases, rather than base 10 which most of us use.
And, because of that, they need to say what base they are using.

..........

 

Yes indeed all very true

But there are other considerations as well as the bare base number, we take for granted and therefore tend to forget.

Firstly there is the positional notation we use today.

For programmers who often split or group digits together to form nybbles, bytes and words etc there is also this grouping convention
At machine level there is also the question of what is used to represent the 1 an 0 symbols. There are several options here, rather more than MigL's on/ off  option.

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6 minutes ago, John Cuthber said:

I'm hoping we can avoid the big endian vs little endian wars.
:-)

If that was directed at me I see no reason for any sort of war, except perhaps the Jersey Battle of the Flowers.

A while back I asked the OP if he understood the significance of 1001.

This only has meaning if we first state the base (in my case 10, in your case something else)

and then it still requires us to agree the meaning of each place.

Furthermore if we have two such numbers say 1111 and 1001 we have the question of the convention about the meaning of putting these two together.

Do we just add them ? If so by what rules ?

Or do we concatenate them to form a new number 11111001 ?

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9 hours ago, studiot said:

I don't know what you mean.

There has been an ongoing cycle of improved theoretical abstraction resulting in improved technology resulting in refined abstraction resulting in.....   since time immemorial.

I think it would help you enormously if you recognised the idea of abstraction and the abstract and its connection to the physical world.
I think it (not) understanding this underlies your (current) difficulties in numbers/bases/units/dimensions/powers  and probably other areas.

 

I "totally" agree with you 100% this post has been of such enlightenment its actually made algebrea simpler for me.

It looks like an intriguing field for creative thinkers and the like...

Your correct about updated technology issues especially dealing with latency issues ie "musical midi keyboards" and "computer response" sounds for music programs....

 

 

6 hours ago, John Cuthber said:

And, for CuriousOne's  (and also any non-trolls) benefit.

Computer programmers are among the people who (fairly) commonly use other number bases, rather than base 10 which most of us use.
And, because of that, they need to say what base they are using.

In particular they commonly use bases 2, 8, 10 and 16.

Binary , Octal, Decimal and hexadecimal.
And they designate them as BIN, OCT, DEC and HEX.
So, for example the number of miles between London and Slough would normally be written as 22.

If we want to be a bit more specific, we need to say that we ae using base 10- known as decimal.

 The number of miles between London and slough  is 22 DEC

That means that the number is two lots of ten and 2 lots of 1

On the other hand, you could also express it in Binary

The number of miles between London and slough  is 10110 BIN
That's

1 times 16 plus
0 times 8 plus 
1 times 4 plus
1 times 2 plus
0 times 1

 

Or we could write it in Octal
The number of miles between London and slough  is 26OCT

That's 

2 times 8 plus
six times 1

OK, back to the joke.

Now, the number of days you have to wait from the start of October before you get to Halloween is 31.

Again, if we are being careful to say what base we use that's 31DEC

 

And consider the number of days into  December you have to go to reach Christmas:25.

It's a base 10 number so we can clarify that: it's 25 DEC

But we could express the number 25 DEC in octal.
It's 3 times 8 plus
1 times 1

So it's 31 OCT.

25 DEC is 31OCT

 

 

Very informative stuff!

Especially the Oct ober and Dec ember analogies..😎

 

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