quote:Originally posted by Jonny T, former cutie: Quick questions: a) Why is there no roman numeral for 0 (AFAIK)? b) How did the mathematicians of the time get round this?

Jonny T.

In (very) short,

a) because they had no concept of zero; b) they didn't.

Luís Henrique

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quote:Originally posted by Jonny T, former cutie: Quick questions: a) Why is there no roman numeral for 0 (AFAIK)? b) How did the mathematicians of the time get round this? Jonny T.

Now, we could fill a library with all the things I don't understand about mathematics, but the way I understand it, this is one of the reasons that higher math functions were never developed by cultures using Roman numerals. Algebra, Trigonometry and Calculus all require the ability to multiply and divide. The absence of a 0 value make this all but impossible. There was an additional problem with the fact that Roman numerals couldn't be separated into columns 1s, 10s, 100s (etc.).

I learned the above from a PBS progam hosted by Jaime Escalante, upon whom the move Stand and Deliver was based. Sr. Escalante credited the Mayans with pioneering the use of the zero value, thus paving the way for development of advanced mathematics.

I have a question for the more math literate. I know from an electronics class, how to calculate certain values using Geometric and Trigonometric formulas. Many of these formulas rely on the fact that a full revolution is 360°. As long as one knows that fact, the formulas work. I'm curious to know, who (or what culture) discovered that a circle = 360°?

quote:I'm curious to know, who (or what culture) discovered that a circle = 360°?

Warning: Nitpicking ahead!

It really doesn't matter who ascribed a balue of 360° to a full revolution. The 360° is't nearly as important as the concept that someone devised a mathmatical method of ascribing a value to an angle.

360 could easily be any arbitrary number, just as temperature is measured in °C and in °F. In one system freezing is at 32, and in another it is at 0. Its just an arbitrary standard in both cases.

posted
Why 360°? Blame those wacky Mesopotamians. The Santa Cruz, Calif. Public Library does: Degrees in a Circle, 360

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phildonner and blitzen
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posted
That great number-meister Pythagoras didn't even consider 1 to be a number!

(Warning: very fine nit-picking) Much of the credit given to the Arabs and Hindus for the discovery of Zero is in fact for the use of the digit zero used as a placeholder, not the number zero representing nothing. (End of nit-picking, resuming the topic)

Using roman numerals, the number CCCVI looks totally different than the number XXXVI, but in a decimal "position based" system, 306 and 36 need to be differentiated by having a "placeholder" in the tens place.

Even in places that used roman numerals, computation was done on an abacus, which has a "position-based" system built right in. And sure enough, there's a way to represent a "zero" digit on an abacus.

It was Leonardo "Fibonacci" of Pisa who noticed that anything you could do on an abacus could also be done using symbols on paper, by using the Arabic numerals, which included a zero symbol.

posted
The number of "angular measurements" in a circle is equal to 2 pi. Very handy...

The Chaldeans (a subset of the Babylonians) came up with 360 for two reasons: it so nicely approximated the number of days in the year, and it was so elegantly divisible by other numbers, thus making it useful for calculations.

One thing in favor of Roman Numerals: they're really handy for ciphering, hidden messages, puns, etc.

("Five hundred begins it, and five hundred ends it. A five in the middle is seen. The first of all letters, the first of all numbers are arrayed to fit in between. When properly read by a scholarly head, the name of a King shall be seen.")

re number theory, some nice folks have said that only primes are "numbers," and the others are merely constructs... But that's getting into theology...

Silas (Holy 217!) Sparkhammer

-------------------- When on music's mighty pinion, souls of men to heaven rise, Then both vanish earth's dominion, man is native to the skies.

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The Vorlon Ambassador
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Speaking of numerical theology...

[oops! I've made an error in this post. See Luis Henrique's post below for the correction. Thanks, Luis!]

Perfect numbers are numbers that can be found by adding the counting numbers together sequentially. (1=1, 3=1+2, 6=1+2+3, 10=1+2+3+4, etc.)

Some theologians have claimed that this is the reason for the six days of creation discussed in the Old Testament. According to them, God could have created the universe instantly, but He took six days in order to recognize the perfection of the number six.

quote:Originally posted by The Vorlon Ambassador: Speaking of numerical theology...

Perfect numbers are numbers that can be found by adding the counting numbers together sequentially. (1=1, 3=1+2, 6=1+2+3, 10=1+2+3+4, etc.)

Those are triangular numbers. Perfect numbers are equal to the sum of its divisors. Six, of course, is a member of both cathegories (1+2+3=6; 6/1=6, 6/2=3, 6/3=2).

Luís Henrique

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Decima Do You Hear What I Hear
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posted

quote:Originally posted by Silas Sparkhammer:

Where's Patrick McGoohan when we really need him?

Silas (Waycross, Thorpe, Littleton and Freeman) Sparkhammer

Obligatory joke: "I am not a number! I am a free man!" (derisive laughter from the current Number Two).

You might want to look at Robert Kaplan's book The Nothing That Is: a Natural History of Zero It will either fascinate you or bore you into a coma.

quote:Originally posted by Decima Do You Hear What I Hear:

Obligatory joke: "I am not a number! I am a free man!" (derisive laughter from the current Number Two).

You might want to look at Robert Kaplan's book The Nothing That Is: a Natural History of Zero It will either fascinate you or bore you into a coma.

Decima "my love of numbers is unrequited" Dewey

I'll go looking for it! Sounds like fun!

There is one interpretation that I kinda like: no one knew what "numbers" really were until we came up with set theory.... Zero is easy: it's the cardinality of the set of things that there aren't any of: square circles, three-legged quadrupeds, likeable trolls, and the like. Then One pops up, out of the fact that the empty set is unique. There's only one of 'em.

And then we all play musical chairs to the tune of the Peano...

Silas ("Clean twos, keen twos, 'God save the Queen' twos, lean twos, mean twos, 'We have the Means' twos...") Sparkhammer

-------------------- When on music's mighty pinion, souls of men to heaven rise, Then both vanish earth's dominion, man is native to the skies.

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Spooky
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quote:Originally posted by phildonner and blitzen: Even in places that used roman numerals, computation was done on an abacus, which has a "position-based" system built right in. And sure enough, there's a way to represent a "zero" digit on an abacus.

OK, I learned how to do this during one of my courses at Uni, but I've forgotten how they did it (it's a bit different to the Chinese abacus). Can you give me a URL that will explain it, 'cause I was fascinated.

IIRC, the Exchequer is named after the wax-tablet abacus that they used to use to calculate taxes even up until ... um ... erm ... it really wasn't all that long ago that they got rid of it, but I can't remember exactly when.

quote:The number of "angular measurements" in a circle is equal to 2 pi. Very handy...

Is that how pi was originally calculated? I thought it was calculated as the ratio of the circumference of a circle to its radius. Or the ratio of its area to its radius squared. Not sure which.

posted
The angular measurement Silas is talking about is the angle you get if you take the radius of the circle and wrap it around the outside and then look to see how far apart the ends are.

Pi is defined as the ratio of the circumference of the circle to its diameter.

posted
French military engineers used (use?) a unit of angle measurement called "grads." 400 grads = 360 degrees. My HP 15c (ca. 1980) calculator has the ability to compute in grads. I am an engineer with > 20 years' experience, and I have NEVER seen a reference (outside of a mathematics textbook) that used grads...
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quote:Originally posted by Radagast: French military engineers used (use?) a unit of angle measurement called "grads." 400 grads = 360 degrees. My HP 15c (ca. 1980) calculator has the ability to compute in grads. I am an engineer with > 20 years' experience, and I have NEVER seen a reference (outside of a mathematics textbook) that used grads...

"Alien Space," a cute play-on-the-floor science fiction battle game by Lou Zocchi, used 400 divisions to the circle. There was a Star Trek variant, too.

Silas ("Gapper Zapper") Sparkhammer

-------------------- When on music's mighty pinion, souls of men to heaven rise, Then both vanish earth's dominion, man is native to the skies.

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quote:Originally posted by Radagast: French military engineers used (use?) a unit of angle measurement called "grads." 400 grads = 360 degrees. My HP 15c (ca. 1980) calculator has the ability to compute in grads. I am an engineer with > 20 years' experience, and I have NEVER seen a reference (outside of a mathematics textbook) that used grads...

Do french 'grads' equate to the mils we use in hte US military system ... Typically used for longer range things like ballistics and artillery...but not so long range that you need to go into minutues of the circle...

War 'way past my math ability' lok edited -- AND TYPING!

Do french 'grads' equate to the mils we use in hte US military system ... Typically used for longer range things like ballistics and artillery...but not so long range that you need to go into minutues of the circle...

War 'way past my math ability' lok edited -- AND TYPING!

The only mils (mills?) I'm familiar with are "thousandths of an inch," commonly used to describe polymer films.

I've never had to use grads. Most calculations I come across are usually degrees, radians almost as often. All are arbitrary, but for trig applications, radians are usually much more convenient.

Alchemy

-------------------- Thinking about New England / missing old Japan

phildonner and blitzen
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posted

quote:Originally posted by Luís Henrique:

Those are triangular numbers. Perfect numbers are equal to the sum of its divisors. Six, of course, is a member of both cathegories (1+2+3=6; 6/1=6, 6/2=3, 6/3=2).

Luís Henrique

In fact, all even perfect numbers are also triangular. (6, 28, 496, 8128, etc.) No one has ever discovered an odd perfect number.

quote:Originally posted by Jay: Um, what about 1? All of its divisors are 1, and 1=1, last I checked.

J "Nitpicker" ay

Nope. 1 is exempt from a lot of number-theory rules. If you allow 1 to be a divisor of another number, then you could violate the uniqueness of prime factorization, since (for example):

6 = 3 x 2 6 = 3 x 2 x 1 6 = 3 x 2 x 1 x 1 etc.

1 is not a "prime number," even though, of course, it quite obviously is.

Silas ("...and I can prove it...") Sparkhammer

-------------------- When on music's mighty pinion, souls of men to heaven rise, Then both vanish earth's dominion, man is native to the skies.

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Jay
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Who said anything about prime numbers?

I was commenting on "No one has ever found an odd perfect number."

quote:Originally posted by Jay: Who said anything about prime numbers?

I was commenting on "No one has ever found an odd perfect number."

OK, gotcha: talkin' about two different things. The definition of a perfect number is that it is the sum of its divisors, in which case 1 *is* allowed, although once only.

(Otherwise, 9 is "perfect," since it is equal to 1 x 1 x 1 x 1 x 1 x 1 x 3 -- giggle...)

By the way, I just got the news that a distributed computing project has popped up with the 39th Mersenne Prime, a 4-million-digit monster. Mersenne Primes are components of perfect numbers, so this is quite a discovery...

Silas (onward toward infinity!) Sparkhammer

-------------------- When on music's mighty pinion, souls of men to heaven rise, Then both vanish earth's dominion, man is native to the skies.

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Tom-Tzu
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posted
I've been thinking about Mersenne primes lately, too, trying to figure out how Mersenne settled on the formula 2^n-1 (where n must itself be a prime number). I did manage to work out why n has to be prime, and it's kinda neat.

Express the number in binary, and what you have is essentially a series of 1s that is n digits long. If n is composite, then that string of 1s can be written as a shorter string of digits repeated several times. And THAT means that the shorter string must be a factor of the larger one, because, for example, 123412341234 is equal to 1234 times 100010001.

So, any string of 1s which is a composite number in length necessarily has at least two non-zero factors. If n has factors a and b, then 2^n-1 has as factors 2^a-1 and 2^b-1.

Okay, I'm probably reinventing the wheel here, and every mathematician with any background in number theory already knows this, but it was still fun for me to discover this on my own...

quote:Originally posted by philmarillion: Brilliant, Master Tom!

I've never seen this technique. The "usual" proof is shorter, but not as elegant, IMO.

For a quick encore, try to show that if a^n-1 is prime, then a must be 2. (Assuming n>1)

Oh, THAT one is easy!

First, note that any string of identical digits cannot be prime, unless those digits are all 1. 22222 is divisible by 2, 77777777 is divisible by 7, and so forth.

Now, a^n-1 can be expressed in base-a as a string of n digits, each of which is a-1. And such a number is necessarily divisible by a-1. If a-1 is not equal to 1, then we have a factor, which means a^n-1 is not prime.

So, for example, just to stay in base 10, 10^n-1 will always be written out using only the numeral 9. 10^5-1 would be 99999. ALL of these numbers would have to be divisible by 9.

Incidentally, this points us at the sole exception to the rule. 9 is itself divisible by 3, but a-1 needn't be composite in all cases. So when n=1, the rule about a^n-1 being prime only if a=2 doesn't apply. a^n-1 can be prime for lots of values of a, so long as n=1.

When I was in school, I took a shot at Fermat's Last. (Didn't everyone?) I used alternate-base arithmetic, and was able to weed out large classes of exponents.

(For instance, if a^n + b^n = c^n, you can use binary arithmetic to show that n may not be divisible by 3... etc. etc. With a few days' effort, I was able to rule out large swaths of numbers -- whereas, of course, there were still an infinite number of possibilities I couldn't touch...)

Silas (love it!) Sparkhammer

-------------------- When on music's mighty pinion, souls of men to heaven rise, Then both vanish earth's dominion, man is native to the skies.

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Tom-Tzu
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posted
Here's another one.

You know how you can always tell if a number is divisible by 3? Add up the digits, and if the sum is divisible by 3, so is your original number. The same is true of 9.

Well, I remember being puzzled at this coincidence, which of course couldn't possibly be a coincidence, so I figured out why it is so.

Let's start with 9. Obviously 9 is divisible by 9, so that's no big deal. Now, you can get to any multiple of 9 by adding 9, right? And that is essentially the same as adding 10 and subtracting 1. Note that adding a 1 and a zero and subtracting a 1 leaves the sum of the digits overall unchanged, so every multiple of 9 will have the same property.

Of course, it also follows that this property will hold for any multiple of n-1 as expressed in base-n.

Fine, but why should this property apply to multiples of 3 in base-10? Because 3 is a factor of 9. Obviously, anything which is a multiple of 9 is also a multiple of 3, right? As well, its digits add up to a multiple of 9. Add or subtract 3 from the last digit (to keep it simple, choose whichever operation doesn't involve changing the 10s column), and the sum of the digits will change by 3. Well, three more or less than a multiple of 9 is necessarily a multiple of 3. QED.

But what impressed me was that this is true in other bases. The general rule is that in any base-n, all multiples of n have the property, as do all multiples of factors of n. So, in hexadecimal, any multiple of 15 (F) has digits which add up to a multiple of 15, any multiple of 5 has digits which add up to a multiple of 5, and any multiple of 3 has digits which add up to a multiple of 3. (Of course, it's trivially true in binary, where all multiples of 1 have digits which add up to a multiple of 1.)

posted
That last is related to the quick arithmetic check called "casting out nines", which generalizes to "casting out (n-1)s" where the number base is n.

To check your arithmetic problem, add up the digits of, say, each of the two numbers you multiplied (repeat until you get a number less than 10). Do the arithmetic problem with the results of the "casting out". Add up the digits in the result of the original problem. These last two should match.