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Several early computer designs regarded a 'word' as representing not an integer, with the bits having values 2^0, 2^1, 2^2, ..., but as representing a fixed-point fraction 2^-1, 2^-2, 2^-3, ...

(For the sake of simplicity in this question I'm ignoring the existence of the sign bit and talk only in terms of positive numbers)

Some examples of this convention are EDVAC, EDSAC, and the IAS machine.

Why was this? To me, having dealt with since the 1970s with machines that have "integers" at base, this seems a strange way to look at it.

Does it affect the machine operation in any way? Addition and subtraction are the same regardless of what you think the bits mean, but I suppose that for multiplication of two N-bit words giving an N-bit result, the choice of which N bits to keep depends on your interpretation. (Integer: you want the "right hand word"; fixed-point fraction, you want the "left hand word").

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    Very early on, it was likely that computers were not considered to be general purpose machines. So if the main task for which a computer was designed involved doing calculations with flractional numbers, prioritizing them over integers would make sense. It seems likely that computers designed for business programs would be more tuned to integers, because money (in the USA) can be treated as pennies, and very little would need to be fractional.
    – RichF
    Apr 1, 2019 at 1:20
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    Not only can be, but must be, to avoid rounding errors that could lose (or create) money. (This also applies to mils or any other smaller fraction of a dollar that might be necessary.)
    – chepner
    Apr 1, 2019 at 20:21
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    Also, remember that one of the primary first functions of computers was to calculate ballistic trajectories, especially for military applications.
    – Ron Maupin
    Apr 2, 2019 at 2:31
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    Note that this is not universal; some early computers used integers, others (Zuse's Z3) used floating point numbers.
    – fuz
    Apr 2, 2019 at 11:11
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    Awesome question. Not something I even realized. However it appears that I am a few years younger than you (b 1969).
    – Andrew Steitz
    Apr 2, 2019 at 20:09

8 Answers 8

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I'd think that it was mostly down to the preferences of John von Neumann at the time. He was a strong advocate of fixed point representations, and early computers were designed with long words to accommodate a large range of numbers that way. You certainly don't need 30-40 bits to cover the most useful integers, but that many were needed if you wanted plenty of digits before and after the decimal point.

By the 1970s though, the costs of integration were such that much smaller word sizes made sense. Minicomputers were commonly 16 bit architectures, and micros 8 bits or sometimes even 4. At that point you needed all the integers you can get, plus floating point had largely replaced fixed point for when you needed decimals.

Nowadays we'd think nothing of using 64 bit integers, of course, but it's a heck of a lot easier to integrate the number of logic gates required for that than it would have been back when they all had to be made out of fragile and expensive vacuum tubes.

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    I'm persuaded by the "preferences of von Neumann" part, since the 3 machines I mentioned had common conceptual roots, but less so by the rest. I agree with the word-size rationale. But given the machine is "fixed point" only, the choice seemed to be between integers, and fractions of magnitude between 0 and 1. Neither seems to me to be better suited to "plenty of digits before and after the decimal point". Either way, the position of the point is purely notional and the programmer needs to keep track of it.
    – another-dave
    Apr 1, 2019 at 12:13
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    Fixed-point doesn't have to be fractions between 0 and 1; it just means the value is an integer scaled by some fixed constant.
    – chepner
    Apr 1, 2019 at 20:25
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    I could perhaps have been clearer on that. Von Neumann points out in First Draft of a Report on the EDVAC that you can do all the calculations with numbers between 0 and 1 and scale the result accordingly, even. You still need those very long words to avoid a loss of precision with the intermediate results though.
    – Matthew Barber
    Apr 1, 2019 at 21:33
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    @chepner I'm aware of 'manual' scaling. But if you read, for example, the EDSAC description in Wilkes, Wheeler, and Gill, then they regard the store itself as holding numbers in the range -1 to +1 (binary point at left), rather than -2^35 to +2^35 (binary point at right) or anything else. That's the point of my question.
    – another-dave
    Apr 1, 2019 at 22:34
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    The First Draft is the clincher for this argument, I think. Thanks for the reference.
    – another-dave
    Apr 1, 2019 at 22:53
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This is not really a hardware issue at all, but just a different way of interpreting the bit patterns. A "fixed decimal point" representation for numbers is still used in some situations, where the full power and flexibility of floating point is unnecessary.

The IBM S/360 and S/370 hardware had decimal arithmetic as well as binary, and IBM's PL/I programming language had both "fixed decimal" and "fixed binary" data types, with an implied decimal point anywhere in the number, though fixed binary was mainly used for the special case of "integers". Fixed decimal was an obvious choice for handling financial calculations involving decimal currency such as dollars and cents, for example, because of the simple conversion to and from human-readable numbers.

Fixed point binary is still used in numerical applications like signal processing, where lightweight hardware (integer-only) and speed are critical and the generality of floating point is unnecessary.

In terms of the computer's instruction set, all that is needed is integer add, subtract, multiply, divide, and shift instructions. Keeping track of the position of the implied decimal point could be done by the compiler (as in PL/1) or left to the programmer to do manually. Used carefully, doing it manually could both minimize the number of shift operations and maximize the numerical precision of the calculations, compared with compiler-generated code.

There is a lot of similarity between this type of numerical processing and "multi-word integers" used for very high precision (up to billions of significant figures) in modern computing.

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    Not just because of the simple conversion to and from human-readable numbers. Fixed point decimal, like BCD on later systems, has the big advantage of avoiding things like 0.30 turning into 0.29999999999999999999 etc.
    – manassehkatz-Moving 2 Codidact
    Apr 1, 2019 at 14:03
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    Fixed-point binary is in widespread use in FPGA work as well. It used to be in heavy use in industry for embedded software generally, before 32-bit microcontrollers with floating-point became available at a reasonable price. I heaved a sigh of relief around 2001 when I could leave that all behind. Then about 4 years back I started working on FPGAs, and I was right back there again!
    – Graham
    Apr 1, 2019 at 16:21
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    Speaking as someone who has actually written a fixed point library in C++ I concur with all points. Integers and fixed points aren't all that different from each other in terms of binary representation. In fact, it could be argued that an integer is just a fixed point type with a fractional size of 0.
    – Pharap
    Apr 2, 2019 at 14:14
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Some early computers did use integers.

Manchester University's ‘Baby’ computer calculated the highest factor of an integer on 21 June 1948. https://www.open.edu/openlearn/science-maths-technology/introduction-software-development/content-section-3.2

EDSAC 2 which entered service in 1958, could do integer addition. https://www.tcm.phy.cam.ac.uk/~mjr/courses/Hardware18/hardware.pdf

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    Well, as I understand it, EDSAC could do integer addition too. Where "the book" says you're adding 2^-35 + 2^-35 to get 2^-34, the programmer is perfectly at liberty to believe he's computing 1 + 1 = 2. It's just in the interpretation of the bits. I would describe EDSAC 2 as replacing fixed point arithmetic for both reals and integers, with floating point arithmetic for the reals, leaving fixed point for the integers.
    – another-dave
    Apr 1, 2019 at 22:43
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Computers came from calculators, and calculators are designed to solve numerical computations, and hence require the decimal point.

Babbage's difference engine ~1754 had 10 of 10-digit decimal numbers.  Its job was computing numerical tables — and printing them, since the copying of the day (by humans) made more mistakes than the mathematicians who made the original calculations.

EDVAC was part of the US Army's Ordnance Department, its job was ballistics computation.

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    But did Babbage say his numbers were between 0 and 10 thousand million (being British, he would not have used "billion" for 10^9), or between 0 and 1 (approximately speaking)?
    – another-dave
    Apr 1, 2019 at 23:01
  • "Babbage's difference engine ~1754 had 10 of 10-digit decimal numbers." Did you mean 1854? Also do you have a source for the number of registers and their size? I've not come across any specifics on the design (Difference Engine #2?).
    – Alex Hajnal
    Mar 5, 2021 at 10:45
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The problems early computers were meant to solve used real numbers. Often these numbers were very large or very small, so you need a scale factor for the computer to handle them.

If the computer natively thinks in numbers from 0.0 to 1.0 (or -1.0 to 1.0), then the scale factor is simply the maximum value of the variable. This is easy for human minds to handle and not very error prone.

If the computer natively thinks in numbers from 0 to 16777215 (or whatever), the scale factor becomes something completely different. Getting the scale factor right becomes much harder and a source of errors.

Lets go with less errors, shall we?

As others have pointed out, the actual hardware is the same for the two schemes, it is just a matter of how humans interact with the machine. 0.0 to 1.0 is more human friendly.

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At the time using fiction point fraction representation seemed like the best solution for handling floating point numbers, because it avoids many of the issues that are present with other formats such as not being able to precisely represent 0.5 and cumulative errors when performing relatively common operations repeatedly.

In time better ways to represent floating point numbers with binary were devised. General purpose hardware was able to process such formats and gave the programmer a free choice to select the one they wanted.

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I am not sure i understand your question right, but if i did then "word" is commonly used not as data-type (like "integer") but as whole hardware register, so 8 bit CPU has 8-bit machine word.

In fixed-point operations, common practice is to store "before point" value in one register, and "after point" value in another register. So, from CPU's point of view they both are words.

What is the main difference between word and integer? word is only a set of bites, it does not represent any digital value, because one and the same word can be represented as different digital values - for example 11111111 word can be -128 Integer or 256 "Unsigned", depending on how you interpretent first bit. Two words can be Double (Integer before point and Integer or Unsigned after point), Long (for 8-bit cpu 16-bit int) and so on, depending on how you use their values in your calculations.

I suppose that "word", in your case, was used to describe the way to access a part of data in memory, meaning that it can not or must not be a whole number, whose other part can be located in different word.

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I suspect it might have something to do with multiplication. For addition and subtraction, of course, the exact position of the radix point is purely a matter of convention. However, multiplication takes two single words and produces a double-word product. If you interpret the two given words as integers, then you have a problem dealing with the result, since it won't fit in a single word. However, if you interpret the two given words as fractions (with radix point up front), the product is also a fraction, and you can just drop the low-order bits and store the high word by itself, and still have a useful result. This is particularly useful when implementing floating-point in software.

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