What really bugs me about math

[Tumbleweed]

I've got a circle whose diameter is EXACTLY X units long. And yet no one alive can tell me what the exact circumference is after some two thousand years of geniuses trying. I'm holding this circle . It has an fixed cicumference. It has a fixed diameter. And yet I can't figure out what they both are!!
And don't get me started on decimals. One third of twelve is an exact four. But using .33333 ad infinitum instead of 1/3 in the equation leads to an inexact number, 3.99999 ad infinitum. This reduces math to proper protocol: multiply FIRST by 1 and THEN divide by three, and NEVER divide the one by three first to get an inexact decimal number.
I wonder how many computers make that mistake?

[joobz]

I've got a circle whose diameter is EXACTLY X units long.
How do you know it's EXACTLY X units long? Are you saying that you have the ability to measure EXACTLY X units?

[Tsukasa Buddha]

I've got a circle whose diameter is EXACTLY X units long. And yet no one alive can tell me what the exact circumference is after some two thousand years of geniuses trying. I'm holding this circle . It has an fixed cicumference. It has a fixed diameter. And yet I can't figure out what they both are!!
And don't get me started on decimals. One third of twelve is an exact four. But using .33333 ad infinitum instead of 1/3 in the equation leads to an inexact number, 3.99999 ad infinitum. This reduces math to proper protocol: multiply FIRST by 1 and THEN divide by three, and NEVER divide the one by three first to get an inexact decimal number.
I wonder how many computers make that mistake?

WTF are you talking about?

(1/3)*12=(1*12)/3=1*(12/3)=4

Should we revive the "Does .999... repeating infinitely equal 1" debate?

(Hint: The answer is yes)

When I plug 1/3 into my calculator and then multiply the answer (.333...) by twelve I get exactly 4.

[ravdin]

You mean that bothers you that pi is an irrational number?

Sorry if the infinities make your head hurt- join the club. But it should make you feel better to know that if you want to calculate the circumference of any circle that fits in the visible universe, 39 digits of pi will suffice for a precision to the width of a hydrogen atom.

[slyjoe]

The exact circumference is pi*d.

Thanks for playing.

[Ziggurat]

I've got a circle whose diameter is EXACTLY X units long. And yet no one alive can tell me what the exact circumference is after some two thousand years of geniuses trying.

Sure they can. It's pi*X. Your complaint is that nobody has all the digits of pi. But considering that it's an irrational number, what were you expecting? You aren't under the mistaken impression that that can ever change, are you?

And don't get me started on decimals. One third of twelve is an exact four. But using .33333 ad infinitum instead of 1/3 in the equation leads to an inexact number, 3.99999 ad infinitum. This reduces math to proper protocol: multiply FIRST by 1 and THEN divide by three, and NEVER divide the one by three first to get an inexact decimal number.
I wonder how many computers make that mistake?

Computers use binary, not decimal, for performing calculations. But many computers use increased precision for intermediate steps of calculations precisely because of these issues, so that the final answer for such a case would indeed typically get rounded to 4 regardless of the order of operations.

[sol invictus]

I've got a circle whose diameter is EXACTLY X units long.

Really? What's X, EXACTLY?

[Madalch]

Really? What's X, EXACTLY?
It's one circle-diameter, by definition.

[sol invictus]

It's one circle-diameter, by definition.

Ah - but then I think I know the circumference too, EXACTLY...

[joobz]

to be fair to the OP, the mistake that's being made is confusing rational numbers with exact measurements.

In Math we know exactly that
Pi*D=Circ.

Since we can write a rational number like 4 with little trouble, it gives the illusion that we can know something is 4(what ever units) exactly. This isn't true.

To be accurate,
In order to measure something exactly 4, we have to know that it is 4.0000(with an infinite number of zeros trailing behind it).
This precision is needed to verify in the real world that that circle is indeed 4 units in circ.

the trick is, if we can measure 4 with infinite precision, than we can measure pi with infinite precision.

So, the paradox that the OP seems to suggest isn't a paradox at all.

[tsg]

I've got a circle whose diameter is EXACTLY X units long. And yet no one alive can tell me what the exact circumference is after some two thousand years of geniuses trying. I'm holding this circle . It has an fixed cicumference. It has a fixed diameter. And yet I can't figure out what they both are!!
And don't get me started on decimals. One third of twelve is an exact four. But using .33333 ad infinitum instead of 1/3 in the equation leads to an inexact number, 3.99999 ad infinitum. This reduces math to proper protocol: multiply FIRST by 1 and THEN divide by three, and NEVER divide the one by three first to get an inexact decimal number.
I wonder how many computers make that mistake?

"Irrational" does not mean "inexact". It simply means the number cannot be represented in the form of $$\frac{a}{b}$$ where A and B are integers.

That it can't be represented in base 10 in a finite number of decimal places doesn't make it inexact either. It just means it can't be represented in the form $$\frac{a}{10^x}$$ where A and X are integers.

[sol invictus]

Of course, pi isn't just irrational - it's transcendental. So not only can it not be written as a ratio of two integers, it is also not the solution to any algebraic equation with rational coefficients.

[Kotatsu]

What totally bugs *me* about maths is how is disproves evolution by mutation and selection. That's really annoying, and I wish it'd stop doing that.

It isn't very funny to quote Kleinman like this, is it?

[joobz]

What totally bugs *me* about maths is how is disproves evolution by mutation and selection. That's really annoying, and I wish it'd stop doing that.

It isn't very funny to quote Kleinman like this, is it?
Kotatsu earns a time out!

No you go sit in the corner, Mister, and think about what you did.:p

[CapelDodger]

Of course, pi isn't just irrational - it's transcendental. So not only can it not be written as a ratio of two integers, it is also not the solution to any algebraic equation with rational coefficients.

As I recall, pi is the limit of an infinite series ...

I'm not helping much, am I :)?

[XBoxWarrior]

It's all God's fault.......

For making Pi such an irrational number? ;)

Planet X is only sooooo big around.

ETA: I like pie

[Molinaro]

:duck:As I recall, pi is the limit of an infinite series ...

I'm not helping much, am I :)?

What's the [infinity-1]th term in that series?

:duck:

[mijopaalmc]

As I recall, pi is the limit of an infinite series ...

I'm not helping much, am I :)?

The most common definition of "transcendental" specifies that the polynomial be of finite degree.

[Jimbo07]

I wonder how many computers make that mistake?

You should be aware that every single digital computer, calculator, instrument, etc. makes a mistake like this. In converting an analog (continuous) signal to a digital signal, your last bit (or least significant bit) in the sequence will have to encode some small value between 0 and 1 as either 0 or 1.

If you want to limit the effect of this error, you have to include more bits. Modern computers can calculate very good numbers, but somewhere, deep in their hearts, this physical limit (by design) exists.

[EHocking]

I've got a circle whose diameter is EXACTLY X units long. And yet no one alive can tell me what the exact circumference is after some two thousand years of geniuses trying. I'm holding this circle . It has an fixed cicumference. It has a fixed diameter. And yet I can't figure out what they both are!!
And don't get me started on decimals. One third of twelve is an exact four. But using .33333 ad infinitum instead of 1/3 in the equation leads to an inexact number, 3.99999 ad infinitum. This reduces math to proper protocol: multiply FIRST by 1 and THEN divide by three, and NEVER divide the one by three first to get an inexact decimal number.
I wonder how many computers make that mistake?Get a circle with a diameter of 7 units and your problem is solved.

(didn't take me 2000 years to work that one out, and genius I ain't)

[CapelDodger]

You should be aware that every single digital computer, calculator, instrument, etc. makes a mistake like this. In converting an analog (continuous) signal to a digital signal, your last bit (or least significant bit) in the sequence will have to encode some small value between 0 and 1 as either 0 or 1.

If you want to limit the effect of this error, you have to include more bits. Modern computers can calculate very good numbers, but somewhere, deep in their hearts, this physical limit (by design) exists.

Whereas something as simple as a circle gets the proportions exactly right every time.

Makes you think ...

Hipassus thought too much, and that was just about the diagonal of a square (root 2). Glub glub ... :eek:

[CapelDodger]

The most common definition of "transcendental" specifies that the polynomial be of finite degree.

You're not helping much either, are you?

[CapelDodger]

It's all God's fault.......

For making Pi such an irrational number? ;)

Planet X is only sooooo big around.

ETA: I like pie

My mother (way more impressive than any god) makes a lemon meringue pie that's to die for. And they're either big enough to go around or she makes two :).

Like any parent, of course, my mother's not entirely rational, but the pie (and the cakes, did I mention the cakes? And the custard slices?) makes that immaterial.

[X]

:duck:

What's the [infinity-1]th term in that series?

:duck:

Infinty-1 has always bothered me.
It's not infinity, by definition.
But if it's not infinity, than infinity is in fact finite.
It makes my head hurt.
So I just pretend it's infinity.
(Sorta like how 00=1)

Oh, and I think the OP is just venting some steam about significant figures.

Said i to pi, "be rational!"
Said pi to i "get real!"

[ddt]

I've got a circle whose diameter is EXACTLY X units long. And yet no one alive can tell me what the exact circumference is after some two thousand years of geniuses trying. I'm holding this circle . It has an fixed cicumference. It has a fixed diameter. And yet I can't figure out what they both are!!
As other posters already commented, the ratio of both numbers is pi, and pi is a transcendental number. That was proven in the 1880s, check wiki on it - I'm too lazy to do your homework. The beauty of math is that you not only get to prove that things do exist, but also sometimes get to prove that things do not exist, or are not possible.

Now, pi is one of the most studied numbers and there exist numerous infinite series which converge to pi. The beauty of math is also that we can calculate of each of these series how fast they converge. So whenever you need to approximate your calculation to a predefined accuracy, you know how many terms of the series you need to take in order to arrive at that accuracy.

And don't get me started on decimals. One third of twelve is an exact four. But using .33333 ad infinitum instead of 1/3 in the equation leads to an inexact number, 3.99999 ad infinitum. This reduces math to proper protocol: multiply FIRST by 1 and THEN divide by three, and NEVER divide the one by three first to get an inexact decimal number.
I wonder how many computers make that mistake?

Computers don't make mistakes, programmers do. A programmer who engages in programming calculations has to take into account the limitations of the floating point numbers of the computer. Floats are also just an approximation of the ideal of the mathematical real numbers: they have a limited accuracy.

Commonly, nowadays processors use the IEEE 754 and 854 standards. Basically the float formats defined by these standards consist of a mantissa part and an exponent part, just as in scientific notation - but then both are binary instead of decimal. Of course the mantissa has a limited number of bits - 48 bits in the 64-bits 'double' format, IIRC.

This means you should be careful how to do operations. To take a basic example, you can't rely on associativity of addition: (a+b)+c may give worse rounding errors than a+(b+c). A good floating point programmer knows this.

As an anecdote: I once worked as developer and support engineer at a company that makes compilers for the C programming language. One customer complained that the expression
(int) pow(2,3)
resulted in 7 instead of 8. The C-function 'pow' is a function which takes two double-precision floating point numbers as arguments, and then calculates the power. So the 2 and 3 were first converted from integer numbers to floating point numbers (no errors yet) and then the power was calculated. Of course, that step may incur some inaccuracy. As it turned out, it gave the biggest possible number smaller than 8 - so it was accurate enough. But the '(int)' means rounding down to an integer, and so he got a 7. Obviously he was a programmer who doesn't know his trade, and doesn't know about the dangers of floating-point arithmetic.

[The Gnomon]

What totally bugs *me* about maths is how is disproves evolution by mutation and selection. That's really annoying, and I wish it'd stop doing that.

It isn't very funny to quote Kleinman like this, is it?

After all, Mathematics is responsible for "fueling" the rise of Hitler.:rolleyes:

[Schneibster]

;3344188']Infinty-1 has always bothered me.
It's not infinity, by definition. Actually, infinity minus infinity is infinity. As a matter of fact, infinity divided by infinity is infinity which implies that infinity minus an infinite number of infinities is infinity. So, no, infinity minus one is infinity. By definition. :D

;3344188']It makes my head hurt.Sorry, I'm probably not helping with that much. :p

[gnome]

;3344188']Infinty-1 has always bothered me.
It's not infinity, by definition.
But if it's not infinity, than infinity is in fact finite.
It makes my head hurt.
So I just pretend it's infinity.

Actually you're closer to correct than you think.

The problem comes from thinking of "infinity" as a number. It isn't. So if you're sticking with definition, by definition infinity minus one doesn't exist because you can't subtract one from something that isn't a number.

Strictly speaking infinity is a concept of "unlimitedness".
So if you take a sequence that increases without limit... and subtract one from each member... that second sequence increases without limit also. You might think of that as "infinity minus one". Which would really not be different from "infinity" itself, you're just got a different starting point.

[Olowkow]

Then of course:

The number of integers is an infinity that is less than the infinity of fractions.

[Schneibster]

So "infinity minus one" is kind of like "blue minus one."

Nice!

[LostAngeles]

Don't forget the different infinities! There are as many natural numbers as there are integers as there are rationals, but there are more reals. Also, there are as many reals in [0,1] as there are in [0,1) as there are reals.

Next week: Axiomatic Set Theory.

[Schneibster]

And don't forget the geometric definitions: points on a line and curves on a plane.

[LostAngeles]

And don't forget the geometric definitions: points on a line and curves on a plane.

As a side note, being shown why the natural numbers are equinumerous with the integers are equinumerous with the rationals is really like (and I'm telling everyone this) when you're high and you realize how amazing your hand is.

It makes sense, but it's something you see regularly now being shown in a whole new light.

And that's when you realize...

...infinity is ****ed up and like a masochist, you love it.

[tsg]

Actually you're closer to correct than you think.

The problem comes from thinking of "infinity" as a number. It isn't. So if you're sticking with definition, by definition infinity minus one doesn't exist because you can't subtract one from something that isn't a number.

Strictly speaking infinity is a concept of "unlimitedness".
So if you take a sequence that increases without limit... and subtract one from each member... that second sequence increases without limit also. You might think of that as "infinity minus one". Which would really not be different from "infinity" itself, you're just got a different starting point.

And is actually an important concept when solving for converging summations of infinite terms.

[bobdroege7]

But there is exactly as many fractions as there are integers, and I can prove it.

[shadron]

I've got a circle whose diameter is EXACTLY X units long. And yet no one alive can tell me what the exact circumference is after some two thousand years of geniuses trying. I'm holding this circle . It has an fixed cicumference. It has a fixed diameter. And yet I can't figure out what they both are!!
And don't get me started on decimals. One third of twelve is an exact four. But using .33333 ad infinitum instead of 1/3 in the equation leads to an inexact number, 3.99999 ad infinitum. This reduces math to proper protocol: multiply FIRST by 1 and THEN divide by three, and NEVER divide the one by three first to get an inexact decimal number.
I wonder how many computers make that mistake?

To add a bit onto what ddt said above, floating point numbers are finite representations. They are contained within a physical limitation of 32 or 64 or some fixed number of bits, so they, by necessity, only approximate any transcendental number, or even any non-trans number that requires more than the fixed number of bits available. Since these numbers are stored in binary, this, by necessity, forces many finite decimal numbers (say 4.1) to be inaccurately stored because their binary representation has an infinitely repeating bit sequence. Many currency values, for example, cannot be completely accurately be stored in a binary floating point number, which is one reason why currency is usually stored as an integer or fixed point value instead.

So, when you add 3.3333333... to 6.666666666... in a computer you get something like 9.999999995296002... The implied inaccuracies when using floating point numbers have been a serious field of study in computer science for 30 years at least, and compensations and computation of inaccuracy statistics are well understood. Hardware floating point units not only detect underflow and overflow during math operations, but also loss of significance and other error modes. Sure, you can ask it to divide by zero, as long as your willing to take an error as your answer.

By the way, in pure math computations (decimal arithmetic, not computers) 3.99999... = 4.0, always.

[LostAngeles]

But there is exactly as many fractions as there are integers, and I can prove it.

I already said that! :p

[Complexity]

What really bugs you about math is that you don't know enough of it.

[sol invictus]

As I recall, pi is the limit of an infinite series ...

I'm not helping much, am I :)?

OK, OK, I should have said of finite degree... anyway almost all numbers are transcendental, so it's nothing very special.

[Elizabeth I]

Everything.

[tsg]

Everything.

That's curable, you know.

[bobdroege7]

I already said that! :p

except for the prove part, got a cocktail napkin?

3 3/1 3/2 3/3

2 2/1 2/2 2/3

1 1/1 1/2 1/3

1 2 3

by one to one correspondence

1 2 3 4 5 6 ect

1/1 1/2 2/1 3/1 2/2 1/3 ect

but you knew that!

[LostAngeles]

except for the prove part, got a cocktail napkin?

3 3/1 3/2 3/3

2 2/1 2/2 2/3

1 1/1 1/2 1/3

1 2 3

by one to one correspondence

1 2 3 4 5 6 ect

1/1 1/2 2/1 3/1 2/2 1/3 ect

but you knew that!

Yes! That's pretty much how you show countability which implies that the set of rationals is equinumerous with the set of natural numbers and we all live in Cantor's Paradise. :D

[sol invictus]

Yes! That's pretty much how you show countability which implies that the set of rationals is equinumerous with the set of natural numbers and we all live in Cantor's Paradise. :D

Well done! The next steps are to prove

a) that the set of non-transcendental irrationals are countable too, and

b) transcendentals are uncountable.

[mijopaalmc]

b) transcendentals are uncountable.

I have no idea how to prove it, but I seem to recall that if A is a countably infinite subset of X, which is itself uncountably infinite, then X/A (also written as X-A) is uncountably infinite.

Thoughts?

[LostAngeles]

Well done! The next steps are to prove

a) that the set of non-transcendental irrationals are countable too, and

b) transcendentals are uncountable.

We did that also. B is easier to do is you show that the algebraic numbers (the non-transcendentals) are a countable subset of R and since the union of countable sets is countable, it implies that the transcendentals are then uncountable, since we know R to be uncountable.

With A, if I remember correctly, any polynomial of degree n and integer coefficients has at most n roots. You can completely determine a polynomial by its coefficients and so, map each polynomial to an n-tuple...

OK, I pulled my notebook.

Basically, the Cartesian product of countable sets is countable and so is the union of those. You can map the set of polynomials by an injection to a Cartesian product of their coefficients and so they are a countable set. The set of the roots of any polynomial of some degree is a finite set and the union of those is therefore countable.

I think I explained that correctly.

So, powerset of the natural numbers is equinumerous with the reals next? :D

[LostAngeles]

I have no idea how to prove it, but I seem to recall that if A is a countably infinite subset of X, which is itself uncountably infinite, then X/A (also written as X-A) is uncountably infinite.

Thoughts?

Yeah, pretty much. Cantor did it by doing counting on the infinite sets which is what I attempted to outline in my above post.

[BillyJoe]

Really, it is infintely irrational to meditate about transcendentals?


...sorry, off I go.

[mijopaalmc]

Can the set difference of two uncountably infinite sets be itself uncountably infinite?

[LostAngeles]

Can the set difference of two uncountably infinite sets be itself uncountably infinite?

Intuitively, I would say yes, but not necessarily so. The difference of the reals and transcendentals would be the algebraic numbers which is countable. The difference of the reals and [0,1] would be uncountable infinite also.

So I don't think that anything can really be said about the difference of uncountable sets, but I could be wrong on this.

[Dragon]

Tumbleweed appears to have been blown away by all this mathematics.

[bobdroege7]

We should apologize for taking his OP and turning into something else.

I am sorry Tumbleweed

[BillyJoe]

I usually love it when that happens, but this time I'm not sure....

...except that it reminded me of the Infinite Improbability Drive. :)

[69dodge]

;3344188']Infinty-1 has always bothered me.
It's not infinity, by definition.
But if it's not infinity, than infinity is in fact finite.
It makes my head hurt.
So I just pretend it's infinity.

I'm not sure what you mean by "infinity - 1". I can think of two possibilities.

If we start with a set that contains infinitely many elements, e.g. the set of positive integers {1, 2, 3, 4, 5, ... }, and then remove from it one element, e.g. 3, infinitely many elements remain: {1, 2, 4, 5, 6, ... }. In this sense, infinity - 1 is infinity. The resulting set isn't identical to the original one---it's missing the removed element---but it still has infinitely many elements, just as the original one did.

We could, alternatively, think of infinity, not as a set containing infinitely many elements, but as a number in its own right. First comes 1, then 2, then 3, and so on. After all of those numbers, there's another number, called "infinity". (Actually, it's usually called "ω" [Greek letter omega].) And we could think of subtracting 1 from a number, not as removing an element from a set, but as moving backward from the number to the one right before it. However, there is no number right before ω. The only numbers before it are finite numbers, none of which is right before it, because between every finite number and ω are greater finite numbers. So, in this sense, there's no such thing as infinity - 1.

[Dave Rogers]

Actually, infinity minus infinity is infinity. As a matter of fact, infinity divided by infinity is infinity which implies that infinity minus an infinite number of infinities is infinity. So, no, infinity minus one is infinity. By definition. :D

Errr.... no. Infinity divided by infinity isn't infinity, it's indeterminate (but it may be infinite). Every member of the first infinite set of values can be mapped onto a member of the second infinite set of values, and an expression hence derived for the value of each term of the resulting infinite set of values. If this results in a convergent series, the result is finite. If it results in a divergent series, the result's infinite. So infinity divided by infinity is whatever you want it to be at the time.

That's how I understand it, anyway.

Dave

[sol invictus]

We did that also. B is easier to do is you show that the algebraic numbers (the non-transcendentals) are a countable subset of R and since the union of countable sets is countable, it implies that the transcendentals are then uncountable, since we know R to be uncountable.

Cheater! You have to prove the reals are uncountable too. This (http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument)
technique is really beautiful, and incredibly simple.

So, powerset of the natural numbers is equinumerous with the reals next? :D

That one might be a bit too hard for a homework assignment... ;)

[Cuddles]

Errr.... no. Infinity divided by infinity isn't infinity, it's indeterminate (but it may be infinite). Every member of the first infinite set of values can be mapped onto a member of the second infinite set of values, and an expression hence derived for the value of each term of the resulting infinite set of values. If this results in a convergent series, the result is finite. If it results in a divergent series, the result's infinite. So infinity divided by infinity is whatever you want it to be at the time.

That's how I understand it, anyway.

Dave

It depends which infinity you're talking about. Aleph1/Aleph0, for example, is Aleph1, for exactly the same reason that Aleph0/10 is Aleph0. So infinity divided by infinity often is infinity, it just depends how big the infinities you're talking about actually are. However, I'm not entirely sure what Aleph0/Aleph0 is, that might be indeterminate.

[Cabbage]

That one (powerset of the natural numbers is equinumerous with the reals next) might be a bit too hard for a homework assignment... ;)
It's not that difficult. It's straightforward to map the power set of the natural numbers onto the real interval [0,1] (which is equinumerous to the reals):

Given a subset X of the naturals, construct a binary number from [0,1] as follows: Place a "1" in the nth position after the radix point if and only if n is in X. For example, the set X = {1, 5, 13} would give the binary number:

0.100010000000100...

(a 1 in the 1st, 5th, and 13th positions, 0 elsewhere).

This map isn't quite one-to-one due to non-uniqueness of binary representations (0.1 is the same as 0.0111111..., for example), but that can be fixed with a little tweaking.

[LostAngeles]

Cheater! You have to prove the reals are uncountable too. This (http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument)
technique is really beautiful, and incredibly simple.



That one might be a bit too hard for a homework assignment... ;)

Sorry! I thought I already mentioned that!

Yeah, the last one involves the Schroeder-Bernstein Theorem, but once you have that and that they're both less-than-or-equal to each other, it follows. My text has a nice proof of it.

Minus the diagram, of course, which the prof. (who wrote the book acknowledged) and proceeded to draw a much nicer diagram.

And that is about where we're up to. We're doing Zermelo's axioms next.

[six7s]

Computers use binary, not decimal, for performing calculations

Assuming that, for some, the talk of compilers, integers and floats might be a tad bewildering... here's a link to a piece that, armed with a pencil and some paper, ought to make some sense to anyone who can count to 100 in decimal and 1024(10) or so in binary :)


Russian Peasant Multiplication (http://www.bbc.co.uk/dna/h2g2/A22808126)

...57 can be written as = 1 + 8 + 16 + 32.

If you want 57 lots of 23, then we add together one 23 (=23), eight 23s (=184), 16 23s (=368) and 32 23s (=736)...

...

It's actually very old - ancient in fact. The Ancient Egyptians used the method and documented it in around 1700 BC, but it may be a lot older than that. In 1960, the Belgian explorer Jean de Heinzelin de Braucourt found a notched piece of bone in the area of Ishango...

How the method got to Russia is a bit of a mystery, but it is thought Western visitors observed it there in the 19th Century. The Egyptian papyri describing the method were rediscovered later. The method is still used in parts of Africa today.

...

In our multiplication example, the computer would be storing numbers 57 and 23 as 111001 and 10111 respectively. When we instruct the computer to multiply them together, it will do it something like this:


Store 57 (111001) in an area of memory. We'll call this the multiplicand area.


Store 23 (10111) in another area of memory. We'll call this the multiplier area.


Clear a third area of memory to hold the result (copy zero into it).

[BillyJoe]

Russian Peasant Multiplication (http://www.bbc.co.uk/dna/h2g2/A22808126)


42! :D


Here's an example which multiplies the numbers 57 and 23 together, but the method will work with any numbers, of course:

1. At the top of a piece of paper, write the first number (57) on the left, and the second number (23) on the right. Mathematicians call these numbers the multiplicand and the multiplier: maybe they should get out more.

2. Halve the first number and write the result below it. Ignore any fractions, so 57 halved is 28.

3. Double the second number and write the result below it, so 23 doubled is 46.

4. Repeat steps 2 and 3 until the left-hand number has reduced to 1. 28 halves to 14, 46 doubles to 92, and so on. Eventually we have 1 in the left-hand column and 736 on the right.

57 23
28 46
14 92
7 184
3 368
1 736


5. Cross out all the rows where the left-hand number is even, so we will cross out the rows 28/46 and 14/92 here.

57 23
- -
- -
7 184
3 368
1 736
1311


6. Add up all the numbers remaining in the right hand column, for the result. We get 23 + 184 + 368 + 736 = 1,311, so 57 x 23 = 1,311.



It didn't seem to work for 43X37, until I realised you have to go down to either 1 (as in the above example) or 0 (in the case of 43X37), and you have to regard 0 as an odd number.
Which raises a question:

Is zero even or odd?

Looking at the fact that every other number is even...
7 6 5 4 3 2 1 0
...zero should be even.

On the other hand, in the above calculation, it is odd.

So that's quite odd!

[BillyJoe]

...and here was I thinking I would have only a series of one-liners for this thread!!!

[Ron_Tomkins]

I've got a circle whose diameter is EXACTLY X units long. And yet no one alive can tell me what the exact circumference is after some two thousand years of geniuses trying. I'm holding this circle . It has an fixed cicumference. It has a fixed diameter. And yet I can't figure out what they both are!!
And don't get me started on decimals. One third of twelve is an exact four. But using .33333 ad infinitum instead of 1/3 in the equation leads to an inexact number, 3.99999 ad infinitum. This reduces math to proper protocol: multiply FIRST by 1 and THEN divide by three, and NEVER divide the one by three first to get an inexact decimal number.
I wonder how many computers make that mistake?

An infinite number of computers.:)

[ddt]

The implied inaccuracies when using floating point numbers have been a serious field of study in computer science for 30 years at least, and compensations and computation of inaccuracy statistics are well understood.
And if you need to implement some standard calculations, you can basically pick up some textbook on it and copy the pseudo-code from the book.

A nice on-line text about the pitfalls of floating-point calculations is What Every Computer Scientist Should Know About Floating-Point Arithmetic (http://docs.sun.com/source/806-3568/ncg_goldberg.html).

Until the advent of the Intel 8087 coprocessor and the IEEE 754 standard - development of which went hand-in-hand - there was the problem, though, that computers had differing floating-point implementations. Supercomputer manufacturer Cray was notorious for changing implementations between machine designs - whatever was faster :).

Hardware floating point units not only detect underflow and overflow during math operations, but also loss of significance and other error modes. Sure, you can ask it to divide by zero, as long as your willing to take an error as your answer.
Error? No, positive number divided by zero is "positive infinity". Likewise for negative. Zero divided by zero is "not a number" :D.

[Molinaro]

57 * 23
= (60 * 23) - (3 * 23)
= 10 * [(120 + 18) = 138] = 1380 - 69 = 1311

I would do that in my head faster than the other methods here.

[ddt]

This map isn't quite one-to-one due to non-uniqueness of binary representations (0.1 is the same as 0.0111111..., for example), but that can be fixed with a little tweaking.
The number of these "doubles" is countably infinite (*), and we're talking about a uncountably infinite set anyways, so they're negligible in the whole picture.

(*) for each decimal where the repetition starts, there's a finite number of possibilities for the preceding decimals, so we're talking here a countably infinite union of finite sets, which is obviously again countable.

[ddt]

We're doing Zermelo's axioms next.
Can't we do Gödel-Bernays for a change? Never got around to read up on them. And ZF is kind of ugly in the way it gets around the Russell paradox.

And after that on to the continuum hypothesis :D. Someone already (unknowingly?) touched on it...

[tsg]

It didn't seem to work for 43X37, until I realised you have to go down to either 1 (as in the above example) or 0 (in the case of 43X37), and you have to regard 0 as an odd number.

I think you may have made a mistake somewhere. I get:

43|37
21|74
10|148
5|296
2|592
1|1184

37+74+296+1184 = 1591

[X]

2 words:
Napier's Bones

[Molinaro]

I think you may have made a mistake somewhere. I get:

43|37
21|74
10|148
5|296
2|592
1|1184

37+74+296+1184 = 1591

43 * 37
=(40 * 37) + (3 * 37)
=(40 * 40) - (3 * 40) + 111
= 1600 - (120 - 111 = 9) = 1591

[shadron]

Error? No, positive number divided by zero is "positive infinity". Likewise for negative. Zero divided by zero is "not a number" :D.

Agreed - I guess we'd want to define what an "error" is, but IEEE has already done that. (I really meant exception condition, but WTH).

[shadron]

;3347557']2 words:
Napier's Bones

I knew you'd get your slide rules in here some how. :)

[X]

I knew you'd get your slide rules in here some how. :)

;)

Although, technically, Napier's Bones are not based on logarithms, and so are not slide rules.

And I unfortunately have not yet managed to find a set to buy. :(

[BillyJoe]

I think you may have made a mistake somewhere. I get:

43|37
21|74
10|148
5|296
2|592
1|1184

37+74+296+1184 = 1591


$#!+!

(oops, is that allowed on this thread?)

My only excuse is that I had just come from a homoeopathy site. :o

[BillyJoe]

43 * 37
=(40 * 37) + (3 * 37)
=(40 * 40) - (3 * 40) + 111
= 1600 - (120 - 111 = 9) = 1591

Let's get an audio link up and check him out for real. ;)

[tracer]

Speaking of infinities:

The Epic Level Handbook for 3rd edition Dungeons & Dragons has tables in it for randomly generating magic weapons and magic armor. If you keep rolling a 96-00 on percentile dice, ad infinitum, it is theoretcially possible to generate a +infinity sword or a +infinity suit of armor.

My question, then, is:

If someone with an infinite AC is attacked by someone with an infinite Attack Bonus, what number does the attacker have to roll on the d20 to score a hit?

[CapelDodger]

Agreed - I guess we'd want to define what an "error" is, but IEEE has already done that. (I really meant exception condition, but WTH).

Trap the buggers, don't let them trap you :).

(I think the modern term is "catch", but it applies just as well.)

[gnome]

Speaking of infinities:

The Epic Level Handbook for 3rd edition Dungeons & Dragons has tables in it for randomly generating magic weapons and magic armor. If you keep rolling a 96-00 on percentile dice, ad infinitum, it is theoretcially possible to generate a +infinity sword or a +infinity suit of armor.

My question, then, is:

If someone with an infinite AC is attacked by someone with an infinite Attack Bonus, what number does the attacker have to roll on the d20 to score a hit?

Can't happen. If at any point you stop rolling in order to play, the bonus is finite.

[Molinaro]

Let's get an audio link up and check him out for real. ;)

That's not what you would hear if I spoke it in the order I actualy did it. I wrote it that way to list the operations I did in a way to show it's correctness.
The sequence of events would be:

43 * 37,
Estimate: 40 * 40 = 1600
Correction to error in estimate from change in 1st number = +3 * 37 = 90 + 21 = +111
Correction to error in estimate from change in 2nd number = -3 * 40 = -120
Add the corrections = -120 + 111 = -9
Correct the estimate = 1600 -9 = 1591

That's not realy the words used when I do it. That just explains the thought thingies going on not involving words/numbers that goes along with the numbers.

Note
1) When doing the 2nd correction we don't use a multiple of one of the original numbers. The estimate value has already been corrected for in the 1st correction calculation and so we use the easy to multiply estimate value we chose.

2) That means we always pick the lower of the original values to calculate first. It's the harder number to multiply. Then after we get the answer, 111 here, we can easily do the other simpler calculation, 3 * 40 =120 here, without forgeting the 1st number.

3) Since we get the smaller number 1st, we can then in one step quickly do the subtraction 111 - 120 = -9, in one quick step before you forget the numbers involves and get all addled with adds.

[CapelDodger]

If someone with an infinite AC is attacked by someone with an infinite Attack Bonus, what number does the attacker have to roll on the d20 to score a hit?

Any competent DM declares a reality-rift before the roll and all bets are off. It encourages diplomacy.

[CapelDodger]

Can't happen. If at any point you stop rolling in order to play, the bonus is finite.

No, the bonuses become undefined. Much like diplomacy.

[LostAngeles]

Can't we do Gödel-Bernays for a change? Never got around to read up on them. And ZF is kind of ugly in the way it gets around the Russell paradox.

And after that on to the continuum hypothesis :D. Someone already (unknowingly?) touched on it...

I can't comment on the first part.

Might have been me who mentioned the continum hypothesis. It was brought up at the end of last class as one of the issues we "currently face" in our stage of studying set theory.

(By, "we," I mean, "my class.")

[ddt]

$#!+!
You're trying to learn Perl? :)

[ddt]

Agreed - I guess we'd want to define what an "error" is, but IEEE has already done that. (I really meant exception condition, but WTH).Trap the buggers, don't let them trap you :).

(I think the modern term is "catch", but it applies just as well.)

:D :D :D

Yes, when you're talking OO exceptions :) But you don't have to catch them, floating point errors just give you some weird (non-) number you can test on. Say, in Java, you'd just test on
x == Double.POSITIVE_INFINITY
when you suspect a floating-point divide by zero might have happened. Whereas with an integer division, you'd indeed get a ArithmeticException which you'd have to catch.

Same idea on the hardware level: most processors generate an interrupt when you do an integer divide by zero, or else they set some flag in the processor status word.

[BillyJoe]

You're trying to learn Perl? :)

No, it's $#!+ with a exclamation mark!

(hint: you have to imagine what letter each subsitute looks like. ;) )

[tsg]

<>!*''#
^@`$$-
!*'$_
%*<>#4
&)../
|{~~SYSTEM HALTED

Transliterated:
Waka waka bang splat tick tick hash,
Caret at back-tick dollar dollar dash,
Bang splat tick dollar under-score,
Percent splat waka waka number four,
Ampersand right-paren dot dot slash,
Vertical-bar curly-bracket tilde tilde CRASH.

[BillyJoe]

(@|| @$$|$$+@^+ NOW!

can't post all in capitals so added this

[lomiller]

If the OP has trouble with pi, I wonder what he/she thinks of i (j for electrical engineers)

[ddt]

If the OP has trouble with pi, I wonder what he/she thinks of i (j for electrical engineers)

Now you're taking up a complex issue!