Base 10 Number System Question

[boc4life]

I'm trying to understand some concepts behind the Base 10 number system, and I have a few questions.

Imagining a scenario where after the number 9 came the "number" A

1. What would the order of numbers be?
...8, 9, A, 10, 11, 12...18, 19, 1A, 20, 21...

Does that look right?

2. Would this number system be described as a "Base A" system?

3. (Unrelated to the imaginary number system) Why is Binary referred to as a Base 2 system, when the number 2 does not even exist within the system?



Do I appear to be way off in my thinking?

[SBR]

1) Yes

2) No it would be called base 11

3) Base X refers to a number system with X digits. So a binary or base 2 system has 2 digits (0,1) and the "regular" number system is base 10 (0,1,2,3,4,5,6,7,8,9)

[boc4life]

[ QUOTE ]
1) Yes

2) No it would be called base 11

3) Base X refers to a number system with X digits. So a binary or base 2 system has 2 digits (0,1) and the "regular" number system is base 10 (0,1,2,3,4,5,6,7,8,9)

[/ QUOTE ]

What confuses me is that for our number system, referred to as Base 10, the number 10 exists on the number line. If the only number system in the world that existed was the Binary system, and the numbers 2-9 did not exist, what would we refer to the system as? Consider the same question for my fictional "A" system.

[vhawk01]

The number ten doesnt really exist though. Its simply a 1 in the tens place and a 0 in the ones place. But I suppose I see where you are going, in that we had the benefit of a developed numbering system before we formally structured it and considered other base systems.

[SBR]

[ QUOTE ]
[ QUOTE ]
1) Yes

2) No it would be called base 11

3) Base X refers to a number system with X digits. So a binary or base 2 system has 2 digits (0,1) and the "regular" number system is base 10 (0,1,2,3,4,5,6,7,8,9)

[/ QUOTE ]

What confuses me is that for our number system, referred to as Base 10, the number 10 exists on the number line. If the only number system in the world that existed was the Binary system, and the numbers 2-9 did not exist, what would we refer to the system as? Consider the same question for my fictional "A" system.

[/ QUOTE ]

You need to put a little more thought into this. The number 2 exists in binary as well its just represented as 10. If a base 2 number system was common then we could give the number 10 a special name.

The number ten doesn't exist any more or less than the number (binary) 10. We just give it a special name.

[FortunaMaximus]

Look at your hands. Primal logic. Ten of 'em, right?

The definition of a base is how many distinct digits it contains.

Base 2 (binary) 0, 1. Two digits
Base 10, standard counting system, 0-9 has 10 digits
Base 16, commonly known as hexadecimal, used in computing.
(0-9, A-F)

Putting "A" in the middle of a base would only expand it, in this case, to Base 11, 0-8, A, 9.

Hope that's helpful. Cheers.

[RocketManJames]

Yes, you're pretty much right that it would be called a Base A system in a world that was based upon a Base 11 system (that is... Base 11 in our Base 10 world).

Base 10 in a world that adopted the Base A system would be Base 12 in our world.

Is this what you're getting at?

-RMJ

[boc4life]

[ QUOTE ]
Yes, you're pretty much right that it would be called a Base A system in a world that was based upon a Base 11 system (that is... Base 11 in our Base 10 world).

Base 10 in a world that adopted the Base A system would be Base 12 in our world.

Is this what you're getting at?

-RMJ

[/ QUOTE ]

I believe this is what I'm getting at. As someone said earlier in the thread, the number 2 DOES exist in binary, it's just represented as 10.

Does this mean that for someone who spoke and thought in binary, their system would also be referred to as a "Base 10", with 10 representing a completely different number than the 10 that we know?


Also, referencing another previous post...Is it likely that the Base 10 system that we use was developed because humans have 10 fingers?

[FortunaMaximus]

It's not only likely, it's the simplest possible explanation. And I can't find a better or even worse one.

[disjunction]

It looks your original question has been answered so I'll throw a couple of more thoughts out there. It sounds like you may not be familiar with this, but all computer people think in Base-16. It is very useful to know. Wife permitting, I am personally going to teach my kids to think in Base-16 or Base-32 and learn the appropriate multiplication tables. This will undoubtedly make them more adept at arithmetic.

The bottleneck in doing arithmetic in your head is memory, and with Base-16, there's simply less to remember. Four-digit numbers are now 3-digit numbers.

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

[_TKO_]

[ QUOTE ]
What confuses me is that for our number system, referred to as Base 10, the number 10 exists on the number line. If the only number system in the world that existed was the Binary system, and the numbers 2-9 did not exist, what would we refer to the system as? Consider the same question for my fictional "A" system.

[/ QUOTE ]

Since we function on a base-10 system, we name all other number systems using base-10 notation. We don't really call our number system "base-10" when we refer to it (at least, I assume most people don't), so it becomes necessary to distinguish other number systems using our own preferred notation. As a previous poster noted, base-2 system has 2 digits in it, and base-16 has 16 digits. So the premise for naming a number system is base-x, where x is the number of digits in the system in base-10 format.

[_TKO_]

[ QUOTE ]
This will undoubtedly make them more adept at arithmetic.

[/ QUOTE ]

How do you account for the additional time required to convert decimal to hexadecimal and back again when they are working with the rest of the world using decimal notation?

[ScottHoward]

[ QUOTE ]
Look at your hands. Primal logic. Ten of 'em, right?


[/ QUOTE ]
thats a lot of hands

[disjunction]

[ QUOTE ]
[ QUOTE ]
This will undoubtedly make them more adept at arithmetic.

[/ QUOTE ]

How do you account for the additional time required to convert decimal to hexadecimal and back again when they are working with the rest of the world using decimal notation?

[/ QUOTE ]

First, they will know how to work in decimal as well, for the really simple problems that are presented in the 20th-century language. There's no reason not to be bilingual.

Second, the goal is to be able to do some problems in your head that you can't do otherwise. Do you think conversion will be a problem for someone who's done these conversions since they were 5? Conversions from hex to decimal, since they'll know the decimal value of each digit, are just a series of additions, and conversions from decimal to hex are just a series of subtractions.

Please tell me you think this will work, I am looking forward to using my children as guinea pigs.

[FortunaMaximus]

[ QUOTE ]
[ QUOTE ]
Look at your hands. Primal logic. Ten of 'em, right?


[/ QUOTE ]
thats a lot of hands

[/ QUOTE ]

LOL. The fingers were implied. Uh, nh.

[_TKO_]

[ QUOTE ]
First, they will know how to work in decimal as well, for the really simple problems that are presented in the 20th-century language. There's no reason not to be bilingual.

Second, the goal is to be able to do some problems in your head that you can't do otherwise. Do you think conversion will be a problem for someone who's done these conversions since they were 5? Conversions from hex to decimal, since they'll know the decimal value of each digit, are just a series of additions, and conversions from decimal to hex are just a series of subtractions.

Please tell me you think this will work, I am looking forward to using my children as guinea pigs.

[/ QUOTE ]

It's not impossible, but I'm wondering if it is actually more efficient. Most, if not all, modern mathematical theories were founded in decimal, so translating those theories to hex would likely be necessary to complete work in hex. Since you've so kindly introduced the analogy of language, I opt to introduce the concept of fluency in a language. Being truly fluent in a language means that you think fluently in that language, which appears to be what you want. But consider the extreme example of you being fluent in English and Japanese with the rest of the world being fluent only in Japanese. It would appear that, on the surface, there is not much benefit to using English. Suppose (and this is purely hypothetical, since I'm not fluent in Japanese) English is a more complex language that allows for superior methods of thinking (like your hex). Still, in an all-Japanese world, translating to English before completing work in Japanese doesn't make much sense. You might as well just complete the work in Japanese. My point is that fluency in English will not benefit you in terms of fluency in Japanese. Thus, I think that fluency in decimal is more important than fluency in hex.

If you do indeed train your children in hex, then I should hope that either you are a hex genius or you know where to find one. Much of my own learning in math and science was accomplished from reading books, all of which used decimal notation.

[SBR]

[ QUOTE ]
all computer people think in Base-16.

[/ QUOTE ]

This isn't true at all. Almost no "computer people" think in base 16. They understand it and can easily convert from base 16 to 10 or to 2 but they don't think in base 16. Trying to teach you children to do arithmetic in base 16 would just confuse them and waste their time when they could be learning much more useful information.

[MrMon]

There are 10 kinds of people - those who understand binary and those who don't.

[diebitter]

No man, there are 3 kinds of people, those that can count, and those that can't.

[SBR]

There's a time for love, and a time for healing.
You can't go back and undo what's been done.
The word of mouth. Time is revealing,
just how far we've let this kingdom come.
Hand to hand, we're finding our way,
And today is just tomorrow's yesterday.
Some will die for you, some will lie to you,
There's all kinds of people in this world.
Turn the world around
Tear the borders down
There's all kinds of people in this world
This kiss of life. The hand of fate
The boy grows up into his fathers' son.
And he learns to love, been taught to hate
To carry on the way it has been done
All our lives it's debts to repay
Maybe someday we can put the past away
Some will die for you, some will lie to you,
There's all kinds of people in this world.
Turn the world around
Tear the borders down
There's all kinds of people in this world
Some will die for you - Some will lie to you,
There's all kinds of people in this world
Turn your fear away - Find a better way
There's all kinds of people in this world
Then in a moment things can change
One look behind and it's never the same
Some will die for you,
Some will lie to you,
There's all kinds of people in this world,
Turn the world around
Tear the borders down
There's all kinds of people in this world

[FortunaMaximus]

[ QUOTE ]
[ QUOTE ]
all computer people think in Base-16.

[/ QUOTE ]

This isn't true at all. Almost no "computer people" think in base 16. They understand it and can easily convert from base 16 to 10 or to 2 but they don't think in base 16. Trying to teach you children to do arithmetic in base 16 would just confuse them and waste their time when they could be learning much more useful information.

[/ QUOTE ]

Agreed. Broken down to the basic idea, you're basically teaching them to convert between number systems faster. I don't see how this is different than temperature or distance conversions.

If you introduce number systems instead, having them work out the squares of 1-10 up to 10-15 powers, show them how certain systems repeat themselves in nature (Fibonacci and rabbits/plants/seashells etc.)

Don't turn them into calculators, please.

Just a view from a late 20's fella with one of his favorite childhood memories being the discovery of an algebra text in the third grade, and basically not being hindered by the "You're 8 years old, you shouldn't be doing polynomials. Go play with action figures instead."

[Borodog]

All number systems are base 10. Do you see why?

[diebitter]

All your base are belong to us?

[SBR]

[ QUOTE ]
All number systems are base 10. Do you see why?

[/ QUOTE ]

All except 1.

[RocketManJames]

[ QUOTE ]
[ QUOTE ]
all computer people think in Base-16.

[/ QUOTE ]

This isn't true at all. Almost no "computer people" think in base 16. They understand it and can easily convert from base 16 to 10 or to 2 but they don't think in base 16. Trying to teach you children to do arithmetic in base 16 would just confuse them and waste their time when they could be learning much more useful information.

[/ QUOTE ]

Ya, us computer types definitely don't think in Base-16. We do think in base 16 for a few specific things... take a bitfield, for example.

You have a sequence of 32-bits (0/1)... each represents something that is turned on or off.

If I said:

Bitfield = 1234567890

It's not too easy to quickly convert to:

32 Bits = 1001001100101100000001011010010

But, if you had the base-16 representation of 0x499602D2

You would be able to quickly break it down quickly as:

4 = 0100
9 = 1001
9 = 1001
6 = 0110
0 = 0000
2 = 0010
D = 1101
2 = 0010

And, now you can quickly see the bits that are turned on and off. This is why Hex is an important representation for computer types. But, we certainly don't think in it when it comes to math.

Maybe there are some freakish computer geeks that do, but neither I nor any of my colleages do.

-RMJ

[FortunaMaximus]

Binary has little character. Literally.

[GMontag]

[ QUOTE ]
3) Base X refers to a number system with X digits. So a binary or base 2 system has 2 digits (0,1) and the "regular" number system is base 10 (0,1,2,3,4,5,6,7,8,9)

[/ QUOTE ]

Not quite. Base X refers to a number system where each place is worth X times the place to its right. There is a difference because it is possible to have number systems with negative or complex bases.

[SBR]

[ QUOTE ]
[ QUOTE ]
3) Base X refers to a number system with X digits. So a binary or base 2 system has 2 digits (0,1) and the "regular" number system is base 10 (0,1,2,3,4,5,6,7,8,9)

[/ QUOTE ]

Not quite. Base X refers to a number system where each place is worth X times the place to its right. There is a difference because it is possible to have number systems with negative or complex bases.

[/ QUOTE ]

Wiki on Radix (http://en.wikipedia.org/wiki/Radix)

[ QUOTE ]
In mathematical numeral systems, the base or radix is usually the number of various unique digits, including zero, that a positional number system uses to represent numbers in a given counting system.

[/ QUOTE ]

[disjunction]

[ QUOTE ]
[ QUOTE ]
all computer people think in Base-16.

[/ QUOTE ]

This isn't true at all. Almost no "computer people" think in base 16. They understand it and can easily convert from base 16 to 10 or to 2 but they don't think in base 16. Trying to teach you children to do arithmetic in base 16 would just confuse them and waste their time when they could be learning much more useful information.

[/ QUOTE ]

If you are an app programmer, or are limited to Java or something, you probably do not need to think in Base 16. If you actually work with the hardware you do. Not the same way you think about decimal on an every day basis, because you are more focused on shifts and masks and stuff. That's not the stuff I want my kid to know. I just want a higher base than 10, I choose hex because it's around.

Most of you are missing my point, which is:

(1) Children are information sponges. They learn super-quick. This will not be confusing, will blend into other lessons, and once known, will be super-useful.

(2) The ability to work with large numbers in your head is super-useful, because you don't always have a calculator. The bottleneck is number of digits. I can multiply 3-digit numbers by each other in my head but have trouble with 4. You can do multiplications in your head in hex that you couldn't dream about doing in decimal, because it requires less digits.

[disjunction]

[ QUOTE ]
[

It's not impossible, but I'm wondering if it is actually more efficient. Most, if not all, modern mathematical theories were founded in decimal, so translating those theories to hex would likely be necessary to complete work in hex. Since you've so kindly introduced the analogy of language, I opt to introduce the concept of fluency in a language. Being truly fluent in a language means that you think fluently in that language, which appears to be what you want. But consider the extreme example of you being fluent in English and Japanese with the rest of the world being fluent only in Japanese. It would appear that, on the surface, there is not much benefit to using English. Suppose (and this is purely hypothetical, since I'm not fluent in Japanese) English is a more complex language that allows for superior methods of thinking (like your hex). Still, in an all-Japanese world, translating to English before completing work in Japanese doesn't make much sense. You might as well just complete the work in Japanese. My point is that fluency in English will not benefit you in terms of fluency in Japanese. Thus, I think that fluency in decimal is more important than fluency in hex.

If you do indeed train your children in hex, then I should hope that either you are a hex genius or you know where to find one. Much of my own learning in math and science was accomplished from reading books, all of which used decimal notation.

[/ QUOTE ]

There have been studies that bilinguals are smarter, based on the bilingualism. There's a myth that sometimes pops up, that there's only so much information in your head you can store, but this simply isn't true. Knowledge begets knowledge. When I took Spanish classes in college, I started looking at some words in English from a different perspective. It's hard to describe, but hopefully you know what I mean.

Also, as far as proofs, in all the math classes I've taken beyond simple algebra, I've never really seen much that depends on decimal. Usually there's a constant k that could be in any base.

[GMontag]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
3) Base X refers to a number system with X digits. So a binary or base 2 system has 2 digits (0,1) and the "regular" number system is base 10 (0,1,2,3,4,5,6,7,8,9)

[/ QUOTE ]

Not quite. Base X refers to a number system where each place is worth X times the place to its right. There is a difference because it is possible to have number systems with negative or complex bases.

[/ QUOTE ]

Wiki on Radix (http://en.wikipedia.org/wiki/Radix)

[ QUOTE ]
In mathematical numeral systems, the base or radix is usually the number of various unique digits, including zero, that a positional number system uses to represent numbers in a given counting system.

[/ QUOTE ]

[/ QUOTE ]

Try reading beyond the first sentence.

From that article:
[ QUOTE ]
Bases work using exponentiation. A digit's value is the digit multiplied by the value of its place. Place values are the number of the base raised to the nth power, where n is the number of digit to the left the units digit.

[/ QUOTE ]

From Non-standard positional numeral systems (http://en.wikipedia.org/wiki/Non-standard_positional_numeral_systems):
[ QUOTE ]
A few positional systems have been suggested, in which the base b is not a positive integer. In these systems, the number of different numerals used clearly cannot be b. For details, see the relevant articles, Golden ratio base, Negabinary, Negaternary and Quarter-imaginary base.

[/ QUOTE ]

[SBR]

I have no clue what your point is, for a person like the OP who clearly does not have a background in mathematics it is far more helpful to provide my explanation since it is far easier to understand and is generally correct.

Your explination was obviously correct however my explination was valid enough from the OP point of view and really didn't need to be corrected.

[_TKO_]

When I took Spanish classes in college, I started looking at some words in English from a different perspective. It's hard to describe, but hopefully you know what I mean.

I know exactly what you mean. But even more beneficial than learning Spanish would be learning the origins of languages. Hopefully you know what I mean.

Also, as far as proofs, in all the math classes I've taken beyond simple algebra, I've never really seen much that depends on decimal. Usually there's a constant k that could be in any base.

Certain digits in the formulas are grounded in base-10, such that you couldn't just sub in a base-2 k where a base-10 k is expected. Also, digits such as 1/2, 2, 4, etc appear in many of these formulas. Of course, that may lend some insight into the fact that a base-2 number system is possibly better than a base-10. Any math professor should be able to tell you that base-10 is a terrible choice for a number system. However, it has become culturally practical in today's world.

[FortunaMaximus]

The Babylonians. They had a sexagesimal system. Yeah, Base-60. However, they used two sub-bases: base 10 and base 6.

More here on Wiki: Babylonian Number system (http://en.wikipedia.org/wiki/Babylonian_numerals)

I'd be inclined to think this is more entrenched in our perception. Note experiments with "internet time" and things like that, that SEEM silly. The hours, minutes, seconds are perfectly fine. Why mess with them, right.

[kazana]

[ QUOTE ]
When I took Spanish classes in college, I started looking at some words in English from a different perspective. It's hard to describe, but hopefully you know what I mean.
I know exactly what you mean. But even more beneficial than learning Spanish would be learning the origins of languages. Hopefully you know what I mean.

[/ QUOTE ]
I grew up trilingual. And as far as I can tell, you do get a better understanding of the basics of languages (romanic based languages in my case) by being used to a few languages spoken around you constantly. It's hard to describe why, but somehow if you're confronted with all those languages it just makes "click" at one point.
Even now, I learn (even completely unrelated) languages a lot faster than most other people that haven't grown up like me. And no, I haven't been studying linguistics or ancient languages.

[ QUOTE ]
Certain digits in the formulas are grounded in base-10, such that you couldn't just sub in a base-2 k where a base-10 k is expected. Also, digits such as 1/2, 2, 4, etc appear in many of these formulas.

[/ QUOTE ]
I don't get this part. Most formulas consist of variables, constants, and (mostly arithmetic) operators. All operators that function for base-10 will work for any other base, too. If any constants are present convert them and you've got your formula adapted. By making your kids work in different bases the convertion should be practically intuitive to them.
I may be mistaken, though. I'd like to see an example where this doesn't work (the formula that works for base-10 only).

[mindflayer]

TKO seems to have the correct answer.. calling your new system a Base A number system is correct, translate that into (common) and we would call it a base 11 number system.
If a base 2 numbering system was the common system used in the world, then we would say your A system was actually a base 1011 number system.

[_TKO_]

[ QUOTE ]
[ QUOTE ]
Certain digits in the formulas are grounded in base-10, such that you couldn't just sub in a base-2 k where a base-10 k is expected. Also, digits such as 1/2, 2, 4, etc appear in many of these formulas.

[/ QUOTE ]
I don't get this part. Most formulas consist of variables, constants, and (mostly arithmetic) operators. All operators that function for base-10 will work for any other base, too. If any constants are present convert them and you've got your formula adapted. By making your kids work in different bases the convertion should be practically intuitive to them.
I may be mistaken, though. I'd like to see an example where this doesn't work (the formula that works for base-10 only).

[/ QUOTE ]

I tried to think of something, but I think the basic forumlas would work fine in Base 2, provided that everything is converted. I feel like fractions would be the toughest part to get around, but probably not to the point where converstion would be possible. I wonder if some formulas could be further simplified in different bases, similar to how working in circular geometry is easier than Cartesian geometry, or the benefit of the shift operation in Base 2 arithmetic.

[kazana]

[ QUOTE ]
I tried to think of something, but I think the basic forumlas would work fine in Base 2, provided that everything is converted. I feel like fractions would be the toughest part to get around, but probably not to the point where converstion would be possible. I wonder if some formulas could be further simplified in different bases, similar to how working in circular geometry is easier than Cartesian geometry, or the benefit of the shift operation in Base 2 arithmetic.

[/ QUOTE ]
Fractions aren't difficult, either. The "/" is only an operator. So 1/2 in base10 would simply be 1/10 in base2.

As for some operations being easier, that could be the case, but then again, working with the factor 10 would be more difficult for any base that's not base10... /images/graemlins/smile.gif

[CORed]

[ QUOTE ]
all computer people think in Base-16.

[/ QUOTE ]
100 buckets of bits on the bus, 100 buckets of bits.
You take one down, short it to ground.
FF buckets of bits on the bus.

[kazana]

[ QUOTE ]
[ QUOTE ]
all computer people think in Base-16.

[/ QUOTE ]
100 buckets of bits on the bus, 100 buckets of bits.
You take one down, short it to ground.
FF buckets of bits on the bus.

[/ QUOTE ]
That song'd be a lot shorter in base2 unless, of course, you start with
"1100100 buckets..."

[jstnrgrs]

[ QUOTE ]
It looks your original question has been answered so I'll throw a couple of more thoughts out there. It sounds like you may not be familiar with this, but all computer people think in Base-16. It is very useful to know. Wife permitting, I am personally going to teach my kids to think in Base-16 or Base-32 and learn the appropriate multiplication tables. This will undoubtedly make them more adept at arithmetic.

The bottleneck in doing arithmetic in your head is memory, and with Base-16, there's simply less to remember. Four-digit numbers are now 3-digit numbers.

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

[/ QUOTE ]

I think base-16 or base-32 would not be that useful since their only prime factor is 2. I would suggest base-12 (or if you're really ambitious, base-30)