I am just wondering what does it mean when somone says something like"3 and 1/2 digit" in case of accuracy of test equipments (or maybe A/D converters). Can somone explain this a bit with some numbers as example?
3 digits would be 0 through 999
Has nothing to do with binary digits, but decimal digits, or rather their representation in 7-segments displays. To display every digit you need all 7 segments, but if for the fourth digit you only have to display a "1" you only need the two rightmost segments, so that can be interpreted as the right half. That was when most DMMs had a maximum reading of 1999. Recently more accurate DMMs became available, having readings up to 3999. If "1" as the highest value for the highest order digit is half a digit, with some imagination you could say that a highest value of "3" is 3/4 of a digit.
Note that for displaying only "1", "2" and "3", you don't need the upper left segment, which a 3 3/4 digit DMM indeed doesn't have for the leftmost digit. It's a small cost saving, but a saving nevertheless.
David L. Jones did a video about Multimeter Counts, Accuracy, Resolution & Calibration.
There he also explains what these half digits are.
To summarize his explaination what 3 1/2 digits mean (in the video 0:30 - 1:30):
A 3 1/2 digits meter can display 1999.
A 4 1/2 digit meter can display 19999 and so on.
The half means that the most significant digit can only go up to 1.
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My best guess with this is that it is in reference to LCD or LED displays.
Some test equipment may well have a "3½ Digit" display. That is, a display with 3 whole digits, and only half of the fourth digit (i.e., a "1").
So the full range of a 3½ digit display would be:
All segments on would give you:
Take this one as an example:
That one's from a 12-hour clock, so there is never any need for the first digit to ever go above 1.
This is a useful marketing term used to explain the nature of a digital display.
It means that the most significant digit can be either 0 or 1.
By adding a relatively low cost display to the system the manufacturer doubles the displayed range. This results in eg multimeters with 2, 20, 200 Volt or mA ranges rather than 1, 10, 100, 1000 ranges. Note that on a 3.5 digit display multimeter the max range on AC volts may be eg 600 Volts rather than the possible 1999 Volts. This is a safety and implementation limitation.
The 3 or 3.5 digit display does not affect the accuracy - but it does affect the displayed apparent resolution. Note that most multimeters have absolute accuracies typically around 1% to 2% on Volt and mA ranges and worse on ohms and Amp ranges. This despite the fact that a 3 digit display has a 0.1% resolution and a 3.5 digit display has a 0.05% resolution. In such cases adding the extra resolution can be useful even though the accuracy is already more than outstripped by the display resolution.
Rarely you may see 3 + 3/4 digit meters - these have eg 0000 to 2999 resolution. This can be extremely nice to have. It gives eg 4, 40, 400, ... ranges. My experience with these is that it often eliminates range changing in typical use when maximum resolution is required with a widely varying signal. These are very seldom seen.
As noted, the term "3 1/2 digit" was coined some time ago to refer to displays that could show three digits 0-9, and a leading digit which could be blank or 1. When some later displays came along with a leading digit that could display 0-2 or 0-3, the terms "3 2/3 digit" and "3 3/4" digit were coined. Note that were it not for the earlier usage of "3 1/2" digit, it would perhaps be more accurate in terms of magnitude to say "3 1/3" digit for leading 0-1, "3 1/2" digit for leading 0-2, and "3 2/3 digit" for 0-3, since log10(2000) is 3.3, log10(3000) is 3.5, and log10(4000) is 3.6, but the terms are what they are.
BTW, a 3 2/3 digit display needs three controllable segments for the left digit (the upper-right segment, the lower-right, and everything else that makes up a "2"); a 3 3/4 digit display needs four controllable segments (upper-right, lower-right, lower-left, and all three verticals). Counting up to 4 would require five segments (split out the middle one), 5 would require six (add the upper left), and seven would require all seven (split the top from the bottom).
All the other answers here are talking about decimal digits on displays. For A/D converters the meaning of accuracy is totally different and is usually given as a fraction of an LSB (least-significant bit), which means that the value of the conversion is accurate to within that numerical amount. This is also captured in the ENOB (effective number of bits) which is also a fractional number -- for instance, an "8-bit" A/D converter will probably only have an ENOB of around 7 bits.
The reason the number can be fractional is due to several things. If it was only due to quantization, and all else was perfect, all conversions would be accurate to 0.5 bits. The reason it's not exactly that is due to other effects such as conversion non-linearity and distortion.
Reading up on ADC terms more may help.