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### Numbering Systems

08/29/2012 9:51 AM

I am learning binary systems by self study,but I am having a problem determining the identification of digits after the Hexadecimal System 0 thru F.

Do they begin to use lower case alpha characters, up to 35? such as Fa for 33, Fb for 34,etc.What about 64 bit systems?What is the proper integer designator for this system?

I am sure the solution is simple,once revealed, but right now I am in the dark.

Any assistance is appreciated.

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#1

### Re: Numbering systems

08/29/2012 10:09 AM

Decimal 165 = Binary 1010 0101 = Hexadecimal A5 (because 1010b = Ah and 0101b = 5h).

Just as in decimal we count 8, 9, 10, 11, so in hex we count E, F, 10, 11 ... 1E, 1F, 20, 21

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#2

### Re: Numbering systems

08/29/2012 10:15 AM

Hexadecimal is base 16, so there are 16 digits needed - 0 thru 9 is 10 and A thru F is 6, 10 + 6 = 16. There aren't any digits after F. A lower case h is often used after hex numbers and a d after decimal numbers to avoid confusion. To represent a number greater than 15d (Fh), 2 digits are used just as 2 digits are used to represent decimal numbers greater than 9, 11 for example.

Therefore,

31d = 1Fh (1*16+15)

32d = 20h

33d = 21h (2*16+1)

34d = 22h

35d = 23h (2*16 + 3)

Good luck

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#3

### Re: Numbering Systems

08/29/2012 11:34 AM

Hex is simply for conveniently representing groups of four binary digits.
Take your long binary number, chop it into groups of four bits and work out each group of four bits into a decimal number, if the decimal number is bigger than nine, you use the appropriate letter to avoid having two characters.
By using 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F we can keep one alphanumeric character to represent any four bits of binary.
And thus two hex digits can conveniently represent an 8 bit binary word.

E.G
1010 is equal to 8 plus 2, which is ten in decimal (which is two digits and thus confusing).
ten in decimal is equal to A in hex.
Hex is really for going for expressing binary, the decimal is just a red herring.

Del

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#4

### Re: Numbering Systems

08/29/2012 12:07 PM

Are you wondering about case? Whether to use upper or lower or mixed case in the way these numbers are written out?

If that's your question - I rather suspect you understand the number system itself and that your question is more one concerning notational conventions; is this the case? (no pun intended) - then typically what is done is to keep whatever case you pick. To wit:

10fa48b

or

10FA48B

either one. If you're, say, a tech writer, look at what your Tech Pubs dept is using already and stick with that. You said you're a student; what does your prof use? Your textbook/handouts?

Consistency is more the rule here. There are no hard-and-fast rules when it comes to writing these numbers out, just conventions.

Historically, the letters are upper-case because Teletype®s had only upper-case letters. They didn't print in lower case. Consequently, printed matter then and later kept the upper-case convention out of momentum. One of those "accident of birth" things.

For my part, I use upper case for another reason: sometimes when I cite a hexadecimal as the first item in an English sentence the uppercase works out naturally, as English sentences typically begin with an uppercase letter. That's just a personal choice, however. If you're working in an established context, best to stick with that.

Hope this helps.

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#5

### Re: Numbering Systems

08/29/2012 12:12 PM

I understand decimal (0--9) Binary (1--0), Octal( 0-77777), Hexadecimal (0-F)

Each integer in Hex has a different unique symbol when you exceed 9.

I also understand that you can represent any number in either system and that the various numbering systems are simply shorthand to represent a basic binary number ,so that you don't have to print a long stream of 1's and zero's.

Hex is Base 16--0--thru F,

So what is base 32: 0 thru U?

What would base 64 be for the number 33:

Would it be V?

For example, Hex 1F24=500 decimal.

Or would you simply repeat the Hex System to get the value?If they do, it seems like a lot of typing when you reach the higher numbers when it could be simplified by a different symbol instead.

There is a gap in my understanding somewhere, and I may not be explaining my question very well.

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#6

### Re: Numbering Systems

08/29/2012 12:31 PM

Base 32 isn't used. Too complicated. But if it were, that's prolly how they'd do it.

Think of it this way: all of those symbols are just that: symbols. And that includes the numerals 0 through 9. They're just symbols. All of 'em. If history had taken a different course (pun intended), we'd prolly be using some other symbols, but we'd still be using them. What matters is that the symbols are unique.

You don't want two identical symbols representing two different things - unless you're my wife and married to me. She does that with pronouns and it drives me stark raving bluddy mad (which is why they stuck me here. You don't think I actually signed-up for this, do you? It's called Therapy lol).

If you want to talk about symbols representing large numbers, you need look no farther than the Roman Numeral system. They've got 'em all, but don't try to do any complex math with 'em. Somebody actually wrote a Chrome calculator plugin wot does arithmetic in Roman Numerals. Wish I had that kind of time. Wait a minute! I do!

One thing those Romans didn't include in their system is the idea of Zero. (Not Zeno, that's some Greek chap who flunked Calculus). What an impact Zero had on Progress (I use the term lightly). There's a book called, simply, Zero, that talks about the history of zero and is worth a read. Very much so. Fascinating history. Check it out.

Let Security know if you have any other questions lol.

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#7

### Re: Numbering Systems

08/29/2012 12:53 PM

Hell is therapeutic?

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#8

### Re: Numbering Systems

08/29/2012 1:03 PM

You gotta read behind the lines lol.

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#16

### Re: Numbering Systems

08/29/2012 1:42 PM

They make you stand in line there?

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#19

### Re: Numbering Systems

08/29/2012 1:47 PM

No, they make us stand on our heads. Don't believe those TV adverts.

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#9

### Re: Numbering Systems

08/29/2012 1:24 PM

Reminds me of my mathematically challenged sister. After she graduated college, she figured out the problem she had with algebra was getting the idea of letters being used as symbols for numbers. To her, letters were supposed to be used to make words, words sentences, sentences paragraphs, etc. So A + B = 6 should have been part of a great novel or something. As a side (side?) note, I think algebra is sometimes taught to young children using apples, bananas, clubs, spades, etc. as symbols.

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#11

### Re: Numbering Systems

08/29/2012 1:32 PM

Algebra's a great way to muddy the water when talking number systems .

A + B = 6 ? wrong. A + B = 15 .

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#15

### Re: Numbering Systems

08/29/2012 1:39 PM

Ok, but what's M + N = ?

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#17

### Re: Numbering Systems

08/29/2012 1:45 PM

Credit for this one goes to CR4 member Yuval...

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#21

### Re: Numbering Systems

08/29/2012 1:50 PM

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#105

### Re: Numbering Systems

09/05/2012 7:37 AM
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#18

### Re: Numbering Systems

08/29/2012 1:47 PM

Base 32? That'll be ... umm ... errr ... lemme see ... oh, yes! M + N = CBA .

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#20

### Re: Numbering Systems

08/29/2012 1:49 PM

About that nap being more productive.... lol

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#66

### Re: Numbering Systems

08/31/2012 5:44 AM

So, somewhere in our vast universe, there are probably creatures with only 2 "fingers" that naturally count in binary, creatures with 8 "fingers" where octal is the norm,and so on.

1 billion in hex would be 369ACA00.

Anyone want to tackle a Googleplex in hex?

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#68

### Re: Numbering Systems

08/31/2012 7:54 AM

Ancient Babylon used Base 60. Sentient millipedes?

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#71

### Re: Numbering Systems

08/31/2012 9:21 AM

No wonder they "Babbled".

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#72

### Re: Numbering Systems

08/31/2012 11:10 AM

Well, there are 360 degrees in a circle. Mariners use points. (they actually WERE pointy sticks on a nav board...) where we would get phrases like "four points off the larboard bow, head North by north east. Lets see..that would mean that they would be working in a base 16 system (I think) Clocks of course work in base 12 and base 60. Architectural story rods work in base 6 (studs per 8 foot span) or base 18 (bricks per 8 foot span) So really, number theory is fairly straightforward, and you can use pretty much ANY base you care to choose. Whatever is suitable for your operation.

Once I figured out that base ten is as arbitrary as any other system, a whole new world opened up to me. Personally, I am partial to octal.

So in what system would a millipede have a thousand legs?

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#74

### Re: Numbering Systems

08/31/2012 11:51 AM

I still feel that due to my time in British schools between 1952 and 1973 gave me (and many others) the basis for learning counting in almost any format.

http://en.wikipedia.org/wiki/Imperial_units

Firstly the coinage, in a Pound Sterling (at that time) was as follows:-

960 quarter pennies

480 Half pennies

240 Pennies, each one of them accurately cut enough to allow usage as weights.

80 Three penny pieces

40 Sixpences

20 Shillings

10 two shilling pieces - Florins

8 two and half shilling pieces - half crown.

4 Crowns

We had similar things with weighing, Imperial:-

16 Ounces = one pound

14 Pounds = one Stone

2 Stones = one quarter

4 Quarters = one hundred weight (112 pounds)

20 Hundredweight = 1 Imperial Ton.

Then with distance:-

12 inches = one foot

3 feet = 1 yard

22 yards = one chain

10 chains = one Furlong

8 Furlongs = one Mile

3 Miles = one League

This only a small simple example.

But as you see, we had to be able to count in various bases, within the same things and to be able to convert from one to the other......good training for computer math.....

About 1969, I had to help a Yale Uni Professor pay his bill in a London restaurant......he could not figure it out!!!!

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#75

### Re: Numbering Systems

08/31/2012 11:59 AM

Great post! Fun.

It is amazing that, even now, a billion in Great Britain is different than a billion in the US. One could hope that at least that would be standardized.

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#76

### Re: Numbering Systems

08/31/2012 12:09 PM

The 109 interpretation is pretty much universal* here, now.

*OK - maybe Britland isn't the whole universe - but it's certainly at the center!

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#78

### Re: Numbering Systems

08/31/2012 12:52 PM

Only the USA dances to a different mathematical tune as far as I am aware......

Its that old "NIH" problem.........

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#77

### Re: Numbering Systems

08/31/2012 12:50 PM

German Billion = 1.000.000.000.000

British Billion = 1,000,000,000,000

US Billion = 1,000,000,000

Look here:-

http://www.jimloy.com/math/billion.htm

The USA is different to most other countries!!! They should "join" the rest of the world!!!!

I always said you Guys could not count!!!!

Joke.

An American comes to a British Doctor in the UK.

American. "Doctor, I am married to a lovely English lady here and she wants longer/more sex with me."

Doctor. "And how often per night do you manage it at this time?"

American. "Between 1 and 3 times..."

Doctor. "That seems a reasonable number of times to me, but I can give you a book and some exercises to help further. Come back each morning and tell me how you did."

American. "Great Doctor, thanks."

Next Day. Doctor. "How many?"

American. "10 times Doctor, but she wants more!!!

Doctor. "Wow, can you?"

American. "Yes"

Next Day. Doctor. "How many?"

American. "20 times Doctor, but she wants more!!!"

Doctor."Wow, can you continue?"

American. "Yes"

Next Day. Doctor. "How many?"

American. "50 times Doctor, but she wants more!!!"

Next Day.Doctor."Wow, can you do that?"

American. "Yes"

Next Day. Doctor. "How many?"

American. "150 times Doctor, but she wants more!!!"

Next Day.Doctor."Wow, HOW can you do that? That is really amazing!!!!!"

American, swaying back and forth in time with his counts "Once, twice, three times........."

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#81

### Re: Numbering Systems

08/31/2012 1:36 PM

#74 is a veritable nexus of British Imperial Units. I love it!

Totally OT GA (hehe)

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#80

### Re: Numbering Systems

08/31/2012 1:26 PM

Q: So in what system would a millipede have a thousand legs?

Well, let's have a look: Hmm..not very promising. Seems there are 10,000 species of millipede covering 13 orders and 115 families (gosh..wonder if my ex is related...), so let's just cut to the chase and pick the bad boy of 'em all: Illacme plenipes. These blokes weigh in at a whopping 375 pairs. None have a thousand legs as far as we know. Still, I can't imagine buying shoes for them all, and you can forget pantyhose.

A: Base 9.0856(...).

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#89

### Re: Numbering Systems

08/31/2012 7:17 PM

You are confusing base number systems with modular arithmatic; anything that has repeating cycles or patterns. All of your numbers above are written in decimal base-10.

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#91

### Re: Numbering Systems

09/01/2012 9:00 AM

It doesn't matter, positional notation is only a way to help it to look "normal". Fact is, if I am counting laps of a race car, and find out that it is 85 seconds in one lap, 45 seconds in a second lap and 52 seconds per lap in the subsequent four laps, then I need to add in base 60 to determine how many minutes it takes to do the race.

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#86

### Re: Numbering Systems

08/31/2012 5:54 PM

When base number theory is presented in math class, the statement is made that ANY number base system can be used and they will give a few examples in an unusual base (i.e. 7, 3) to demonstrate that it is really possible and ensure that the student understands the concept.

In real life, where "practical" and "useful" are important terms, the only useful numbering systems are: 10=decimal, 2=binary, 8=octal, and 16=hexadecimal. That is it. I have never seen any others. The reason for these should be obvious but maybe not to everyone.

(10) Because we have ten fingers.

(2) Only important once computers were invented because that is how the memory works. (If someone invents a tri-state memory chip then a base-3 system would be useful.)

(8) This shortens the string of characters (3 bits) and still only uses one numeric digit without any special characters.

(16) This shortens the string of characters (4 bits) and only uses a few extra letters. Two hex digits represents a byte (8 bits, 0-255). Four hex digits represents a full word (16 bits, 0-65535). This is much shorter and easy for the human eye to detect changes.

Any number base higher than 16 is impractical because the number of characters required increases exponentially. English only has 26 letters and we generally do not count with those. Who would want to go higher?

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#87

### Re: Numbering Systems

08/31/2012 5:58 PM

The Babylonians?

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#88

### Re: Numbering Systems

08/31/2012 6:25 PM

This forum has led me to many areas in search of answers, and in my recent research, I have discovered:

There is a base 64 number system,using upper case alphabet and lower case alphabet characters, plus 2 other symbols, usually + and /, as shown in a link from #24 above.

There is also tristate logic that has been around for many years.

Bus Enable is the third state.

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#94

### Re: Numbering Systems

09/01/2012 9:33 AM

and then there is single sideband...

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#95

### Re: Numbering Systems

09/01/2012 12:02 PM

Now you are talking Ham radio or CB's.But there are now a multitude of broadcast methods,besides the old familiar am/fm,ssb.Digital being one of those.

In reality, digital transmission is still analog, it is just broken up into pieces of a certain sequence that are interpreted into data useable by the computer on the receiving end, which in turn transforms it into intelligible data for human consumption.And yes, everything inside every computer is still just one's and zeroes.(Unless you are talking about Quantum computers,that have an infinite number of states that exist between one and zero!).

Anything besides one or zero is simply there for our convenience.

Once upon a time all programming had to be done with single bits.

Programs were very small, and had to be well thought out and very compact.

As programs became larger, better ways of getting all of those one's and zeroes into the computer were developed, but still, memory was at a premium,and processor speed was very slow,so there was still a premium placed on efficiency and size of the program.

The various numbering systems are simply shorthand methods to these bits.

Now the evolution is at the stage where memory is virtually unlimited,speed is astronomical and GUI's have replaced the hard work of having to manually input every command. Just to construct an icon used to require many lines of code,with every bit position identified.At a certain level, this is still occurring,but we don't have to worry about it as a user.The programming guys, the ones that actually work with ones and zeroes, on machine level, that roll their brains in on a wheelbarrow every morning do that for us.

They are the guys in the trenches doing all of the grunt work, and modern society owes them a great debt for all of our modern conveniences.

There are programmers that use various programming languages to accomplish an assigned task,but they are using the tools made by the geniuses at machine level.

The emphasis on these programmers now is to "Get a working product out the door,and we will fix the bugs as they pop up."

Competition is a 61tch!

Hence all of the bugs in new software and the afterthought revisions.

I do not wish to stray too far from the original subject, so I'll stop while I am still on topic.

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#96

### Re: Numbering Systems

09/01/2012 10:23 PM

carries information. Just not digital information.

A remarkable invention in any case.

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#98

### Re: Numbering Systems

09/02/2012 4:16 AM

I understand SSB. Half of an rf carrier wave is wasted when it goes thru the detector diode .Someone decided to save a lot of bandwidth and power and only send one half of the normal signal,either upper or lower, thus doubling the number of channels.A great idea indeed, and it seems obvious and simple in retrospect, but that is the way of many great ideas."Gee,wish I had thought of that!"

We live in an analog world, and I prefer analog where possible,like vacuum tube amplifiers instead of digital solid state renderings of the same material.

The problem with digital in some cases, is the chopping up and reassembly of the original material, no matter how elaborate, is still not the same.The gaps may not be noticeable,but they are there nonetheless.Some numbers will not digitize to infinite accuracy,so an approximation must be made.It is easy to cut a pizza into 3 pieces, exactly equal, but it is not possible to represent 1/3 in decimal or binary exactly.The numbers are rounded off.This rounding off may not be significant after a certain number of decimal places, but the effect is a lack of ambiance,of true "reality".

The younger generation will not notice this, for they have no point of reference, and us old timers are fading away into the foggy mist of the past and will soon be just a photo on a wall, and our descendent grandkids kids will ask"Who was that?"

And they will reply "That was your great great great great grandfather.He lived in an analog world.They actually built things by hand back then, and walked outside in the sunlight.And there were flying things in the sky called birds,and things that were actually alive in the ocean, terrible monstrous things,before we learned how to control it for our purposes.And people actually ATE things that came from the water, they did not have universal food like we do.I can't remember his name though."

And so it goes, not always upward, but always onward.

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#99

### Re: Numbering Systems

09/02/2012 4:20 AM

I went for broke and bought a used Icom wot transmits on Neither Sideband (NSB).

I don't even have to turn it on.

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#102

### Re: Numbering Systems

09/02/2012 4:51 AM

My netbook seems to be bristling with upper sideband transmitters, but I don't know how to connect the antennas.

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#103

### Re: Numbering Systems

09/02/2012 4:57 AM

(touché!)

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#92

### Re: Numbering Systems

09/01/2012 9:24 AM

Well put. I totally agree and gave you a GA to negate that OT, so you are back to 0 again.....

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#12

### Re: Numbering Systems

08/29/2012 1:35 PM

europium and Del have it. You're having trouble grasping 16 symbols (as would any of us), imagine 32 or 64! You are right that the main use for base16 is to make handling large binary numbers easier. A long time ago (yes I remember), octal was used to handle 3 bits at a time. I suppose that was to eliminate the need to agree on new symbols. But since all(?) computers used multiples of 4 (now 8) bits in the processors, hexadecimal came into general use. What took 3 octal digits can be expressed in 2 hexadecimal digits.

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#22

### Re: Numbering Systems

08/29/2012 1:50 PM

Yeah, but only Del got a GA.

Serves me right, I suppose (I knew I shoulda given the OP an example using Mersenne Primes )

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#67

### Re: Numbering Systems

08/31/2012 6:28 AM

Thought only Poms were supposed to whinge .

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#82

### Re: Numbering Systems

08/31/2012 1:40 PM

Well, you guys did colonise this joint you know. Take some credit for a job well done!

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#23

### Re: Numbering Systems

08/29/2012 3:00 PM

1F24 is not 500 decimal !!!
500 decimal is

1 1111 0100 in binary

taking this as blocks of 4 as in my first post gives
0001 1111 0111 this becomes
1 F 7 in hex

I think you are guilty of overthink .
I once heard someone giving a huge ly complicated explanation and confusing a student.
The thing to realise is it's just done for convenience. Forget the decimal if possible and work between binary and hex or vice versa. Only convert to decimal if necessary.
Why? Because binary is what we use when actually use manipulation the bits in the electronics, driving ports turning things on and off etc. Hex is just a convenient way of writing binary.
Del

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#26

### Re: Numbering Systems

08/29/2012 3:24 PM

Um, I think it's 1F4h

1 * 256 = 256

15 * 16 = 240

4 * 1 = 4

256 + 240 + 4 = 500

I think something slipped in your comment between lines 3 and 5

Never mind - no point in beating a dead horsecat. One thing about this place - there's no point in checking your work, there's plenty of others going to do it anyway. Right?

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#27

### Re: Numbering Systems

08/29/2012 3:57 PM

I just miss-copied the binary which perfectly illustrates why we need hex!
thus with a swish of his cape Del turns disaster into triumph as scampers away unscathed
Del

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#28

### Re: Numbering Systems

08/29/2012 3:59 PM

Where'd you buy that Hex Inverter? I want one!

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#30

### Re: Numbering Systems

08/29/2012 4:22 PM

Guy called Schmitt's got some 7414's going cheap ...

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#31

### Re: Numbering Systems

08/29/2012 4:24 PM

Would that be that trigger-happy TTLator and counter-hex maestro Jacques Schmitt?

Or was it 'hex counter.' Do they come in six-packs? Counters, I mean.

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#58

### Re: Numbering Systems

08/30/2012 4:36 PM

I don't know Jaques Schmitt.

Or maybe I do. Double negatives are always tricky.

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#33

### Re: Numbering Systems

08/29/2012 6:38 PM

You are correct.That was a typo on my part,caused by fat fingers on slim keyboard.

It is interesting that they still use Hex when 64 bit computers are now the norm.

Admittedly,Hex is easy, once accustomed to it, but it looks like a longhand way to handle 64 bit numbers, especially if the number is really large.

Even handheld calculators can convert from binary to Hex to Decimal, etc.

Anyway, I yield to the powers that be, I am just a newbee, and , as someone said, guilty of "Overthink"

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#34

### Re: Numbering Systems

08/29/2012 6:48 PM

Never, ever think of yourself as "just a noobie." That's the worst thing you can do.

Think of yourself as "someone with a fresh pair of eyes."

There might be nothin' new under the sun, but I rather suspect most of it has yet to be seen - and only by unencumbered eyes.

"Seek the wisdom of the ages, but look at the world through the eyes of a child."
~ Ron Wild

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#59

### Re: Numbering Systems

08/30/2012 6:23 PM

"There might be nothin' new under the sun, but I rather suspect most of it has yet to be seen"

That's very good. I'm going to use that.

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#60

### Re: Numbering Systems

08/30/2012 6:31 PM

If you think it's very good, why not rate it with a GA?

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#61

### Re: Numbering Systems

08/30/2012 6:38 PM

Because, while used as an encouragement to the new user, it was not a direct answer to the original post. This leads me to thinking it is in an "off topic" area. GA for off topic doesn't seem to flow from my thinking.

If I have breached proper site etiquette, let me know. That is certainly not my intent. I am more than willing to adapt to the current environment.

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#63

### Re: Numbering Systems

08/30/2012 6:57 PM

You certainly haven't breached any site etiquette that I'm aware of (or should I say "of which I'm aware" - there are those among us who'll jump on you from a great height for breaching the Rules of Grammar ).

The problem is that GAs and OTs shouldn't interact in the way they do; an answer can be both "good" and "off topic", but there is no mechanism on CR4 for expressing that unambiguously.

Best recourse is (IMHO) - if you enjoyed it (including just finding it funny), or thought it insightful or useful in the context of the thread - give it a GA.

Feel free to make up your own rules .

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#73

### Re: Numbering Systems

08/31/2012 11:24 AM

Reward gives no reward!!!!!

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#84

### Re: Numbering Systems

08/31/2012 1:49 PM

Sure you aren't thinking of 'Just_Desserts'?

(do those come with sprinkles?)

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#38

### Re: Numbering Systems

08/29/2012 7:09 PM

1F24 Would equal 7972 in Decimal.

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#43

### Re: Numbering Systems

08/30/2012 1:51 AM

On the contrary mon ami, Hex is easy to change from 64 bit 1 & 0 to hex or decimal.

1010|0011| =8 bit

i normally us 2 to the power method if the most right is the LSB (Least Significant Bit)

that is 2 to the power 0 (=1) then the next one is 2 to the power 1 (=2) etcetera.

If you do all the 64bits like this it gets tedious, so just doing it in groups of 4 is more convenient.

so the right 4 bits

23 22 21 20

0 0 1 1 = 2 + 1 =3 dec and 3 hex

left 4 bits

1 0 1 0 = 8 + 2 = 10 decimal and A hexadecimal

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#69

### Re: Numbering Systems

08/31/2012 8:11 AM

You lamented: "a typo on my part,caused by fat fingers on slim keyboard."

Not a problem. Think about it:

• Whole new and completely unforeseen branches of novel mathematics have been unwittingly spawned this way. For instance, 2 + 2 = 5 for very large values of 2. This fact would never have been discovered were it not for fat 'fingers on a slim keyboard'.
• Were it not for 'fat fingers on slim keyboard', millions of Facebook users wouldn't have any Friends at all,
• Microsoft writes most of their software that way (though I tend to suspect they prefer to use their thumbs). For instance, Windows was supposed to be called Panes In the Arse.
• My fingers are huge. Did you think my writing this post was an act of volition?

And so forth
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#90

### Re: Numbering Systems

08/31/2012 7:50 PM

I suggest that you try several test problems in base16 and base64. Use the same values but converted to the different systems, then you will see the advantage of having fewer characters to work with and keep a sense of value for.

111d x 150d = ?

Would you like to pay ABCDh for a sandwich or just Aa-64?

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#24

### Re: Numbering Systems

08/29/2012 3:15 PM

You seem to have a couple misunderstandings.

An integer is a number with no fractional part. 1 is an integer. 502 is an integer.
So when you write Each integer in Hex has a different unique symbol when you exceed 9, you should really write "Each digit in hex has a unique symbol: 1 through F."

The value of a digit in a particular place is consistent in numbering systems. Starting with the rightmost place, the value of that place is the digit times the base0. The next to the left is the digit times the base1, the next the base2, etc.
So 500d in hex is not 1F24. It is 1F4: (1 x 256) + (F x 16) + (4 x 1).

Large numbers in hex will always require fewer digits than the same number in decimal will.

Another convention you may see is Ox preceding hex numbers (instead of h after).

Base 32 and Base 64 are not generally used (but take a look at this.)

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#25

### Re: Numbering Systems

08/29/2012 3:17 PM

Ok ...
1F4 ... but I was close (ish)
It's hard to do counties with these paws.
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#64

### Re: Numbering Systems

08/30/2012 10:35 PM

My polydactylic cat operates in base 6 or 12, depending on whether one or two feet!

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#29

### Re: Numbering Systems

08/29/2012 4:22 PM

Now we all know how a mime counts...

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#32

### Re: Numbering Systems

08/29/2012 4:35 PM

Yeah, but what comes after mime?

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#53

### Re: Numbering Systems

08/30/2012 11:49 AM

Revolution Mime?

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#83

### Re: Numbering Systems

08/31/2012 1:43 PM

Tem. (sigh)

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#35

### Re: Numbering Systems

08/29/2012 6:49 PM

THANKS! That explains it perfectly.I was on the right track, I just didn't know how to ask the question properly.

According to your link,: " Base 64 is also occasionally used in computing, using as digits "A-Z", "a-z", "0-9", plus two more characters, often "+" and "/".

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#62

### Re: Numbering Systems

08/30/2012 6:56 PM

Hey Bosen Mate Higgs,

Thanks for serving your country and covering our *!\$\$.

Keep you head down and stay safe brother.

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#36

### Re: Numbering Systems

08/29/2012 7:02 PM

I dunno 'bout that. I get...

0x7FEFFFFFFFFFFFFF = 1.7976931348623157 x 10308

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#10

### Re: Numbering Systems

08/29/2012 1:27 PM

Also if you are working with large numbers, don't sweat it, use your window calculator.

on start menu below click window start button.

In "search for programs and files" dialog box, type "Calc" you will see windows calculator link.

When calculator pops up, click view, and select "Scientific"

Type in dec # you want converted to hex, click "Hex" tick box to see, or click binary tic box, type in binary 1s and 0s #, click "Hex" tick box to see binary # converted to Hex. Etc.

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#13

### Re: Numbering Systems

08/29/2012 1:35 PM

I used to teach night courses in C and C++ programming to corporate weenies wanting to brush up on their skills. I absolutely forbade the use of calculators in class until I was satisfied my students understood what that calculator understood and why.

Your recommendation is good, and a time-saver for people who need it, but there's nothing like understanding what you're doing. Prematurely using a calculator in the learning process is crippling. I know, because I know kids who cannot fathom the concept of long-division on paper. My kids can take square roots on paper because dad's one of those Auld Skool types who insist on knowledge acquisition, not just learning something so they can regurgitate it on a TACS test. That's just animal training and dumbest damn thing ever to crawl out from under a rock, IMHO.

Gotta go. Gonna be late for my daily rabies shot lol. Take care.

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#37

### Re: Numbering Systems

08/29/2012 7:03 PM

I am working my way thru a digital instruction book, and I can now trace a keystroke from the key thru every gate to the display output, of a 7 segment display.The logic is simple, but when combined in the proper combinations,gives amazing results.

I realize this is very basic, but I am old school too, and I want to know from the ground up how stuff works.I have a long interesting road ahead.There is so much information available now!

I also bemoan the loss of effective teaching being replaced by rote learning,which is a sad commentary on parenting in our society, the non corporal punishment in schools and at home, and the emphasis on quotas instead of a real education being required to graduate.

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#40

### Re: Numbering Systems

08/29/2012 11:58 PM

Good. So now you can get started on something MUCH more interesting...

Boolean Algebra.

But before you go....

Grey Code

Try not to tear your hair out....

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#42

### Re: Numbering Systems

08/30/2012 1:39 AM

Interesting - I never knew about Gray Code. So it's not actually based on a "base" but rather on the mechanical limitations of switches. I have a vision of all digital electronic devices being driven to a state of chaotic breakdown and becoming suicidal when someone suggests switching to Gray Code to make things work faster.

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#46

### Re: Numbering Systems

08/30/2012 7:22 AM

The limitations of switches weren't quite the motive for developing Gray Codes. As only one bit changes state between any pair of adjacent symbols, Gray Codes are more robust codes in terms of signal-to-noise ratio than are other codes. Gray Codes are also handy if you want to minimize switching transients that may occur when, say, a whole gaggle of bits change state simultaneously, ie, from 0xFFFF to 0x0000 or vice versa, for example, in systems where such transients may introduce problems. Gray Codes are still used and not that much of a pain, really (IMO).

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#49

### Re: Numbering Systems

08/30/2012 8:40 AM

I am familiar with Grey Code, it is used in position resolvers for radar, among other things.It is called an ABSOLUTE encoder,versus a relative encoder.It generates a unique code for every position and is not affected by power loss,whereas a relative encoder simply generates the same pulse train constantly when in motion, and no signal when stopped.

I am studying Boolean Algebra now, and it is very interesting.A very convenient way to express binary circuitry.Just a matter of understanding the symbology and operands.

Rules.Rules.And more rules.Everything is rules.Learn the rules, you can play any game,and like any game, the more you play,the better you get.I hope to get into the amateur leagues with my pursuit of digital.I am still striking out more often than I hit,but I am getting better.

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#50

### Re: Numbering Systems

08/30/2012 9:49 AM

"...and one company to rule them all, and in the courtroom, bind them." - Your Friendly Neighbourhood Fruit Company, Cupertino, California

"Just think that if Eve had offered Adam Linux instead, we wouldn't be in the damned mess we're in today." -Grognak the Barbarian

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#51

### Re: Numbering Systems

08/30/2012 10:17 AM

"Straight" binary could be used for an absolute rotary encoder, but it would be prone to errors when reading the position near a point where two or more bits changed value at the same position - the worst case being at the 360°/0° point, where the result changes from all 1s to all 0s. Due to slight variations in the mechanical alignment and electrical characteristics, the values read during the transition could be anything between all 1s and all 0s.

Gray code (note the spelling - named after Frank Gray) circumvents this by having only one bit change at a time, so the position is always read correct to the least significant bit (after conversion back to binary).

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#97

### Re: Numbering Systems

09/02/2012 3:55 AM

"'Straight' binary..." The Establishment's answer to Gay binary?

'Gray' Code is spelt with an 'r' right?

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#100

### Re: Numbering Systems

09/02/2012 4:42 AM
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#101

### Re: Numbering Systems

09/02/2012 4:44 AM

YESSSSS!!!!!

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#52

### Re: Numbering Systems

08/30/2012 11:07 AM

My mistake...that would of course be GRAY CODE, not "GREY CODE".

I love machine language....there are no vowels...grrr.

And yes, it is used on radars, and elevator systems, and things that make sounds like star trek sound effects. Its kind of fun to be adding (say angles) using gray code. I never mastered it...I always just converted them into standard numerical binary and did an end around carry (or whatever operation I needed to do) and then converted it back to gray code. Its really nice because there are, like NO transient spikes from counting up from (say) 7 to 8 ...binary would spike the power supply with this count, 0111 to 1000 and gray code would go from 0100 to 1100, (same thing of course)

You can get a converter here if you really want one.

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#65

### Re: Numbering Systems

08/30/2012 11:36 PM

Machine language you say?

AAC -Alter All Commands

AWP -Argue With Programmer

BB -Branch on Bug

BD -Branch to Data

BLP -Boot from Line Printer

CAF -Convert ASCII to Farsii

CCB -Consult Crystal Ball

CPB -Create Program Bug

CRASH -Continue Running After Stop or Halt

CU -Confuse User

DAB -Delete All Bugs

DBTP -Drop Back Ten and Punt

DBZ -Divide by Zero

DC -Divide and Conquer

DDS -Delaminate Disk Surface

DSUIT -Do Something Utterly, Indescribably Terrible

DWIT -Do What I'm Thinking

EBRS -Emit Burnt Resistor Smell

EDP -Emulate Debugged Program

ETI -Execute This Instruction

EWM -Enter Whimsy Mode

FPT -Fire Photon Torpedoes

FMP -Finish My Program

FRL -Fudge Result of Last operation

GDP -Grin Defiantly at Programmer

GREM -Generate Random Error Message

GSI -Generate Spurious Interrupts

GSO -Generate Stack Overflow

HELP -Type "No Help Available"

HHB -Halt and Hang Bus

HSC -Halt on System Crash

IAND -Illogical AND

IDI -Invoke Divine Intervention

IPI -Ignore Previous Instruction

IRPF -Infinite Recursive Page Fault

JNL -Jump when programmer Not Looking

JRAN -Jump RANdom

KCE -Kill Consultant on Error

KEPITU -Kill Every Process In The Universe

KSR -Keyboard Shift Right

LAP -Laugh At Programmer

LOAR -Log Off At Random

LOL -Laugh Out Loud

LRO -Lose Result of Operation

MAN -Make Animal Noises

MWAG -Make Wild-Assed Guess

NBC -Negate By Clearing

NIH -No Instruction Here

NMI -Negate Most Integers

NOP -Needlessly Omit Pointer

NPC -Normalize Program Counter

PUGH -Pack Up and Go Home (Unconditionally)

PUNT -PUNT (recursively)

RD -Randomize Data

RPSW -Randomize Program Status Word

RU -Recurse Unconditionally

SAI -Skip All Instructions

SFT -Stall For Time

SHABM -Shift a Bit More

SHB -Stop and Hang Bus

SIO -Scramble Instruction Order

SMR -Skip on Meaningless Result

SRBO -Set Random Bits to Ones

SRBZ -Set Random Bits to Zeroes

SRBI -Set Random Bits then Invert

TDRB -Test and Destroy Random Bits

TOS -Trash Operating System

TPONR -Turn Power Off for No apparent Reason

UAI -Use Alternate Instrucction set

UOP -Useless OPeration

UUBR -Use Undefined Base Register

VVM -Vaporize Virtual Memory

WBR -Wait then Branch Random

WF -Wait Forever

WP -Wake Programmer

WSWW -Work in Strange and Wonderous Ways

WUD -Work Until Demo

XIO -eXecute Invalid Opcode

XPSW -eXecute Program Status Word

YAU -Yell At User (synthesis plugin required)

ZAR -Zero Any Register

ZBR -Zero and Branch Random

ZEOW -Zero Every Other Word

ZPM -Zap Physical Memory

ZSR -Zero Stuff at Random

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#70

### Re: Numbering Systems

08/31/2012 9:16 AM

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#79

### Re: Numbering Systems

08/31/2012 1:01 PM

Opcodes of course! Please don't tell me the 'machine language' to which you refer was literally 'ones and zeros'!

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#93

### Re: Numbering Systems

09/01/2012 9:27 AM

Well, yes of course it was!!! Ones and zeroes that is. Dedicated flight computers only know whether airplanes or are on the ground or not, or whether flaps are deployed or not, and so forth. Mostly a series of Yes-No decisions combined with some basic logic no more complicated than "are we in the air -yes, is the arm switch slected on-yes, is the trigger selected on -yes, are the guns loaded or empty -loaded, THEN send a signal to the engine igniters to restart the engines in case they choke out from the smoke from the guns.

We would use ascii from time to time to keep it all straight, but I don't remember any opcodes. (though I think we used phrases like "input error corrected, pilot replaced" "IEPR" sort of a YAU code) But then, I left the world of airplanes just when 'puters were coming into their own, causing more problems than they were solving. I developed a deep antipathy towards the blessed things because they were always impossible to field repair, and it was too difficult to fault find the complex aircraft systems with all those black boxes which seemed to have stray gemlins inside them. I would hope that things would be different these days, but I doubt it.

You would think that the amount of engineering that is lavished on these graceful sky traveling miracles of twenty first century technolgy that some effort would have gone into designing them to be easily fault found. Rows of test jacks so that you can monitor the system without taking plugs off the "black boxes", even my car has a plug into which you can plug in a diagnostic computer. But airplanes....well, electrical snags still account for 40 percent of down time. You would think something would be done about that by now.

I WISH that back in the day, we had the time to come up with those cute opcodes. I like the "programmer asleep" code. I know they are all (or mostly) tongue in cheek, and they gave me a chuckle to read down the list.

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#39

### Re: Numbering Systems

08/29/2012 10:51 PM

Totally agree. Try giving one of these youngsters a few pieces of change along with a bill AFTER they've rung up the sale on the cash registers. They can't make change unless the register tells them how much. If something totals \$4.19, I try to give them the 19 cents, or even worse, 20 cents in order to get rid of a pocket full of change. For the vast majority, they can't do it in their head, because they never learned basic mathematics. In most public schools, calculators are used. I've seen some of the "smarter" kids wait until you give them the money BEFORE they enter total tendered into the register!!!

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#14

### Re: Numbering Systems

08/29/2012 1:36 PM

With the Win 7 calculator, 'scientific' doesn't do it - you have to select 'view -> programmer' (or use the shortcut ALT+3).

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#41

### Re: Numbering Systems

08/30/2012 12:00 AM

When I learned computing HEX didn't appear to exist. Probably because FFFF didn't "feel" like a memory location. Anyone remember 173000g or 773000G?

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#44

### Re: Numbering Systems

08/30/2012 4:11 AM

Octal was ideally suited to the PDP-11 instruction set, with basically 8 registers and 8 addressing modes. My first work with computers (may years ago) was to program an LSI-11 to control an isotope emission body scanner. I did it in machine code (or if you like 'hand-assembled assembler'), in octal, entering the code (stored in 4k core memory) initially using a teletype before progressing to a VT52 CRT terminal.

After a while it was pretty easy to do the 'assembly' without referring to the manual (tho' counting the offsets for jumps was a bit of a pig).

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#45

### Re: Numbering Systems

08/30/2012 4:17 AM

By 'eck lad I remember the Digital Equipment Corporation's PDP9
Oh the thrill of toggling those paddle switches
Del

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#48

### Re: Numbering Systems

08/30/2012 7:46 AM

What really thrilled me was when DEC was considering using three-state symbols in an experimental project.

Complete the following sentence:

Binary digits are to 'bits' as trinary digits are to ______.

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#47

### Re: Numbering Systems

08/30/2012 7:39 AM

I am sure that you had all the answers you need, but I still feel, from your first question, that, maybe there is a point to clear:

At the start of the PC history, they used 8 bits registers, which is 8 binary digits, to address the memory. Up to date, 8 bits registers are used still. Also, the Hexadecimal representation was used from the begining.The biggest number (Positive integer that could be represented was 255 decimal (FF in hex). Therefore, 2 registers were used to get to the number 65536 (4 digits in hex, FFFF). This became the 16 bit PC. After that, you have the 4 registers PC which equals 32 bit and the 64 bit (8 register address) PC (the operating system must be designed to use these addresse).

All the above has nothing to do with the numbering system bases which is a mathematical exercise: Octal = base 8 (0 -- 7), 8 bit registers could use this base for representation, but is rarely used. Hexadecimal 16 (0 -- F) each 2 bits can be represented by one symbol. Higher number bases are used in speciality cases as shown by the link provided...

My contribution here is to make sure that you were not linking the 8 bits, 16 bits, 32 bits and 64 bits computer memory addressing schemes with the need to have different numbering bases to be able to use them.

I hope I did not misread you.

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#54

### Re: Numbering Systems

08/30/2012 12:45 PM

Mate;

You are confusing number BASE systems with computer "bit" binary operating systems.

There are no alpha-numeric chatacters after "F" in hexidecimal.

The values are: F, E, D, C, B, A, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 HEX and corespond to : 15, 14, 13, 12, 11, 10 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 Decimal

Hexidecimal is a BASE 16 numbering system that allows assignment of "weighted" values to each bit within a byte of binary DATA in a page frame of computer memory.

Binary system bits are used at the Transistor-Transistor-Logic (TTL) machine system level to execute transistor gate "ON" (1) and "OFF" - (0) that correspond with assigning the system weighted values (in this case BASE 16 value) to the respective memory addresses.

This allows logic function to be executed within the computer/machine by addition subtraction, multiplication, and division of the weighted values within the memory addresses being accessed by the programming.

A 32 bit system has eight (8) bytes with four (4) bits of binary.

A 64 bit system has sixteen (16) bytes with four (4) bits of binary.

This translates into the actual number of bits within a page frame that is available of access by the program per line/row entry.

Search this on wikipedia under number conversions for better understanding and also look at computer machine language for a better understanding of the 32 and 64 bit systems and their respective capability of program execution and machine configuration use.

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