Calculating Machine: Newsletter Challenge (01/04/11)

GA and that's all I have to add --subtract

Though we know that to be a fact as far as A+B = A--B. A calculator may come up with a syntax error.

I tried with calculator, answer is given without syntax error...

Mine does

I choose the machine that does division

Addition can be easily done by hand or mentally

Subtraction likewise

Multiplication can be hand done by successive addition or halving/doubling

Division by machine would be the most useful as it is not easily done manully

GA: I agree, see answer #24 below.

Yep... that was what I remember being taught. GA to your #24 as well... I really need to read the whole thread before I post.

JavaHead

GA.

It is the one machine that can perform any of the 4 operations with common knowledge taught at the basic algebraic level. No knowledge of binary required, which not all are taught, no knowledge of shifting required, no knowledge of 2's Complement required, no requirement of converting the original given values into another format.

Just need to know the formula, order of operations, and how to add or subtract '1' in your head.

Older&wiser, congratulations on your first GA! Welcome to CR4.

JavaHead

Well, A - B = A + (-B). So addition isn't too much different from subtraction. Multiplication and division are repeats of the first two.

(I heard in just recent news about the reintroduction of the soroban to Japanese math classes.)

How do you enter a negative number?

Doesn't A + B = A -(-B) also require entering a negative number?

Maybe it's just a tomayto/tomahto thing?

_{(Dances around issue of smuggling in 2's or 10's complement.)}

Depends how the [-] key operates, and if there are 'bracket' keys. 0-B should enter -B. Without a [-] (or 'change sign' key) I can't see any way of entering a negative.

_{(also steering well clear of complements - it would take a lot of pencil-and-paper work (or a subtract function?) to generate them)}

Precisely. Unless you can enter a negative number none of the machines is fit for all purposes. If you can enter a negative number the addition and subtraction machines are equally efficient. Both of them can be used for multiplication by either repeated addition or repeated subtraction (possibly involving a negative number), and both of them can be used for division by repeated addition or repeated subtraction (possibly involving a negative number). Where is the problem?

By subtracting it from zero

You explained much better than I was going to. GA. I was going to simply state that at the most basic level a computer always adds for each and every function. You said that's how it was done "back in the day", but I think this still holds.

You're correct (to a degree), so GA, - but how useful (to quote the challenge question) would your calculator be?

Shall we discuss Turing machines?

Oh wow, paleontology--I haven't seen one of those fossils in years!

About 3 (?) years ago, Scientific American did a retrospective on slide rules, complete with a paper cutout. A nice reminiscence for us old-school slipstick operators.

Great post! This is how computers did addition, subtraction, etc. early on. Then they invented the ALU, which uses fewer transistors.

Incidentally, some logical constructs are functionally complete, meaning you can implement any logic function using just them. The NAND function is one of them. You could theoretically implement an entire CPU using only NAND gates. Each 2 input NAND gate can be implemented with 2 CMOS transistors.

This is a bit off-topic, but I found this link that goes to the HP calculator museum. There are java scripts there that simulate old HP calculators, including the HP-25 I had in college:

http://www.hpmuseum.org/simulate/simulate.htm#java25

the machine has no doubt about 9's complement

I agree it is the adding machine. After using the abacus in school the first machine I used was the Facit mechanical calculator. Basically it was add only machine but could be used for subtraction, multiplication and division.

Not sure but there is a problem with the wording.

"Which machine is the most useful in performing any one of these four mathematical operations."

Should not it read which machine is most useful in performing all of these four mathematical operations.

So I say they are equally useful in performing the single operation they can.

The only used add and used the two's compliment to subtract (if my rusted memory serves me right)

on a 3 bit register sign plus 2 bit value

0ne would have

3 = 011

2 = 010

1 = 001

0 = 000

-1 = 111

-2 = 110

-3 = 101

-4 = 100

To subtract 2 from 3 one would add 110 to 011 and get 1001 , loose the left bit and get 001 = 1

Maybe the bit on the left was the positive , I must find my HP assembler book.

the one that "adds"..

The one that only adds, Addition is the basic of all basic operations.

Subtraction is just addition of negative numbers, multiplication is just a repeated addition, and division might looked at as successive subtraction.

Since nothing was said about brackets and negative number entry, I would assume that they are allowed with no exception for the 4 machines. Logic is to take it as is with no furthure assumptions from my side.

Randall,

A very interesting answer - worth a GA , certainly.

I will try and squirrel this away somewhere.!

Sleepy

Good answer. I like your answer better than mine (#34).

if the machine divides - she cannot add or subtract

can't work

Which part of this:-

-----------------------------------------------------------------

And that A+B = {(A/A) + (B/A)}/(1/A) i.e. {1+(B/A)}/(1/A)

Similarly A-B = {(A/B) -1}/(1/B)

Then all we need to be able to do to achieve the two simpler tasks is to add or subtract 1.

-------------------------------------------------------------------

didn't you understand?

the machine can add, subtract, multiply or divide - but only one of them!

A+B = {(A/A) + (B/A)}/(1/A)

contains two of them!

OK?

If you read further in each equation in Post 24 (and similarly 50), the required addition/subtraction is further equated to an expression requiring that only 1 needs to be added/subtracted. The person, not the machine, does this.

I select one only adds.

Let me know when you find one that works with Roman numerals, Uncle!

I've had one on my palm for about 9 years:-

http://www.aldweb.com/articles.php?lng=en&pg=18

I just don't know how I managed without it.

I really want an RPN one though.

The one that adds.

computer can perform all the operations and these operate on adding only, so answer should be adding machine.

I think it is the calculator with a multiplying function.

I would choose a machine that adds distances along a linear or logarithmic scale (your choice depending upon the operation desired). Aka - a slide rule.

The one that only adds.

It can add negative numbers (subtract), add repeatedly (multiply), add negative numbers repeatedly (divide).

Assuming there is no sign change key but there is a decimal cabability, the one which subtracts would be the most useful

for adding, (A+B) start with zero and ignore the negative signs

0-A-B is the same as -(A+B)

subtraction (A-B) is obvious (A-B)

Multiplication (A*B) similar to the adding for multiplication discussed in other entries but again start with zero and ignoring the negative signs. if B has three significant digits for example 273

0-A*100-A*100-A*10-A*10-A*10 ... A*10-A-A-A

division (A/B)

Approximate (A/B) to nearest single decimal value for example 20000/43 is between 100 and 1000

start with A and subtract (in this case) 100*B watching when the resultant is less than 100*B (after 4 iterations 2800 is less than 4300) if the resultant is greate

multiplier times B until the resultant is less than the multiplier time B (6 iterations of -430 result in 220) continue with each lower multiplier as wanted.

Of course if you have a log table beside the subtracting calulator, the multiplication, division and power options are very easy. (reminds me of the slide rule)

None of them unless the operator has a sound knowledge of basic mathematics.

Addition is the only function computers do. As stated above, subtraction is adding the inverse, etc.

Computers just do it really fast.

This was taught when I first was learning about computers 30 years ago back in the age of the stone abbacus. Still holds true today.

And my RPN did not like the a + b = a--b.

The one that adds is the most useful. It is how computers add, multiply, subtract and divide. Multiplication is repititive addition. Subtraction is nine's compliment addition. Division is repititive nine's compliment addition. This is how mainframe computers from the 60's performed their arithmetic computations. Things may have changed by now, but this worked well for the IBM 1401 and 360 models. As well as Control Data machines.

As of this post it seems the majority are leaning towards Addition with Subtraction coming in a close second. There was one vote for multiplication but with no supporting dialogue and three for division.

But, let's add some critical thinking and evidence…

Reading into the OP challenge, I am assuming that by "most useful" one might presume the machine that requires the operator to do the least amount of side work manually, or requires the least amount of work in total. So, let's assume the each machine can only perform a single process at a time (enter 1st value, enter process, enter 2nd value, equals) and is able to assign negative/positive attributes.

Given two random integers (269 and 71) what is the operator to machine ratio of processes performed for all 4 functions performed on all 4 machines? To clarify, an operator process is something the individual has to do mentally or on paper, for example reference post 24 - having to add or subtract '1' while using the division machine would be an operator process.

Adding Machine:

269 + 71 = 0:1 operator to machine ratio

269 - 71 = 0:1

269 * 71 = 0:70 (for example 269+269, 538+269,…)

269 / 71 =

Which ever machine can perform all 4 functions with the least amount of processes, of both the operator side process and machine process, would be the most efficient and, thereby, the most useful. Or, assuming all machines require at least some operator processes, whatever machine requires the operator to do the simplest processes. For example - having to add or subtract '1' 10 times as compared to having to find the 9's complement of '269' once, manually.

So, should we 'weight' the process ratios based on complexity?

Regardless, for those touting a specific machine, please work out the most efficient process ratio for that machine with the provided numbers (so we're all on the same page) for each mathematical process.

We'll then compare and see what we come up with.

JavaHead

An interesting offer Java -- I'll take it up for 297 / 71 = 0: ?, since I opted for the division machine (Post #3). Now in the case of simple fractions division is intuitive or requires only a few steps as 1/4 = 0.25, even 17/32 =0.53125. However in general, such as in dividing 297 by 71, the first operation gives 4. But there is a remainder of 13 ! ! !. Since the fraction 13/71 is irrational, the process continues indefinitly - that is there are potientaly an infiniite number of steps. Thus going by the criteria that you stated, the solution of 297 / 71 = 0: infinity, handily beating out the other three machines.. Like 4.183098592......etc.

He does have a good approach to this... but we need to reread his proposal...

what is the operator to machine ratio of processes performed for all 4 functions performed on all 4 machines?

I'm not sure doing the 9's is valid if the machine can't initially subtract.

Neither do I accept "just multiply A and B then add C x times" if the machine can't initially multiply.

However; assuming the 'simplest machine and operator';

Add is A+B

Multiply is A+A+A ... to B steps.

Divide is add the divisor B until the sum approaches A, and would exceed A if added again, thus producing a count C of adds and a fraction. (remainder/ B)

Subtraction? wind it backwards? (Which would also do divide).

So "simplest machine" would seem the key to doing all four.

But simple + does 3 of 4

The one that adds.

add!!!

Good explanations from Randall (24) and PassingtonGreen (50). Another vote for division.

Subtraction:

can simulate addition, but addition can not simulate subtraction; multiplication is multiple addition and division is multiple subtraction

A+B = A-(0-B)

A-B=A-B

n*A=A+A+...+A+A (n times)

A/B=A-B-B-...-B-B (n times)

Subtraction is the addition of a negative. Multiplication and division can be performed by adding logarithms (or adding a negative logarithm). I'd go for the adding machine.

Addition is the most useful function. All other operations are achieved by stacking addition operations or adding negative numbers.

Barely Definitions. Single-operation 'machines', 'calculating'.
But I'm convinced it is enough to postulate that none of these 4 kinds is feasable to replace any of the other.

I reduce to 2 inputs and 1 output, inmiddle 1 of the 4 kinds of operation.
And I state that single-operation means, disassambly of the unit to recycle parts to another kind of operator is not possible.

First I presume analog machines
An operating amplifier as comparator substracts, its inputs are + and -.
To perform addition the - input must be an inverter preconducted to.
A voltage controlled amplifier performs multiplication, also possible would be a circuit, that outputs exp(log(A) + log(B)), A, B inputs (as by slide rule).
The compound machine for division is a voltage controlled attenuator. But the log/exp method also is imaginable.
I find no exchanges possible, e.g. add: as 'sub' - the inverter cannot be defeated, as 'mult' or 'div' - no way, it could do the add in the log/exp variant, but the unit might not be opened.

Than digital calculating machines, dual system presumed
'add' and 'sub' can't be substituted by the other, this is because of the complement used to represent negative numbers in a steady range of numbers. The TWO's complement of A is INC(NOT(A)). I dont know of other complements in different based systems, if base is not a power of 2 I dought that a NOT operator exists that makes sense. Like the analog units there is no way to defeat the complement-builder or add it.
To multiply by 'add' a count control is not available. 'mult' does it in a more sophisticated way but it can't add nor substract various numbers. Similar the division by 'sub' connot be done without count control and comperator.

Cascading can even build 'add'(A,B) by 'sub'(A,'sub'(0,B)), thus 2 'sub's. Or 'mult'(A,B) by 'div'(A,'div'(1,B)). But this is not enough for "any of these 4 ..." I presume.

see post 54

I dont know of other complements in different based systems

9s complement. If you want to subtract say a 6 digit number from 1 million: then, just regard 1 million as 999,999 +1. For example

1,000,000 minus 123,456 =

999,999 + 1 minus
123,456

=

876,543 + 1 = 876,544

In your notation that would be INC(nines complement(123,456))

This is why 1s complement works in computers.

I would choose the addition machine as every other mathematical operation can be acheived by addition.

Since multiplication and division are higher order functions, I believe addition and subtraction engines can perform their tasks when properly operated. Since addition and subtraction are subsets of each other through the use of reversal of signing, either can perform the same job. But addition may be a better choice only since the human mind finds it more difficult to operate double negatives. In my mind, an addition calculating machine would be the basic answer.

I think the question is poorly phrased, the option is only between addition and subtraction, as the question says Which machine is the most useful in performing any one of these four mathematical operations? All the operation are solved with addition A*B = A1 + A2 ... +Ab [ A1 = A2 ... Ab ]

A/B = C = B1 + B2 ... +Bc [ B1 = B2 ... Bc ]

Adding a number to itself result is equivalent to shift left (effectively doubles the number). Doing this n-1 times (n = the effective number of digits the machine can handle) and adding the carry in the right side of the number every time it overfows is equivalent t divide-by-two (derived just by additions). Now with these tree functions add,double,divide by 2, I think you can implement almost any function. I would vote for addition. Subtraction is identical. S.M.

I agree with reply # 8 the adding machine was around a long time and used for the other functions

division

I am curious why so many find division so 'unique'

I'm beginning to think it's because it traditionally division 'reduces to zero' in method appearance, so achieving by 'adding to parity' is simply not 'permissable' in the formal psych.

As written Q. 2392/46 asks how many 46's in 2392?

Ok - there is a 'multiplying process' (adding) which you then subtract - called 'long division' - and you eventually (mostly, or sometimes arrive at, or near, "zero").

To arrive a parity is actually a more 'direct answer to the question' a fraction poses.

Direct Method; add 46, until parody is met.

I added it 52 times - got 2392 = 2392

A. 52

Then again I did ROSMSL when the teacher said "now we will move on to Improper Fractions" and nobody had actually grasped the question a fraction is asking.

But, equally; it has been said my 'visualising' of math is "past weird", however I suspect a great many of you will 'multiply to parity' when doing division "in your head".

How long does it take you to add 46, 52 times using a calculating machine?

Put yourself in the position of say Kepler, or Newton; you're offered one of the four machines: which would you take?

Um - tricky moving goal post question

Time is now a factor? Ok;"All's fair in love and CR4";

How about; less time than it would take your handle pulling monkey to add or subtract with a dividing machine? And with no parity concept, to multiply 2 x 2

But my handle pulling monkey can do all, provided he/she can '1111' (keep count), when it gets really matrical

Oddly; on "the position of say Kepler, or Newton..."

The slide rule was invented around 1620–1630, shortly after John Napier's publication of the concept of the logarithm. Edmund Gunter of Oxford developed a calculating device with a single logarithmic scale, which, with additional measuring tools, could be used to multiply and divide. The first description of this scale was published in Paris in 1624 by Edmund Wingate (c.1593–1656), an English mathematician, in a book entitled L'usage de la reigle de proportion en l'arithmetique & geometrie. The book contains a double scale on one side of which is a logarithmic scale and on the other a tabular scale. In 1630, William Oughtred of Cambridge invented a circular slide rule, and in 1632 he combined two Gunter rules, held together with the hands, to make a device that is recognizably the modern slide rule. Like his contemporary at Cambridge, Isaac Newton, Oughtred taught his ideas privately to his students, but delayed in publishing them, and like Newton, he became involved in a vitriolic controversy over priority, with his one-time student Richard Delamain and the prior claims of Wingate. Oughtred's ideas were only made public in publications of his student William Forster in 1632 and 1653.

And - as you know - it principally adds and anti-adds logs. [though some also have liner scales on edge for 'measurement', so add and subtract' "normally" can be achieved with a similar liner rule 'held together with the hands']

And just down a bit, in Wiki, is Advantages the first of which is more or less the concept I was alluding to.

But, moving on; I guess lucky Isaac had a machine that added stuff and poor old Kepler just had a pair of "dividers" to add or anti-add units against parity with a quantum [and keep count].

So what would I choose were I Kepler? Well I have the adding machine that Isaac had!

Given you can conceptualise parity as the same as 'null', Aka zero, then as JD said - like post #1? - it's all adding. Aka "keeping count" or monkey 1111

It's the 'conceptualising' that actually counts.

"Which machine is the most useful" (from original question)

I do not think time is a new element. The time to do something is part of its usefulness.

I do agree that more definition could have been put into the original post, but that probably would have resulted in many fewer responses.

What do you win if you lose the chess game?

maybe indulgent smiles from the parole board?

In vino veritas; in cervisia carmen; in aqua E. coli.

-

in tanga women?

look ahead that carry

the adding machine

The one that adds.

addition and subtraction both useful; the counter for multiplication and division has to add/Increment the countervalue by 1 or subtract/decrement by -1

who can see a difference in usefulness?

It is dead simple for the person (not the machine) to add/subtract 1, and then use the division machine to do the "heavy lifting" for all four operations. Adding a step or two to an operation beats the living dickens out of repeated addition or subtraction. That's why the division machine is best, hands down. Why are people having such difficulty in seeing this? (Once again, see posts 24 and 50.)

Also take a look @ Post #3

Saying addition and subtraction are "dead simple" kind of goes against the nature of the question, which is "which one operation is most useful for performing any (and all, presumably) of the four operations".

How do you add two numbers with a division machine again?

Reread post 91 again, please. I said that adding or subracting 1 is dead simple, not addition and subtraction in general. And I also said where to find how to use division for all four operations; namely posts 24 and 50.

Maybe division isn't such a bad choice after all.

With all due respect, especially those who create the challenges, where ambiguity is present, or the bench marks open, they should be revised/addressed.

E.g. I could argue that knowing my multiplication tables, obviates the use of all these machines. So to an extent the value of any depends on what operation I would see as 'more time absorbing' or 'error prone' - I.e. it's a purely personal view without 'operator ability' specified as a common denominator.

I would go so far as saying there cannot be a 'correct answer' to this - with out that fixed.

Describing a proposed method and actually demonstrating it are obviously different things. I think there really needs to be a competition amongst the various proponents of various single-operation calculating machine. I propose a competition using the following two numbers: 113 and 355. Being a mechanical engineer who can only think in base 10, initial and final results need to be in base ten. We will count the number of steps needed to complete all of the following calculations.

113 + 355 = 468

113 – 355 = -242

113 x 355 = 40115

355 / 113 ≈ 3.1416

(This last operation gives us a very good approximation to Pi.)

Every time the machine does a calculation counts as one point.

Every time the human has to do an operation such as write down a number on a piece of paper, and every step in a manual add, subtract, multiply, or add, counts as two points. Manual steps cost more points because we are slower and make more mistakes. I also mean every step in an operation.

Manually calculating 5/2 would take the following steps: 1) write 5, 2) write 5, 3) divide 2 into 5 and round down, 4) write 2 on top of long division, 5) write 4 below the 5, 5) subtract 4 from 5, 6) write 1 below the 4, 7) write a 0 ,8) divide 10 by 2, 9) right down the .5 for a total of nine steps. Lowest number of points total for all four calculations wins.

Required accuracy is five significant digits.

Winner gets lots of Good Answer Points.

By the way, I am clueless about how to actually do the calculations this way.

Thanks,

Jim