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Is it likely the primary numbering system will be replaced by base 2?

find_the_fun

Active member
Feb 1, 2012
166
I took a course in computer networking and the prof claimed that in elementary school they will soon start teaching base 2 - i.e. binary instead of decimal. I find this hard to believe as, it's my understanding, that the reason base 10 was adopted is because we have 10 fingers to count with.
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,143
I took a course in computer networking and the prof claimed that in elementary school they will soon start teaching base 2 - i.e. binary instead of decimal. I find this hard to believe as, it's my understanding, that the reason base 10 was adopted is because we have 10 fingers to count with.
I highly doubt the base 2 thing. I would think it would be more likely to use base 6 (2*3) or base 30 (2*3*5). They have just the amount of alphabet to use on the base 30, so I doubt we'd use any higher. When I'm bored I typically count in base 20. It's surprisingly easy to do, in fact.

-Dan
 

soroban

Well-known member
Feb 2, 2012
409
Hello, find_the_fun!

I doubt that binary numbers will prevail . . . on this planet anyway.


I'm convinced that our base-ten system arose from the number of our fingers.

Before numerals were invented, I'm sure we indicated quantites with fingers.

Ask a small child "How old are you?"
You may get the answer "This many" with a display of fingers.


I have explained to my students that, with a little thought,
. . we could have established a base-five system.

With, say, our right hand, we can indicate the quantities 0, 1, 2, 3, 4, and 5.
Then one "hand" would represent the quantity "5".

With our left hand, we can indicate the number of "hands".

So with three fingers on the left hand, we have "3 hands".
With four fingers on the right hand, we have "4 fingers".

Hence, with (3 left, 4 right), we can express "nineteen".

And with both hands we can express any number up to 30.


With "binary fingers", we can express any number up to 1023.
 

CaptainBlack

Well-known member
Jan 26, 2012
890
I took a course in computer networking and the prof claimed that in elementary school they will soon start teaching base 2 - i.e. binary instead of decimal. I find this hard to believe as, it's my understanding, that the reason base 10 was adopted is because we have 10 fingers to count with.
Ask this "professor" when they thinks we will switch to non-QWERTY keyboards (or why VHS prevailed over Betamax).

Then ask them how this relates to the possibiylity of a change to the common numeration system.

You might also mention that the word "moron" passed through my mind with reference to their claim.

CB
 
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Bacterius

Well-known member
MHB Math Helper
Jan 26, 2012
644
The reason base 10 is the primary numbering system is because we count in everyday life in base 10. Simple as that. How is came to become so popular is irrelevant to the matter at hand (though it probably has to do with our fingers, yes).

Nitpick: when you're counting on your fingers, you're not using base 10, but actually base 1, where:

0:
1: |
2: ||
3: |||
4: ||||
and so on..

As for binary, it is the least efficient integer base (after base 1) to write down on paper, because it can only store 1 bit of entropy per digit, whereas base 10 can store $\log_2(10) \approx 3.3$ bits of entropy per digit. Of course, you could use base 100 and store much more information, but you'd need 100 distinct digit symbols to differentiate between them which is a pain (for instance, in base 16 we use the ordinary digits in addition to letters A to F).

This means binary will never be adopted by humans. It is simply inefficient when we have more digit symbols available (when we can store more information in a single digit). On the other hand, binary is useful precisely because:
- it can only take two values, which is how modern transistors work
- it can only take two values, which is the basis of boolean algebra

Now if you mean teach base 2 as general knowledge, then yeah, sure, but then why not teach the general thing and educate them about positional notation and show them that base 10 isn't some mystic base but is no more special than base 5 or base 991?
 
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CaptainBlack

Well-known member
Jan 26, 2012
890
Now if you mean teach base 2 as general knowledge, then yeah, sure, but then why not teach the general thing and educate them about positional notation and show them that base 10 isn't some mystic base but is no more special than base 5 or base 991?
Ahh... but that is not news, binary has been taught in early secondary education for at least 45 years (or to put it another way, I learned about binary at school in maths class before I was 15).

CB
 

Bacterius

Well-known member
MHB Math Helper
Jan 26, 2012
644
Ahh... but that is not news, binary has been taught in early secondary education for at least 45 years (or to put it another way, I learned about binary at school in maths class before I was 15).

CB
Of course - so did I (well, I learned about it in my own time when I started programming, when I was 11 or 12). I was just reading between the OP's question's lines and adding more information to my post.
 

Deveno

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
here's the trade-off:

a smaller base has a smaller "look-up table" (fewer sums and products that need to be memorized) at the cost of more operations per integer. for example: 21 x 3 in base 2:

10101 x 11 =

(10000 + 100 + 1)(10 + 1)

= (10000 + 100 + 1)(10) + (10000 + 100 + 1)

= 100000 + 1000 + 10 + 10000 + 100 + 1

= 11111

the only "memorized" sums and products needed are:

0 + 0 = 1
1 + 0 = 0 + 1 = 1
1 + 1 = 10

0*0 = 0
0*1 = 1*0 = 0
1*1 = 1 (four of each)

but 21 takes 5 digits to store (since two of the digits are 0, it's not as bad as it could be).

in base 10:

21 x 3 =

(20 + 3)(3) = 20*3 + 3 = 60 + 3 = 63

(i am omitting the sub-step 20 = 2*10 because our usual "positional" way of using arabic numerals automatically takes care of this).

as a result of a larger "look-up table" (our 100 memorized sums and products (each)), we greatly reduce the number of steps we have to take.

i think it doubtful we could maintain look-up tables of larger than 1000 easily (so we're looking at a base of < 31 for common-place use), that's a rather steep learning curve for arithmetic. base 12 probably would actually be an improvement, allowing easy calculations of halves, thirds, fourths and sixths. base 2 just takes up too much processing time, it's like a handicap ramp instead of stairs: easy going, but a longer walk.

but, at the end of the day, computation is about utility, and if there's no clear advantage to choosing a different base than 10, it doesn't seem like its worth the cultural overhaul to get it done. as long as one realizes 123 isn't really a number, but a "code-name" for a number, it's all good.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
I recall having trouble with other bases in 7th grade, until my father told me to think of an odometer (the old analog type) with wheels having a different number of numerals on their circumference. Then it made perfect sense.
 

CaptainBlack

Well-known member
Jan 26, 2012
890
... but, at the end of the day, computation is about utility, and if there's no clear advantage to choosing a different base than 10, it doesn't seem like its worth the cultural overhaul to get it done. as long as one realizes 123 isn't really a number, but a "code-name" for a number, it's all good.
There may be a clear advantage to another system, but you will still not be able to make the change because of the cost barriers, both money and and mental against it. That was the whole point of the reference to qwerty - there are better keyboard layouts but you will not get a change in what most people use because of the size of the barriers that have to be surmounted to get from here to there.

CB
 

find_the_fun

Active member
Feb 1, 2012
166
The same prof also made the comment: When a new technology comes into replace an old one, the owners of the old one work really hard to bring it up to par as a result it doesn't get replaced. Does anyone agree or disagree with this statement? I can't think of any examples off the top of my head but the prof was referring to how the new version of IP addressing (IPv6) will have so many avaialable IP addresses (enough to cover about 3 devices for every 1m^2 on the surface of the Earth) so IPv4 will be revamped and people will forget about IPv6.