I was wondering why a computer would need binary code converters to convert from BCD to Excess-3 for example. Why is this necessary can't computers just use one form of binary code.
Some older forms of binary representation persist even after a newer, "better" form comes into use. For example, legacy hardware that is still in use running legacy code that would be too costly to rewrite. Word lengths were not standardized in the early years of computing, so machines with words varying from 5 to 12 bits in length naturally will require different schemes for representing the same numbers.
In some cases, a company might persist in using a particular representation for self-compatibility (i.e., with the company's older products) reasons, or because it's an ingrained habit or "the company way." For example, the use of big-endian representation in Motorola and PowerPC chips vs. little-endian representation in Intel chips. (Though note that many PowerPC processors support both types of endian-ness, even if manufacturers typically only use one in a product.)
The previous paragraph only really touches upon byte ordering, but that can still be an issue for data interchange.
Even for BCD, there are many ways to store it (e.g., 1 BCD digit per word, or 2 BCD digits packed per byte). IBM has a clever representation called zoned decimal where they store a value in the high-order nybble which, combined with the BCD value in the low-order nybble, forms an EBCDIC character representing the value. This is pretty useful if you're married to the concept of representing characters using EBCDIC instead of ASCII (and using BCD instead of 2's complement or unsigned binary).
Tangentially related: IBM mainframes from the 1960s apparently converted BCD into an intermediate form called qui-binary before performing an arithmetic operation, then converted the result back to BCD. This is sort of a Rube Goldberg contraption, but according to the linked article, the intermediate form gives some error detection benefits.
The IBM System/360 (and probably a bunch of newer machines) supported both packed BCD and pure binary representations, though you have to watch out for IBM nomenclature — I have heard an old IBMer refer to BCD as "binary," and pure binary (unsigned, 2's complement, whatever) as "binary coded hex." This provides a lot of flexibility; some data may naturally be best represented in one format, some in the other, and the machine provides instructions to convert between forms conveniently.
In the case of floating point arithmetic, there are some values that cannot be represented exactly in binary floating point, but can be with BCD or a similar representation. For example, the number 0.1 has no exact binary floating point equivalent. This is why BCD and fixed-point arithmetic are preferred for things like representing amounts of currency, where you need to exactly represent things like $3.51 and can't allow floating point error to creep in when adding.
Intended application is important. Arbitrary precision arithmetic will require a different representation strategy compared to the fixed-width registers in your CPU (e.g., Java's BigDecimal class). Many languages support arbitrary precision (e.g., Scheme, Haskell), though the underlying implementation of arbitrary precision numbers varies. I'm honestly not sure what is preferable for arbitrary precision, a BCD-type scheme or a denser pure binary representation. In the case of Java's BigDecimal, conversion from binary floating point to BigDecimal is best done by first converting to a String — this makes such conversions potentially inefficient, so you really need to know ahead of time whether float or double is good enough, or whether you really need arbitrary precision, and when.
Another tangent: Groovy, a JVM language, quietly treats all floating point numeric literals in code as BigDecimal values, and uses BigDecimal arithmetic in preference to float or double. That's one reason Groovy is very popular with the insurance industry.
tl;dr There is no one-size-fits-all numeric data type, and as long as that remains the case (probably the heat death of the universe), you'll need to convert between representations.
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This Flash game has a lot of players including me and some friends. We noticed the same thing can run differently for different people. The math in the simulation is definitely to blame. Whether the cause is in hardware, OS, browser, 32-bit/64-bit, etc. is not really known. But with the combinations we have to test with, we've gotten 5 distinct end results from the same simulation starting conditions, and can likely get more.
This makes me wonder, does Actionscript have a floating point math specification? If so, what does it say about the accuracy and determinism of the computations?
I compare to Java, which differentiates between regular floating point math with the Math class and deterministic floating point with the StrictMath class and strictfp keyword. Both are always within 1 ulp of the exact result, this also implies the regular math and strict math always give results within 1 ulp of each other for a single operation or function call. The docs are very clear about this. I'd expect other respectable languages to have something similar, saying how accurate their floating point computations are and if they give the same results everywhere.
Update since some people have been saying the game is dishonest:
Some others have taken apart the swf and even made mods for it, they've seen the game engine and can confirm there is no randomness. Box2d is used for its physics. If a design ever does run differently on subsequent runs, it has actually changed due to some bug, usually this is a visible difference, but if not, you can check the raw data with this tool and see it is different. Different starting conditions as expected get different end results.
As for what we know so far, this is results on a test level:
For example, if I am running 32-bit Chrome on my desktop (AMD A10-5700 as CPU), I will always get that result of "946 ticks". But if I run on Firefox or Internet Explorer instead I always get the result of "794 ticks".
Actionscript doesn't really have a math specification in that sense. This is the closest you'll get:
https://help.adobe.com/en_US/FlashPlatform/reference/actionscript/3/Math.html
It says at the bottom of the top section:
The Math functions acos, asin, atan, atan2, cos, exp, log, pow, sin, and sqrt may result in slightly different values depending on the algorithms used by the CPU or operating system. Flash runtimes call on the CPU (or operating system if the CPU doesn't support floating point calculations) when performing the calculations for the listed functions, and results have shown slight variations depending upon the CPU or operating system in use.
So to answer our two questions:
What does it say about accuracy? Nothing, actually. At no point does it mention a limit to how inaccurate a result can be.
What does it say about determinism? Hardware and operating system are definitely factors, so it is platform-dependent. No confirmation for other factors.
If you want to look any deeper, you're on your own.
According to the docs, Actionscript has a catch-all Number data type in addition to int and uint types:
The Number data type uses the 64-bit double-precision format as specified by the IEEE Standard for Binary Floating-Point Arithmetic (IEEE-754). This standard dictates how floating-point numbers are stored using the 64 available bits. One bit is used to designate whether the number is positive or negative. Eleven bits are used for the exponent, which is stored as base 2. The remaining 52 bits are used to store the significand (also called mantissa), the number that is raised to the power indicated by the exponent.
By using some of its bits to store an exponent, the Number data type can store floating-point numbers significantly larger than if it used all of its bits for the significand. For example, if the Number data type used all 64 bits to store the significand, it could store a number as large as 265 – 1. By using 11 bits to store an exponent, the Number data type can raise its significand to a power of 21023.
Although this range of numbers is enormous, it comes at the cost of precision. Because the Number data type uses 52 bits to store the significand, numbers that require more than 52 bits for accurate representation, such as the fraction 1/3, are only approximations. If your application requires absolute precision with decimal numbers, use software that implements decimal floating-point arithmetic as opposed to binary floating-point arithmetic.
This could account for the varying results you're seeing.
Not sure if the title of my question makes sense, so bear with me. I'd like to find a system for representing single digit numbers with as few bits as possible. There is a method called "Densely packed decimal" (https://en.wikipedia.org/wiki/Densely_packed_decimal) which would be my ideal solution, but I wouldn't even know if that's possible or how I could implement it without further research or guidance from a guru.
The next best thing would be to be able to use a 4-bit addressing system to represent digits, but once again I'm not sure if that is even possible.
So! Barring implementations of the above methods/systems, I could settle for a 1-byte data type which I could use to represent pairs of two integers. Is there a 1-byte data-type in Fortran, or does it not allow for that level of control?
There is a 1 byte datatype in (almost) every programming language. It is the character. It is actually the definition of a byte, that it can hold a default character.
There is also a 1-byte (strictly speaking 1-octet) integer type in Fortran, accessible as integer(int8) where int8 is a constant from the iso_fortran_env module (Fortran 2008).
Both can be used to implement such things. Whether you will use division by other numbers, xoring, or Fortran bit manipulation intrinsic functions https://www.nsc.liu.se/~boein/f77to90/a5.html#section10 (probably the best option) is up to you.
Does anyone know how the binary systems is better than decimal when doing arithmetic? It was a question in a test and I can't seem to find a good answer anywhere...
In specific:
Explain the benefits of using binary rather than decimal under the following topics-
integer addition
integer multiplication
integer division
floating point storage
Any help would be greatly appreciated.
Actually, it's not necessary for human to use binary system. With the topic, I think your question is more like why computers use it?
Modern computers are based on the electronic circuits. We use a simple 2-state gate to represent information. The basic element in circuits is transistor, which shows high and low voltage, representing 1 and 0 respectively. For a complex circuit, a given input can produce some outputs. If we see that from a high level, it shows the operation of 0's and 1's.
The circuits can also be used for storage, which consists of 0's and 1's, because of the 2-state.
Computer uses the binary system both for storage and operation. If we change it to a decimal system, we have to find a component which has ten state, representing 10 numbers. Besides, we also have to design operation between these basic numbers.
Until now, electronic circuits make it perfect with binary system.
Although this question is not obviously related to a program, I think that it is quite interesting and it will help me with a program I am working on.
My question is this:
Computers are binary systems and have 3 fundamental operations available to them: AND, OR, and NOT (as I understand it), from which all of its other functions are derived. I can understand how the system can perform arithmetic on binary numbers using these operators, but how can the system then convert these numbers into decimal for the user without using the conventional operators (ie. +, -, *, /)?
You have BCD or IEEE Floating point to deal with decimal in the binary system, there are other specifications, but these are the most common and the IEEE is the one computers use nowadays i think.
I'm trying to understand the differences between these two systems and their impact on C programming.
From what I've learned from Wikipedia:
both systems are used to represent negative numbers
one's complement applies bitwise NOT to negative number (the system has +0 and -0)
two's complement does as in step 2 and adds 1 (eliminates +/-0)
Am I missing something else?
My questions:
which architectures support which system? What is the most common these days (1's or 2's complement)?
in what sense should we consider these systems when programming in C? Does it mainly make sense only in embedded world?
Thanks in advance!
Most systems nowadays use two's complement, since it lets the computer do the same exact operation for addition/subtraction without caring about the particular sign of the number.
When you're programming, the arithmetic works regardless of the system used -- the range of the data types are defined by the language, so if it says a type will work in the range -2^31 to +2^31 - 1, then it'll work regardless of the notation. You need to be careful when working with individual bits or bit shifts, though -- those won't behave like power-of-two arithmetic in non-two's complement systems (although you're not too likely to encounter such systems, and probably never will, if you're just working with PCs).
The only advantage of ones'-complement notation for integers is that it allows conversions to and from sign-magnitude form to be performed without a carry chain. Building a computer with a set of blinkenlights that show each register's value in sign-magnitude form will be much more convenient if the registers use ones'-complement form than if they use two's-complement form. If one wanted to use separate storage latches for the blinkenlights and the CPU's registers, the easiest way to accommodate things would be to have one circuit which translates two's-complement to one's-complement or sign-magnitude form, and then have each register write simultaneously store the two's-complement value in the register while updating the blinkenlight latches with the sign-magnitude value. Latching circuitry is sufficiently expensive, however, that if registers are being built out of discrete latches anyway, adding some circuitry to the ALU to make it use ones'-complement, and then feeding the lights from the CPU's "real" registers, may be cheaper than including an extra set of latches for the lights.
Over the last few decades, of course, the relative costs of different circuit elements have shifted to the point that it would be absurd to have lights wired to directly report the state of a CPU's registers. Consequently, the practical advantages that ones'-complement designs might have had in the past are no longer applicable.