About a tyrant called "=".
I got involved in an interesting thread on symmetry and read the following, in my eyes very wise words:
The symmetry of what's the left hand side (LHS) and right hand side (RHS) of the "is equal to" sign, =, is here discussed. It's pretty obvious that = is justified if both sides are the same.
Still, when looking at mathematical equations, there are quite different things on both sides of the = sign. Only trivial cases like x=x look symmetrical. Most equations are not reducible to the trivial example.
So in what sense are both sides equal? Is it only an equality of a quantity or a number?
You can write a condition on energies, say that the kinetic energy equals potential energy. The quantities are the same on both sides, Joule, that is. Dimensional analysis is, by the way, a useful tool if both sides of a = sign are consistent. In the equation of two energies, this is obvious but in complicated expressions it comes in handy and you can even use it to anticipate.
So equality in units is important as well as the numerical equality. What else can = be applied to? It can be used to equalize sets, for example. Or to define a function, f(x)=y. The sign has the power to equalize. It equalizes two expressions on both sides of it. There are variations: <, >, ?, ?, ?, ?, ?, ?, and ? are cousins of the almighty equalizer but less powerful in the sense that they allow more numericals. Still, it requires both sides to be numerical or sets. Or don't they?
There is another set of nephews: ?, ?, ?, ?, ? each comparing the two sides of an equation and meaning that they are similar or about similar and about equal. ? means "identical to" but is not often seen. There is :=, which means equal by definition, and last but not least, my favorite, ?, not equal to. A bastard child that can be applied to most things in the world.
Is math powerless without =? Is the = the tyrant who equalizes both sides to his advantage, and if so, what's the advantage? Or is he unifying different sides and offering understanding and peace? Is unifying the same as equalizing? If both sides are equal, are they unified? Enough questions. Whatever thoughts related to this thread are welcome!
In mathematics, it is often said that the left hand side of the equation represents the very same thing as the right hand side, a specific mathematical value, or object. In reality, the two sides express two distinct things, with an equality between these two. When two things which are different, are said to be equal, the difference between them has already been excused in that judgement of equal. So we now have a second level of excusing differences for the sake of symmetry, the excuse which exists right at the level of producing the equation.
The symmetry of what's the left hand side (LHS) and right hand side (RHS) of the "is equal to" sign, =, is here discussed. It's pretty obvious that = is justified if both sides are the same.
Still, when looking at mathematical equations, there are quite different things on both sides of the = sign. Only trivial cases like x=x look symmetrical. Most equations are not reducible to the trivial example.
So in what sense are both sides equal? Is it only an equality of a quantity or a number?
You can write a condition on energies, say that the kinetic energy equals potential energy. The quantities are the same on both sides, Joule, that is. Dimensional analysis is, by the way, a useful tool if both sides of a = sign are consistent. In the equation of two energies, this is obvious but in complicated expressions it comes in handy and you can even use it to anticipate.
So equality in units is important as well as the numerical equality. What else can = be applied to? It can be used to equalize sets, for example. Or to define a function, f(x)=y. The sign has the power to equalize. It equalizes two expressions on both sides of it. There are variations: <, >, ?, ?, ?, ?, ?, ?, and ? are cousins of the almighty equalizer but less powerful in the sense that they allow more numericals. Still, it requires both sides to be numerical or sets. Or don't they?
There is another set of nephews: ?, ?, ?, ?, ? each comparing the two sides of an equation and meaning that they are similar or about similar and about equal. ? means "identical to" but is not often seen. There is :=, which means equal by definition, and last but not least, my favorite, ?, not equal to. A bastard child that can be applied to most things in the world.
Is math powerless without =? Is the = the tyrant who equalizes both sides to his advantage, and if so, what's the advantage? Or is he unifying different sides and offering understanding and peace? Is unifying the same as equalizing? If both sides are equal, are they unified? Enough questions. Whatever thoughts related to this thread are welcome!
Comments (64)
I think the concept missing from your OP is balance. Equal means both sides are balanced.
in Computing := means 'becomes equal to.'
So a program variable is given an initial value using :=
For example, using camel casing:
personName:= Jinty
The textual variable personName becomes equal to Jinty
OR 'is instantiated' to Jinty
= is more about balance than symmetry.
A tonne of feathers is not symmetrical in size to a tonne of steel but they are both balanced in weight.
:ok:
I hadn't looked at it that way yet. I had an equality of masses in mind, but your example shows what it actually is. Literally a balance! The numerical values are equal, but the stuff on one side is different from the other indeed. Balanced but not equal. M(feathers)=M(steel). Equality=Balance...
:up: :up: :up:
To question them is what philosophers do.
I think it is the balance that matters most, yes, rather that whats on either side of =
Balance....imbalance.....return to balance
chaos...combination....order....entropy..chaos...combination....order.......
waveforms have balance points or tipping points, in-between a single crest and trough, so you have:
balance.....up crest.....tipping point(or perhaps another balance point)....down crest.....balance.....down trough.....tipping point......up trough.....balance. There are many symmetries in waveforms as well.
The concepts of Balance, Imbalance, Chaos, Order are certainly very important in Philosophy, I think.
Indeed!
Balance
In short: an equation, by which I mean two sets of symbols connected by an '=' sign, constitutes only part of a complete logical sentence, not a complete sentence in itself.
The equation needs to be included in a sentence in order to be meaningful. Consider the following:
(1) sin x = tan x cos x
(2) sin x = 0.5
Item (1) is an 'identity', and is true for all x except odd multiples of 90 degrees. So we need a sentence to give it meaning, such as:
"For all real x except odd multiples of pi/2 we have sin x = tan x cos x"
Identities like this allow us to simplify and sometimes solve algebraic problems by substituting the expression on one side for the expression on the other. They do not say the expressions are the same. They are not, as they use different symbols and require different steps to evaluate them. But the identity says that, subject to any constraints imposed by the enclosing sentence, both sides will deliver the same value when evaluated, regardless of the values given to any pronumerals ('variables').
Things like Item (2) are typically just referred to in practice as an 'equation' (a more constrained meaning than I gave that term above) and is used to find allowable values of x. An example might be:
"Indiana Jones worked out that the sun would shine onto the crypt only when the angle of the sun from the vertical was 60 degrees, where sin x = 0.5."
Note that here the equation does NOT say that sin x is always equal to 0.5 but rather, that the thing we are interested in (illumination of the crypt) occurs when sin x has that value.
Conclusion: we can't identify a meaning for '=' without further context, including at least the full sentence in which the 'equation' occurs. That meaning will vary depending on the context.
Your conclusion is correct and it would be more helpful to write your second equation as a question, or a statement to be actioned, such as:
(Find values of x such that) sin x = 0.5 (and both sides of this system will be balanced)
To me, that's what sin x = 0.5 means in my head. sin x = 0.5 is just 'shorthand' is it not?
At the heart of it, it sounds like the question is about the absolute linguistic unit versus the relational. We find that 'equals' has a meaning in the mathematical community that is more intentionally defined than the one in our casual usage. Do we hold fast to those meanings and furtively ignore their linguistic significance in favour of reproducing completeness, or accept the synchronic meaning freely as separate and referential to a mathematically absolute definition? I think in any feedback system we have to understand there is an underlying dicrete-ness to its idealization; that – excluding special cases – something can be determined as one thing and simultaneously negate another, not solely in a formal sense but also in terms of content.
When we perceive we can negate as well as posit; create something that cannot have being at the same time we imagine some other aspect of it as 'being there.' In our outer world there is thought to be one set of physical laws that negate others, but there could be other physical laws present not meeting certain conditions of sensibility.
I digress...
b) 6 = 12 ÷ 2 = 2 × 3 (is the same as)
c) Left to the reader as an exercise
d) Ditto
.
.
.
3 + 5 = ?
The expected answer is '8'. But suppose we write:
3 + 5 = 5 + 3
No point, red cross, wrong answer. But why? 3 + 5 = 8 is true, and 3 + 5 = 5 + 3 is also true. 3 + 5 = 5 + 3 tells us something that 3 + 5 = 8 does not, namely, that addition is commutative. So it carries new information, just as 3 + 5 = 8 carries new information. We could also write "3 + 5 = x + 3 + 5 - x, for any number x", showing that there is an infinity of solutions to the problem. This will either get us sent home early or promoted to the fast-lane maths class, depending upon our teacher's mood.
Perhaps the tyranny is not the "=" sign itself but the baggage of expectations we carry around with regard to its application.
From reading other comments, where you gave some info about your background, your Maths is far beyond mine. I noticed that N=1/N seems to only have the solution N=1, is this the only solution?
Quoting Cornwell1
The thing about equating kinetic energy with potential energy is that it seems to involve some kind of category mistake to describe the two as equal. One is a measure of the actual movement of a thing, while the other is a measure of a thing's capacity to move. Since a cause is required to transform the potential to actual, then if we express the two as equal we neglect the reality that one is temporally prior, and the other posterior. This temporal difference implies that the supposed equality between them neglects an important fact.
I wondered about that too. Seems only 1 and -1 seem to be the answer. Maybe N can be different things. So that it's inverse is equal to itself. An inverse matrix?
Yeah, another doh! moment for me, to add to my ever-growing collection.
Thanks for correcting the oversight. Same to
So apart from Crowell1's inverse matrix question, it seems to me that all
N=1/N suggests is positive = positive and negative = negative
I think if you write positive = negative, you do require supporting context such as
Balanced/equal in quantity but opposite in 'charge?', 'Polarity?', 'magnetic attraction?'
Which would be more accurate or are they all equally valid.
I've always considered +ve and -ve numbers to owe their existence to the existence of +ve and -ve charge, and its electro/magnetic components.
Indeed. Potential energy is different from kinetic energy. Potential energy is defined in force fields. Kinetic energy is caused by a force. So is potential energy but in a "reverse mode". By pulling or pushing, it is stored or extracted. The kinetic and the potential can be transformed into one another or be compared in value. While kinetic energy is always positive, potential energy can be both, depending on the gauge. Is kinetic energy somehow gauge dependent too?
So, values balance, the "things" are different.
Quoting universeness
I had too look up "doh":
DOH
Also found in: Dictionary, Thesaurus, Medical, Financial, Encyclopedia, Wikipedia.
Category filter: Show All (22)Most Common (1)Technology (3)Government & Military (9)Science & Medicine (3)Business (2)Organizations (4)Slang / Jargon (6)
AcronymDefinitionDOHDeliriously Overcome with Hilarity (chat/internet)DOHDepartment of Health (various locations)DOHDirectorate of Health (various locations)DOHDepartment of HealthDOHDukes Of HazzardDOHDepartment of Hydrology (various organizations)DOHDivision of HighwaysDOHDetroit Opera House (Detroit, MI)DOHDepartment of HousingDOHDepending on HeelsDOHDestination Option HeaderDOHDocument Operations HandbookDOHDepartment of Highways (Thailand)DOHDate of HireDOHDeclaration of Helsinki (medical ethics; World Medical Association)DOHDays on Hand (inventory)DOHDoha, Qatar - Doha (Airport Code)DOHDefenders of Honor (gaming, Counter-Strike: Source Clan)DOHDouble Over Head (waves)DOHDepartmental Overhead (USACE)DOHDNS (Domain Name System) over HTTPS (Hypertext Transfer Protocol Secure)DOHDepartment of Hell (gaming)
I think the last one doesn't apply in your usage. But I got the feeling! :smile:
Ha Ha.....Now that was an impressive list. :lol:
It is a common sound made by the great philosopher Homer Simpson, from The Simpsons cartoon show.
He says Doh! anytime his brain refuses to work properly.
:lol:
Basically, yes.
Math is all about =.
You see, math has emerged from a need to picture reality around us.
Calculation, addition and substraction has been something that we need for practical uses. Math hasn't developed from some existential philosophical interest, but how to solve practical problems. Even animals can count: if a bird sees three men going to a barn and two of them later come out, the bird can count that one of the men is still inside the barn. And calculation, computation, is all about something is equivalent to another.
Now people might think that < or > would be different, but basically it's applying the same logic. The real opposite to this is the non-computable.
And what does and can mathematics say about the non-computable?
Only that it exists. That there exists mathematical entities that are non-computable. And that's it.
And what do we do when we face a problem that is non-computable?
Well, we surely don't use math to solve it. In fact, we likely don't approach the problem as if there would be one certain true solution for it and that one can deduct it somehow. And to describe it we don't make any mathematical models or formulas, but for example use narrative.
Certainly n=1/n has solutions n=1 and n=-1. The other meaning I have in mind is quite different. Hint: I have written hundreds of mathematical programs in BASIC. :cool:
I wrote many programs in my very early days as a teacher in BBC BASIC.
Having to number every code line was fun eh?
Would it not just be
100 INPUT N
110 LET N=1/N
120 PRINT N
So do you mean, that such a program would just display fractions rather than provide values for N which make the equation balance?
A code line containing N=1/N is an assignment expression, not an equation.
This single line shows already your optimistic outlook on life! Great! :smile:
So by N=1/N you mean the new N becomes the inverse of the old, for example 3 becomes 1/3?
Yes but you covered this in the OP with the composite symbol :=
Early programming languages did not distinguish between = and := or
Equals(=) and becomes equal to(:=)
In words the code line:
110 LET N=1/N
would be 'Let the numeric variable/container called N become equal to (or contain the answer to), 1 divided by the content of variable N after the line 100 INPUT N has been executed.
BUT this is not the correct mathematical use of =
= is not an ASSIGNMENT operator is is a BALANCE operator
So the equation N=1/N in words is
Find 'anything you like' to put into the algebraic variable N so that both sides of this 'equation' are balanced.
So more modern programming languages use := to act as the assignment operator and the use = as a true/false operator, such as
Condition: IF N=1/N THEN
Action if true: PRINT N
Action if false: PRINT 'inputted value does not balance the equation.'
N=1/N was a trick question. Sorry :cool:
I now use Liberty Basic. I used Virtual basic until one morning I opened my computer and discovered that Microsoft had deleted the language. Over the years I have tried a number of languages, Pascal, Fortran, Mathematica, C++, etc. But it seems the more sophisticated they are, the more they cater to the popular applications in math. My interests are about as as far from popular as one can be.
I enjoy the challenge of programming a complicated and unusual math process from scratch. Click on my image to see an example.
I haven't had to use required numbering in some time. Neither VB nor LB require them.
I have the idea that the equals sign represents something uniquely powerful about the ability to reason, and furthermore that it is often taken for granted.
Notice that the expressions "the same as" or "is" are roughly equivalent to "=". In all such cases, whether you're saying that "this thing is the same as that thing" or "28+2=30", you're employing a judgement about identity and/or meaning. In maths there is no room for disagreement about what "=" means, as maths deals only with defined abstractions.
However in cases of practical judgement, where we say that one thing is ‘the same as’ another thing, there may be room for disagreement or interpretation. But the point is, all such judgements rely on rational abstraction. This is something everyone does automatically, as it is intrinsic to the nature of thought and speech. Not only maths, but also logic, wherever the expression ‘is’ is used, exemplifies this capacity. ‘That shade of green is very like the Irish hills’. ‘Income inequality is the cause of social conflict.’ In all cases we’re making judgements about equivalences, which we look through, which is what we bring to bear on any such judgements.
I think this is the source of the idea of ‘mathematical certainty’, which is that only insofar as something can be expressed numerically can it be described with certainty. Mathematical physics, then, achieves its astonishing degree of mathematical rigour and accuracy because the objects of physics are highly amenable to quantisation, their attributes are describable entirely in terms of numerical values in practical terms. I believe that is the source of the prestige of physicalism in contemporary culture; it’s through mathematical physics that an enormous number of astounding scientific discoveries have been made, which has made physics paradigmatic for knowledge in general. It is the ability to reduce and abstract to numerical values that is behind this ability. And that relies at every point on judgements of equivalence.
So - agree that “=“ is all-powerful although would not necessarily concur that this power is tyrannical.
Quoting jgill
I assumed by 'my image' you were referring to the icon which takes you to your profile page but when I went there, I could find no code example
Used many programming languages throughout my career as well. Mostly using Python at the moment.
All sounds good to me!
O0ps, sorry, I meant the icon is a product in the complex plane of a BASIC program I wrote . In this instance, the program created the unexpected demon from a coupled pair of differential equations: dz/dt=f(w,t) and dw/dt=g(z,t),where the functions involved contained sines and cosines.
During the time I worked I knew a number of mathematicians who would have little to do with computers. A bit surprising since numerical analysis was popular then, and some of them were actually researching in that topic. I did too, but computers were primitive and numerical analysis sought to speed up computations, even if the mathematician couldn't speak the CS language.
My initial encounter with computers was a graduate math course in numerical analysis taken in 1962. We wrote short programs, turned them in to someone behind a window where IBM cards would be punched, and finally after a day or so, run through a machine the size of a large room. Then we would find we had made a mistake, and would repeat the process over several days.
It was not a pleasant experience.
And you can do this at home nowadays? Progression can be great!
Why I tend to think people here are much younger than they are? I thought the same of universeness, who was a computer teacher. Nice that you intend to learn that still! I have read quite some stuff of you here. Sounds fresh and young, breaking with the "established" science. Anti-matterialistic (I'm a materialist myself, but I think all matter is litterally charged with unexplainable stuff). Keep it up!
Good luck, my friend. :smile:
Haha! That's it! We are miracle-charged! Made in heaven!
Yeah, I used to entertain my students with stories about the early days of computing.
Opcodes and operands, assembly codes, big valves switching on and off for 1 and 0.
punch cards (I had some metal ones to show them) and punch tape etc. Pages of daisy wheel printer paper, connected pages of thin paper, perforated on both sides, full of nothing but 1's and 0's with an error in one 16bit stream, on the 31st page of a printout of 100 pages. A 1 that should have been a 0.
They began to understand how error checking was quite hard in those days.
A friend at uni did his thesis on manipulating fractals. He produced some amazing looking flora type images based on ordering chaotic fractals using algorithms. Good stuff.
:grin: :heart:
Ha Ha. Now, that's a brilliant piction(my word for picture with caption)! aint it the truth! best post I've seen since joining this forum.
Still laughing....... :lol: :lol: :lol:
Ha! "Science piction" movies. When subtitled.
What's the difference between = and :=? Say [math]f(x)=x^2[/math] and [math]f(x):=x^2[/math]. And what's the difference with ?, "identical to"?
Yeah, but I just stole the word from the board game called 'Pictionary,' so I can't claim to be the originator.
Quoting Cornwell1
It depends, are you asking about what these mean in maths or computing or both?
The way they work is the same in both fields but their 'workings' are represented/emulated in Computing in a different way than they are in maths.
I don't know of a programming language that would have a code line, such as
f(x):=x^2
To me, with my computing hat on, this line is an incorrect use of the := operator.
With my maths hat on, it would also be incorrect to write this because use of := means that you are not writing an equation, you are writing an assignment expression. So it only makes sense if you are simply trying to say "I am assigning x^2 TO BE A FUNCTION OF X"
In computing, it would be more like:
REM subroutines
f(x);
LOCAL z;
LET z=x
LET z:=z*z
RETURN z
REM main program
INPUT x
y:= CALL f(x)
PRINT y
REM end program
In maths, as you know, the function f(x) = x^2 would graph as a parabolic curve.
A loop would be used within the above program to create the same thing on a computer screen.
So f(x) = x^2 to me in words says "a function/operation/process which can be equally represented by x^2 or the function of squaring a range of numbers, one after the other.
A function of x SUCH THAT we will square each value of x. Is to me, a balanced statement.
The := operator is only needed in computing because = is more commonly used as a true/false operator.
I have never come across the symbol ? in my maths or computing experience that I can recall.
but to me, identical means no difference at all. I don't see any difference between
x = x and x ? x
Perhaphs jgill, can offer better insight here, than I.
Quoting Cornwell1
[math]2x=3-4x[/math]
Conditional equation, true only for select values of x
[math]{{\left( x+1 \right)}^{2}}\equiv {{x}^{2}}+2x+1[/math]
Identity, true for all or nearly all values of x
The latter isn't used much anymore.
Quoting universeness
Nope. You did a great job, buddy :cool:
:grin:
I posted a lnk to a documentary about Claude Shannon a few weeks back. Shannon is widely credited with coining the term 'bit' for 'binary digit' (although he always says one of his co-workers did it first.)
Anyway, never mind, my question is, did the earlier computers, such as ENIAC, not operate on binary digits? Shannon's paper was published in 1948 and ENIAC commenced operations in 1945. My knowledge of maths and computer science is rudimentary but I'm finding it hard to imagine how computers could operate on anything other than binary digits (leaving aside quantum computers, which I know I'll never fathom).
Earlier computers represented numbers in decimal. Here's an example. So, consider adding 4 to 128. 8 + 4 modulo 10 = 2. So only the bottom two vacuum tubes would be switched on in the third column. 8 + 4 > 10, so carry 1 to the second column. 2 + 1 modulo 10 = 3. So only the bottom three vacuum tubes would be switched on in the second column.
Quoting Wayfarer
Quantum computers use qubits which are able to represent bit values (0 and 1) in superposition.
You are correct, they don't. All computers work on the basis of the binary system, including quantum computers, they just use Qbits which have three states instead of two (the third state is based on quantum entanglement, so the system is still basically binary).
You could use say trinary but parity checking and error checking would be a lot more complicated.
Consider an input cable carrying an electric signal.
A binary digit 1 is recorded if a voltage on the wire is >0 and < 5 volts.
This happens within a set time pulse.
If no voltage is detected on the line then a 0 is recorded.
This is a simplistic view as there are other elements such as start bits and stop bits etc
If you introduce a trinary system then you need to represent 0,1 and 2. A three state system.
So you would have to have a range of voltages which represent a 2, say 5v....<8V.
Things get much more complicated due to this.
any loss in voltage (due to weather, interference, degradation, surge etc) can turn a 1 into a zero or a 0 into a 1) Most of these errors can be caught by parity checking (which I won't go into for now)
But if you have a third state, then a 2 can fall to a 1 or a 0 etc, much more complicated.
So binary is best.
Oh, I forgot to type, I don't know who coined the phrase BInary digiT (bit). I've never heard of Claude Shannon, so there you go, you know stuff I don't about a subject I have an honors degree in and taught for 30 years.
I think the earliest stand alone computers worked on the same principle, for the same reason, that more than two states would have added too much complexity. I think this was even the case with the difference engine created by Babbage.
Thanks both for the illuminating answers. That top reference hits the nail right on the head.
Claude Shannon is often mentioned on this forum. He wrote a paper on the transmission of electronic information and error correction which is considered one of the fundamental papers of the information age. His work made compression possible (among many other things). But it also has deep philosophical implications, according to a lot of people anyway. And last but not least he was a fascinating character, very humorous and known for building crazy inventions, like a mechanical calculator that operated on roman numerals. You can read more about him here. Also the documentary I mentioned in the other thread is really an excellent production and provides great insight into his life and character, if you can find it online somewhere.
@Andrew M - 'The ENIAC required 30 vacuum tubes to store the decimal 128. Ten for the 1, ten for the 2 and ten for the 8. In addition, it had to turn on 11 tubes.'
No wonder it was so enormous! :yikes:
:up:
Quoting Wayfarer
Numbers can be stored in a binary representation about 3 times more efficiently than in a decimal representation (since 2^3 is approximately 10), so 999 (1111100111 in binary) would require 10 vacuum tubes. The miniaturization improvements are really to do with the hardware, e.g., billions of transistors on a chip compared to the space required for vacuum tubes.
Quoting universeness
A qubit has two measurable states (0 and 1), but a potentially infinite number of superposition states. Entanglement is a different phenomenon and requires two or more qubits.
Yes I do get that, I learned about moore's law and microprocessing quite some back, but still, the cumbersome nature of the early vacuum-tube models were due at least in part to their processing logic.
I remember reading quite some years ago that a 'musical christmas card' - the type that plays a tinny carol when you open it - contains more computing processing power than existed in the world in 1946.
Reminds me of Marvin, the android from Hitchhiker's Guide to the Galaxy. A brain the size of a planet, and they used him to open doors...
Don't you mean more inefficiently? 999 requires 10 tubes in binary, but 3 in decimal. Or do you mean 3 times 10 tubes (10 for each decimal digit)? 8 requires 3 tubes in binary (well, 1-7: 000-111, 8 is 1000), while it needs 8 tubes in decimal? Or is there one tube for 0-9, like one for 0-1? Isn't binary then 5 times as efficient? I mean, if you need 10 tubes for a decimal digit and 2 for a binary, you need 5 times as many tubes. 999 needs 3 tubes though while in binary it needs 10. You need 30 tubes for all decimals 1-999 though and only 10 for the binary version. Is that last to which you refer?
Quoting Andrew M
Isn't the number of superimposed states 2^n, where n is the number of electrons entangled?
Surely using vacuum tubes, switching on or off, to represent decimal, would be a completely different system. You would need to represent 10 states 0-9. In such a system. 999 could be stored using 3 of the 'state 9' tubes. Representing 999 in binary needs 10 tubes. How is 10 tubes more efficient than 3 tubes.
I understand that such decimal representations would not work due to complexity of decimal arithmetic.
The circuitry needed for binary arithmetic is much simpler but I don't see your efficiency argument.
[math]\sin (x)\approx x,\text{ }\sin (x)\cong x\text{ }for\text{ }x\approx 0[/math]
or the more mysterious [math]\operatorname{s}(n)\sim t(n)\text{ for large }n[/math]
???
The latter is the least well-defined symbol. The former means approximately, which of course requires clarity.
Yes - see the ENIAC example link in my initial post (which showed 30 tubes).
Quoting Cornwell1
I'm referring to the different states that a single qubit can be in. Every point on the surface of the Bloch sphere (where the sphere represents a qubit) is a potential pure state.
Quoting universeness
Yes, but there are 30 tubes in total, compared to 10 tubes for binary. See the ENIAC example link in my initial post.
There are infinite superpositions of spin up and spin down, for one electron. After a measurement the state tends to evolve to a state with both up and down equally present. There are only 2 states involved in the computing. There potentially infinite states, that's true, but you make it sound if this infinity is part of the computing power.
After measurement, the spin state will be definitely up or down, and not in superposition.
Quoting Cornwell1
The point of quantum computing is that computation can be done on superposition states. It's just the final measurement that reduces (collapses) to one of the two basis states (0 or 1).
Quoting Cornwell1
That wasn't the point of my comment but, sure, I think it's a part of it. Consider modelling nature - the more available states, the more accurately and precisely it can be modelled.
Yes, but it not stays in that state, and two states are necessary for quantum computing. There was a computing done in which about 70 (I don't remember the exact number) qubits were involved, facilitating 2exp70 possibilities in parallel.
Had to look these up again to remind me what they represented.
The most advanced maths I taught was Higher and Advanced Higher but I had not taught either for at least 20 years as I was full time Computing after my f9rst 10 years.
? : almost equal to
? : asymptotically equal to
? : approximately equal to
I have never seen these used as operators in computing but I'm sure they are used by some programming language for various purposes.
sin(x) almost equal to x, ok but to start with? does this not depend on which units are used?
sin(90 degrees) = 1 but sin(90) = 0.89399 or am I going in the wrong direction as regards your intended points.
Ah, I see what you mean now!
As the valves became tiny transistors, a decimal system would require 30 transistors to represent the range 0..999 but can such a problem not be overcome using something like floating-point representation? So 999 stored as a 9 vaccum tube and use the 3 vaccum tube to represent the repetition? You would need a method to separate this from storing 93 but....
I think the main reason for not using decimal is still the complexity of decimal arithmetic compared to binary arthmetic.
Sin(x) is a real number, not a degree; this is in radian measure. And it's true only near zero.
The three symbols you cite are used in a looser sense. I use them to mean approximately. The symbol in s(n) ~ t(n) means "behaves like", so its a bit loose also. Here's an example:
[math]\frac{n!}{{{n}^{n}}}\sim \sqrt{2\pi n}\cdot {{e}^{-n}}[/math]
This is a form of Stirling's formula.
Yes, and I see now that you're referring to qubit coherence. I found an interesting summary of the historical and projected improvements in coherence times for quantum computing systems:
Quoting Quantum coherence times, 2000-2040
A representation that was later used was binary-coded decimal. That required four transistors per digit, for a total of twelve transistors for a 3-digit decimal.
So sounds like bcd reduces inefficiency significantly from 30 vs 10 to 12 vs 10 valves/transistors.
So again, I think the strongest reason for using binary and not decimal in computing is the complexity of decimal arithmetic compared to binary arithmetic and the error handling which would be required if you used 10 states instead of 2.