Can Alternative Number Systems Offer Greater Advantages for Problem Solving?

[iggybaseball]

I do not know a lot about math theory, however I know that we (America) (I'm not sure if it's different for different parts of the world) use a base 10 number system. I also know that there have been other number systems invented. Can number systems other than base 10 be more beneficial for specific problems? If so, could you give an example? Thanks.

 

[B-Con]

The first obvious example is base 2, what computers use. Computers really don't have a way of storing "part" of a value, so they deal with litteral "on" and "off" values, which are 1 and 0 respectively. Base 2 is very helpful in this arena because it allows computers to do a lot of math using just on and off signals. (Base 2 is often called "binary".)

A natural extention of base 2 is base 16 (since 2^4 = 16). This base (in computer terms called "hexadecimal") is used to abbreviate base 2 numbers since not everyone wants to have to write out all the 1s and 0s, so every hexadecimal digit corresponds to 4 binary digits.

You'll occationally see base 8 in use (2^3, go figure), but it's fairly uncommon. I believe it's most practical use is in certain biology aspects.

There is also the natural base, e, which serves as a fundamental base number that makes recursive appearances in calculus and aids in making some calculations simpler. 'e' has no rational decimal approximation (like pi), it's equivilent about 2.78...

Overall, though, the number base you choose to use for something is up to you, and it's all about convenience. Realistically, it doesn't matter what base you use, it's still the same number. The only difference is how you say it, and how easily you say it. Some bases may be faster for computer math calculations, while others may be more consise, while others may aid in solving math problems; it's just about what helps.

 

[SGT]

The only reason we use base ten is that we have ten fingers in our hands. In the antiquity some people used their fingers and toes to count and got a sort of base 20 system. This can still be seen in French that uses quatre-vingts (four twenties) to denote 80.
In ancient English they also used the word score to denote 20, so 60 could be called threescore.
Mayans also seemed to use a base 20 number system and Babylonians used a base 60 one. This is still useful in the measurement of time and angles with 60 seconds to the minute and 60 minutes to the hour or degree.

 

[mathman]

Historical note (computer base). Internally, it has always been base 2 for the reasons discussed above. In the early days of computers (up to around 1960 when IBM came out with 360 series), people used base eight as the shorthand for binary representations (3 bit numbers), while at the same time 6 bits were used for character representations. When the 360 series was introduced, IBM changed to 8 bits for character representations, so that 4 bits (hexadecimal) became the shorthand for binary representation.

As you can see in both the old and current systems, 2 "numbers" are used for each character.

 

[HallsofIvy]

Some American Indians, along the Pacific coast used a number system based on 4 (it was not an actual place value system so wasn't, technically, "base 4") because they counted using the spaces BETWEEN fingers.

 

[Hurkyl]

I feel the need to point out that we're not talking about different number systems -- we're talking about different numeral systems.

No matter what base we use, we're still talking about the same number system. We're just discussing different notations for numbers, in particular, how we use numerals.

 

[DaveC426913]

I was always fond of base 1.

I'll bet pretty much all of you have used it regularly at some time in your lives.

Hint: it's used a lot when scoring card games.

 

[DaveC426913]

Base 12 is probably the most useful base for a convenient numbering system. With one exception (5), it is divisible by the first 6 counting numbers. That pretty much covers all the common dividing needs we have.


(BTW, note that Base 60 is the smallest base that is divisible by all 6 of the first counting numbers. I wonder if the base 60 numbering system is a coincidence?)

 

[topsquark]

Base 12 is probably the most useful base for a convenient numbering system. With one exception (5), it is divisible by the first 6 counting numbers. That pretty much covers all the common dividing needs we have.


(BTW, note that Base 60 is the smallest base that is divisible by all 6 of the first counting numbers. I wonder if the base 60 numbering system is a coincidence?)

I personally count large numbers (such as the number of graduates walking across the stage) using base 20. It's amazing how high you can count on just your fingers and toes!

-Dan