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**bossk171****Member**- Registered: 2007-07-16
- Posts: 305

This has been bothering me for some time. It's a very common conversation amongst mathematicians that terms like "complex" , "irrational" and "imaginary" bias math doers against them. By making them somehow unapproachable or non-existent. That's true, the vocab we use to describe numbers is very biased, but I'd like to speak (complain?, rant?) about another more sinister bias:

The decimal system. 10 is a very uninteresting number mathematically. It seems arbitrary to choose it as the base to our numbers. And we are all guilty of letting it affect us, some examples:

When my mom turned 50 I thought that all of a sudden she was very old (half of a century!) but on further reflection, 50 is a really insignificant number. Who cares? On a side note, 49 seems much more exiting, being a square and all...

My family often teases me for cooking something in the microwave for 27 seconds or 33 seconds instead of an "even 30." But why? what does it really matter? What makes 30 any more relevant than 29?

Yesterday I watched an episode of Psych on TV in which all the characters were attending their 13th reunion. It was an ongoing gag that they were having a 13th and not a 10th or 15th.

I could go on forever, but I won't. You should have the point by now.

I understand the need for a base "something" number system, it would be very unrealistic (and impossible) to have a different symbol for ever number. But I propose 10 is a far too arbitrary number to base a number system on.

Are these just delusional ramblings of a mathematical snob? Has anyone else ever felt this way? Most importantly, what would you choose as your base if you had the ability to reset the number system? And if you would leave it the same, why? Let's assume for this discussion that math could be "reset" without a "transition period" i.e. no one would have to relearn how to count... that way a discussion of a societal reaction doesn't cloud the overall mathematical discussion.

There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

Last night, coincidentally, I had a dream that involved the beauty of the number ten.

It involved some 2-D geometrical figures all touching one another in some shape.

And the gist of the dream or the part that meant the most to me was when I thought,

"Wow, so the strange number ten is all coming together, wow!".

Now unfortunately, as with many dreams, I cannot recall the details exactly,

but if the dream is right, there is something very special about the number.

And it wasn't the simple ten-pin arrangement, I know that...

It was profound in the dream, but possibly imaginary.

**igloo** **myrtilles** **fourmis**

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

10 = 1 + 2 + 3 + 4. 1 is a point, 2 is a line, 3 is a plane, 4 is a solid. 10 is a "triangle number". 10 was thought to be the perfect number by Pythagoras.

Numbers which are divisible by 2 and 5 can be recognized by the ones digits. Divisibility by 3 also has a "nice" test in base 10. These are the three smallest primes (and hence "most common") primes.

Smaller bases involve long numbers, very hard to write (100 = (1100100)_2) while large bases suffer from having to memorize the many different symbols. 10 is a pretty nice compromise.

Ten is the number of **digits** on your hands. How do you teach children to count? With their fingers of course. Do you think it is just a coincidence that "digit" and "finger" are synonymous?

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**TheDude****Member**- Registered: 2007-10-23
- Posts: 361

I believe ten is the base of our modern number system because we have 10 fingers (barring any deformities). Imagine living 10,000 years ago with no concept of a number system. You travel to a nearby village and want to buy some sheep. How are you going to describe how many sheep you want to buy? Probably by holding up one finger for each sheep that you want. The problem comes if you want 12 sheep. To solve this you flash all ten fingers once, then hold up two fingers. What you've just done, without knowing you've done it, is count in base ten. My guess is that from there people were so used to counting in tens that by the time a formal number system was implemented they decided to make it in base 10 for simplicity's sake.

As for what number I would use as a base, I'd go with 16. Theoretically 1 would be a nice number base because arithmetic operations would be quite simple (e.g., to add 2 numbers together just append them: 000 + 0000 = 0000000), but it would have obvious practicality issues (you couldn't easily record numbers of any reasonable size). Two could be interesting since it's the smallest prime number and the only even prime number, not to mention it's wide use in modern technology, but it would suffer the same practicality issues that 1 does.

Like using 2 for a number base, 16 would mesh nicely with our use of bits in electronics. It is a perfect square, and it's root is also a perfect square. Being a power of 2 it has only a single prime factor. It would be far more practical from a storage point of view than 1 or 2 (and slightly moreso than ten), but not so big that the number of symbols needed to represent all of the digits would be cumbersome.

Wrap it in bacon

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,685

Great discussion!

I always have the desire to make people understand that the decimal system is just arbitrary ... that "50" is just a way of representing the underlying number. Like "cat" is just a way of representing a real cat.

It reminds me of the closing scenes of "Predator" ... the Alien presses something on its wrist and you see these symbols flashing. Schwarzenegger suddenly understands it is a countdown timer and makes haste out of there. I wonder what base it was in?

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**bossk171****Member**- Registered: 2007-07-16
- Posts: 305

John E. F: I very much know the feeling of dreaming about math. Many times I have "solved" a math problem in my sleep only to wake and either not remember the solution or (worse) remember it and realize it makes no kind of sense in the real world. I wonder if your profound realization was in fact profound or not.

Ricky: I had forgotten that 10 was a triangular number. However I don't feel like that enough to make it relevant. 9 is a square, and frankly I think square numbers are much more interesting than triangular numbers. I wonder if your considering 10 to be a "nice compromise" is societal conditioning talking. I'm curious, if you had to pick a new base, what would you choose?

TheDude: Base 16 is really interesting. My choice was going to be Base 8 for many of the same reasons, it's interesting that we both went in the same direction with our thought patterns.

MathIsFun: That's a really interesting direction you're coming in from... discussing symbolism over the actual mathematics. It reminds me of a conversation I was having with my father in which I said that few people visualize addition "correctly" I said most people visualize the number one and the number 1 make the number 2 not 1 object and 1 object make 2 objects... am I being clear here?

This is an interesting discussion, and the "no right answer" aspect makes it approachable for almost everyone.

There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

bossk171 wrote:

But I propose 10 is a far too arbitrary number to base a number system on.

It is NOT arbitrary! The base-10 is based on the number of fingers that you (or any other tetrapod vertebrate) have on your hands. Theres nothing arbitrary about that.

Still, just to let you know, other bases have been used at various times by various groups. The Babylonians used base 60. The Celts used base 20 the counting system in many Celtic languages (including Cornish, Manx and old Welsh) is based on 20.

As for more uncommon bases, the Hindi system, I gather, is almost like base 100 the Hindi language has a virtually different name for each number up to 100. Still stranger counting systems exist:

Ndom base 6: http://www.sf.airnet.ne.jp/~ts/language … /ndom.html

Nimbia base 12: http://www.sf.airnet.ne.jp/~ts/language … imbia.html

Huli base 15: http://www.sf.airnet.ne.jp/~ts/language … /huli.html

while the Alamblak system has no base at all; the Alamblak language only has words for 1, 2, 5 and 20 and all counting is based on these: http://www.sf.airnet.ne.jp/~ts/language … mblak.html

And, last but not least, computer scientists use base 2.

*Last edited by JaneFairfax (2008-07-29 02:51:46)*

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

TheDude wrote:

Like using 2 for a number base, 16 would mesh nicely with our use of bits in electronics. It is a perfect square, and its root is also a perfect square.

Dude, the practical usefulness of a particular number base is not because it is a perfect square, but because of its factorizability. For practical-sized bases, therefore, the best bases are 60, 72, 84, 96 and 108, each having 12 factors. The Babylonians probably chose 60 because it is the least of the 12-factor bases and therefore most compact and economical to use. More importantly, 60 is divisible by 10 the magic number for pentadactylous species like *Homo sapiens*.

The next base with more than 12 factors is 120, which has 16 factors. But the larger the base, the more unwieldy it is, and the more difficult to keep track of whats going on mentally. Therefore 60 is the most practical of the practical bases to use.

*Last edited by JaneFairfax (2008-07-28 13:32:00)*

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**bossk171****Member**- Registered: 2007-07-16
- Posts: 305

Wow that Alamblak system is crazy, good find Jane.

10 is arbitrary, I have 24 ribs perhaps base 24 is more appropriate? I have 2 ears, perhaps we should go back to considering binary? I have 4 limbs, maybe base 4? I guess the number 10 isn't so arbitrary as the fingers (or toes) being what we use as to count.

And before you try to explain the difficulties of counting on your ribs (I wouldn't really know where to start) that's not really my initial point, it's just my counterpoint.

I feel that 10 is very *human*. Which is arbitrary. I've always thought that math should transcend humanity and should be an unbiased and perfect system. I feel (perhaps wrongly) that humanizing it cheapens it, subtracts from its overall beauty. As MathIsFun pointed out, a cat would still be a cat if there were no people around to name it, just as mathematics would still exist.

This subject has jumped rather abruptly into the realm of philosophy which is all the more fun.

*Last edited by bossk171 (2008-07-28 13:26:00)*

There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

Consider this I just learned:

((2 * cos(36 degrees) * 2) - 1)^2 = 5

and remember a circle or apple pie divided into 10 parts is 36 degrees.

And the cosine of 36 degrees is half of phi if phi is the golden ratio they used to build the length and width of the parthenon.

(Is that true about the parthenon?)

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

I just drew a ten-pointed star drawn with

10 straight lines inside a unit circle.

Each line is the length of phi, and each

line starts where the other line ended.

I gotta go to bed; gonna visit an old friend

tomorrow up north...

**igloo** **myrtilles** **fourmis**

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**kylekatarn****Member**- Registered: 2005-07-24
- Posts: 445

JaneFairfax wrote:

And, last but not least, computer scientists use base 2.

Oh yeah!

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

10 is arbitrary, I have 24 ribs perhaps base 24 is more appropriate? I have 2 ears, perhaps we should go back to considering binary? I have 4 limbs, maybe base 4? I guess the number 10 isn't so arbitrary as the fingers (or toes) being what we use as to count.

And before you try to explain the difficulties of counting on your ribs (I wouldn't really know where to start) that's not really my initial point, it's just my counterpoint.

But you need to realize that you are committing a logic fallacy called a Straw Man. You are distorting Jane's argument, and then arguing against that distortion. Doing such doesn't mean it was done on purpose, but it was done none the less.

What you posted would be valid if Jane was saying, "We use base 10 because we have 10 fingers." But that isn't her argument. Her argument is, "We use base 10 because we have 10 fingers which are in a rather ideal place for counting things and showing to other people."

Unless you can show me a natural way to show someone how many carrots you need with you limbs, your counterpoint doesn't stand up against the argument.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**TheDude****Member**- Registered: 2007-10-23
- Posts: 361

JaneFairfax wrote:

TheDude wrote:Like using 2 for a number base, 16 would mesh nicely with our use of bits in electronics. It is a perfect square, and its root is also a perfect square.

Dude, the practical usefulness of a particular number base is not because it is a perfect square, but because of its factorizability. For practical-sized bases, therefore, the best bases are 60, 72, 84, 96 and 108, each having 12 factors. The Babylonians probably chose 60 because it is the least of the 12-factor bases and therefore most compact and economical to use. More importantly, 60 is divisible by 10 the magic number for pentadactylous species like

Homo sapiens.The next base with more than 12 factors is 120, which has 16 factors. But the larger the base, the more unwieldy it is, and the more difficult to keep track of whats going on mentally. Therefore 60 is the most practical of the practical bases to use.

Whew, 60 digits? My gut instinct tells me that a system with that many digits would be unwieldy. Apparently the Babylonians managed, but I also suspect that math was restricted to their more advanced students rather than being widely taught.

Admittedly, 16 was a rather greedy choice for me since I work with computers and have an attachment to powers of 2. If we're looking for a base with a lot of factors how about 12. It has half of the factors of 60 with only one-fifth as many digits.

As for 10 being an arbitrary number, I think bossk is saying that there's not much about it that's special from a numerical standpoint. It's not completely arbitrary since, as you noted, we have 10 fingers. It also has some interesting properties that Ricky pointed out, but from a purely mathematical point of view it's not very special and probably is not ideal.

Wrap it in bacon

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**bossk171****Member**- Registered: 2007-07-16
- Posts: 305

Ricky wrote:

10 is arbitrary, I have 24 ribs perhaps base 24 is more appropriate? I have 2 ears, perhaps we should go back to considering binary? I have 4 limbs, maybe base 4? I guess the number 10 isn't so arbitrary as the fingers (or toes) being what we use as to count.

And before you try to explain the difficulties of counting on your ribs (I wouldn't really know where to start) that's not really my initial point, it's just my counterpoint.

But you need to realize that you are committing a logic fallacy called a Straw Man. You are distorting Jane's argument, and then arguing against that distortion. Doing such doesn't mean it was done on purpose, but it was done none the less.

What you posted would be valid if Jane was saying, "We use base 10 because we have 10 fingers." But that isn't her argument. Her argument is, "We use base 10 because we have 10 fingers which are in a rather ideal place for counting things and showing to other people."

Unless you can show me a natural way to show someone how many carrots you need with you limbs, your counterpoint doesn't stand up against the argument.

I didn't realize that was called "Straw Man" very interesting. You are right of course, I didn't mean to do that on purpose, but my understanding of her point was flawed and by extension my response was.

But I feel like the rest of my post holds true. I still maintain that it feels very human... and more importantly: that it shouldn't.

TheDude gets my point exactly. Mathematically 10 is very boring. I'm thinking about those really lame energy beings from classic Star Trek that are super evolved have no physical form (and hence no fingers). I wonder what base they'd use?

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

Here is a unit circle

with 10 phi length

lines. phi is the

amazing spiralling

rectangle number,

where squares go

in a spiral getting

smaller and smaller.

1.618034 = 1/0.618034 approx.

The radius of the unit circle

is one.

Notice I drew it by hand so

the top two points are too

close together, sorry.

I know the lengths are phi

because the cosine of 36 degrees

is phi/2 and the horizontal lines

in this drawing go from 180-36 degrees

to 36 degrees, and the bottom

horizontal line goes from

180+36 to 0 - 36 degrees.

*Last edited by John E. Franklin (2008-07-30 09:54:27)*

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

Here's an alternative to base-10.

Web Page:http://www.mathpages.com/home/kmath165.htm wrote:

Another interesting type of

number system is the "factorial system", where the denominations are

1, 2, 6, 24, 120, etc, and the nth digit is in the range from 0 to

n. This works because of the identity1*1! + 2*2! + 3*3! + ... + k*k! = (k+1)! - 1

This system is more "universal" than any particular geometric system

because it doesn't make use of any special "base". Every number is

used as the base of one of the columns. Of course, with this system

you need to keep inventing new digits to write larger numbers. On the

other hand, with just digits 0-9 you can express numbers from 0 to

3628799.

0=0

1=1

10=2

11=3

20=4

21=5

100=6

101=7

110=8

111=9

120=10

121=11

200=12

201=13

210=14

211=15

220=16

221=17

300=18 (because the 3 is in the 6's place and 3x6=18)

301=19

310=20

311=21

320=22

321=23

1000=24 (because the 1 is in the 24th's place)

1001=25

1010=26

1011=27

1020=28

1021=29

1100=30

1101=31

1110=32

1111=33 (This is 24 + 6 + 2 + 1)

1120=34

1121=35

1200=36

1201=37

1210=38

1211=39

1220=40

1221=41

1300=42

1301=43

1310=44

1311=45

1320=46

1321=47

2000=48

2001=49

2010=50

2011=51

2020=52

2021=53

2100=54

2101=55

2110=56

2111=57

2120=58

2121=59

2200=60 (this is 48 + 12)

2201=61

2210=62

2211=63

2220=64

2221=65

2300=66

2301=67

2310=68

2311=69

2320=70

2321=71

3000=72 (this is 24x3)

3001=73

3010=74

3011=75

3020=76

3021=77

3100=78

3101=79

3110=80

3111=81

3120=82

3121=83

3200=84

3201=85

3210=86

3211=87

3220=88

3221=89

3300=90 (this is 24x3+6x3)

3301=91

3310=92

3311=93

3320=94

3321=95

4000=96

4001=97

4010=98

4011=99

4020=100 (this is 24x4+2x2)

*Last edited by John E. Franklin (2008-08-19 16:46:28)*

**igloo** **myrtilles** **fourmis**

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