
Hi Tom  you've touched on a topic that's close to my heart. As you mention, there are some numbers that are easy to represent in decimal (like 0.1) that cannot be exactly represented in binary (the equivalent to 0.1 is an everrecurring pattern of 1s and 0s  since we don't have infinatly large fields in which to store these numbers, we have to truncate, which leads to errors).
Of course, there are some numbers that are difficult to specify in decimal (base10)  take 1/3, for example, which equares to 0.333333r (recurring). This would be easy to specify in certain other bases. Similarly, there are some values that are easy to represent in base2 that cannot easily be represented in base10.
You mention one reason for using BCD is when you are accepting input from decimalbased devices or providing output to decimalbased devices. There is another big reason  in many countries it is mandated by law that any financial calculations performed on a computer must return EXACTLY the same results as if they had been performed with pencil and paper. This is so that everyone knows exactly what will happen with regard to rounding values and so forth. The only way to achieve this is for the calulations to be performed using some sort of base10 representation, like BCD. Check out some blogs I wrote on this topic a few years back: Binary Coded Decimal (BCD) 101 and BCD coming to an FPGA near you
@Tom: You said There are plenty of ways to represent "real" numbers in a binary system.
The "real" numbers confused me for a moment because in mathematics a real number is a value that represents a quantity along a continuous line  including integers, fractions, and irrational numbers such as the square root of two. But I think that when you said "real" numbers you meant "numbers in the real world" .. is that right?
@Tom: I would guess that the three most common are probably Binary Coded Decimal (BCD), Floating Point, and Fixed Point.
I guess I would have said: Integer (unsigned and signed), Fixedpoint, Floatingpoint, and BCD. Also, integers may be considered to be a special case of fixedpoint (i.e., fixedpoint without the fractional part).
Traditionally FPGA designers worked predominantly with integer or fixedpoint (with some BCD for hardward acceleration targeted at financial applications / highperformance computing). More recently, they are being used to implement floatingpoint calculations for things like beamforming and radar applications  Altera has some real cool stuff in this regard that helps you map a floatingpoint pipeline onto their DSP slices and programmable fabric.
Oooooh, I LOVE this stuff. A lot of folks never think about this. In the case of an 8bit binary value, for example, we can consider it to represent different things, like:
UUUUUUUU An 8bit unsigned value (0 to 255 in decimal)
SIIIIIII A signmagnitude value (127 to +127; includes 0 and +0)
SIIIIIII A two's complement value (128 to +127; only one 0)
In the case of the signmagnitude format, the signbit just represents + (0) or  (1). In the case of the two's complement representation, the sign bit represents a quantity: 0 = 0, 1 = 128).
The same thing applies to BCD. If we take an 8bit field, for example, thsi can be used to represent:
UU An unsigned value from 00 to 99
SU A signmagnitude value from 9 to +9; includes 0 and +0)
SU A ten's complement value (50 to +49 in decimal)
Still on nine's and ten's complements  as you noted the way we perform a subtraction in binary is based on the fact that performing the operation (a  b) is the same as performing the operation ((a + (b)). In binary, the b is the two's complment of +b, and we can easily generate this by inverting all of the bits in b (to generate the one's complemet) and then adding 1. It's also easy to do thsi in hardware, because we already have an adder, so we just invert all of the bits in b and then we set the carryin to the adder to 1 (which equates to the "add 1").
The cool thing is that we can do much the same thing with nine's and ten's complements.
For those who want to learn more, you can check out my book How Computers Do Math. Also, you might find the following two columns to be of interest:
A minus B=A+NOT(B)+1(Part 1)
A Minus B=A+NOT(B)+1 (Part 2)
Re: What's real?
tomii
12/25/2013 12:14:02 PM
No, I actually meant the wealth of numbers that are on the real axiz of the complex plane (ow, that makes my head hurt). A decimal is a fraction (maybe), if you can just figure out what it is. Now, I'm not trying to say any of those ways you can represent the real numbers is correct, because we're almost always going to lose something somewhere.
Like I keep saying, evaluate your alternatives, and choose the method(s) that best match your needs/requirements. My way most assuredly is not the best way, and your mileage (like my own) may vary.
@Tomii: One interesting point I forgot to mention is that you can use BCD representations to represent anything you want  so you could use BCD to represent integer, fixedpoint, or floatingpoint values  now my head hurts :)
Re: Why use BCD?
AZskibum
12/26/2013 3:18:44 PM
Nice presentation of BCD arithmetic, even if it is not often used anymore. I'm looking forward to your installments on fixed vs. floating point.
Multiply is a series of additions? Divide?
KarlS01
12/26/2013 4:45:10 PM
Is multiply by 1000 done with 1000 additions and divide by 1000 done with 1000 subtractions?
Also, RISC does not have decimal/bcd instructions so computers have another way of doing decimal.
Here is a little thought going back to scientific notation and a little C# evaluation run on a PentiumD.
// this was done with 8.5, 5.25 and 8.5, 5.33 inputs double scale = 8.5 * 5.33; // 44.625 45.305 double scale1 = 8.5 * 5.33 * 1000; // 44625.0 45305.0 int scale2 = 85 * 533; // 44625 45305 int scale3 = scale2 / 1000; // 44 45 int scale4 = scale2 % 1000; // 625 305
It is a matter of formatting yhe output to include the decimal point between the divide and modulo result when done on a PC. FPGA needs hardware to handle input and output formatting.




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