A few scanning tips
The video monitor and the scanner have no problem making any RGB color at any pixel, but printers cannot do that. The printer is a very different kind of device, and has very sophisticated ways to kludge a rather crude image.
First, B&W printers (Color printers next page)
B&W printers do NOT print shades of gray. They use black ink or black toner, and they can print only Black. To simulate gray in graphics, they print halftones. With a magnifying glass, you can see halftones in the images in any book, magazine or newspaper. Halftones are arrays of dots arranged in a grid, say 6x6 or 8x8 to represent each image pixel as a shade of Gray. For dark gray, more grid dots are black. For light gray, more grid dots are white. (More modern methods used for color in magazines vary the size of the dots instead of the ratio of light/dark dots.) The printing graphics software and driver can specify different halftone grid sizes for different effects. For example, a good laser printer might print 600 dpi, or it might print 128 shades of gray, but it cannot do both at the same time. If a larger grid is used, more shades of gray are possible, but less resolution is possible.
1x1 shows 2 shades (black or white, 600 dpi Line art) **
** 1x1 is not halftones, it is simply called "line art" mode.
6x6 shows 37 shades of gray, reducing image resolution to 600/6 = 100 lpi.
7x7 shows 50 shades of gray, reducing image resolution to 600/7 = 85 lpi.
8x8 shows 65 shades of gray, reducing image resolution to 600/8 = 75 lpi.
10x10 shows 101 shades of gray, reducing image resolution to 600/10 = 60 lpi.
This material is not about prepress, but to quickly mention lpi, the printing industry's term for resolution is lpi (Lines Per Inch), a measure of printed image resolution-like detail. Magazines typically use 133 or 150 lpi images, newspapers are often 85 lpi, and highest quality art books 200 lpi. A "line" is one row of grid cells.
For prepress, or commercial ink press printing, color or grayscale, the rule of thumb for scanning resolution is
Scaled dpi = (lpi x 1.5) x (printed image width/original photo width)
That is the obligatory formula for prepress, we see it everywhere. Lpi is lines per inch, used to create screens for commercial ink press printing. 1.5 is a minimum, and often we see it as (lpi x 2.0) as the upper limit for commercial requirements. 2.0 is the upper limit of usefulness, not a goal. However, editors do often ask for 300 dpi (the theory being that it is better to have too many pixels than too few). If scanning for an imagesetter to create a screen for magazine publication, the lpi formula above is very valid.
Magazines 133/150 lpi - scale to 225 to 300 dpi.
Newspapers 85/100 lpi - scale to 150 to 200 dpi
But most of us are probably not doing that. Normally we don't know any lpi specification for our home and office printing jobs, and lpi is not applicable to them, because our inkjet printers work with a different error diffusion dithering technology anyway. However, reasonable dpi guidelines for scanning for printing will be offered later, so we do know a ballpark number for dpi for our printed images.
This formula's ratio of (printed/original) size also applies to our home scanning resolution requirements. It is needed because if we scan a 300 dpi image, but later expand the image on the printed paper to twice the original size (twice the inches), we would then have only 150 dpi in the larger image on paper. When the printer expands the image, the pixels are farther apart than before. So to double the printed size, and yet maintain 300 dpi for the printer, we should scan at 600 dpi to allow for the size increase, so we still have 300 dpi in the expanded result. That's why the formula above has the multiplier of (printed size / original size).
The standard rule for prepress is that we must scan for the capability of our specific output device, using scanning resolution dpi = lpi x 1.5. The extra 50% is to accommodate the printer driver's resampling when it rotates the halftone screen (hypotenuse at 45 degrees is 1.414 length). A 2.0 factor may give slightly better quality, especially for grayscale images, but most say 1.5 is plenty. 2.0 is an upper limit, a maximum, not a requirement or goal. The right range is 1.5 to 2.0. Lasers and ink presses do use halftones and lpi, but inkjets use a different dithering method.
However, one big problem with lpi is that we cannot find lpi mentioned in our printer's specifications, inkjet or laser, because lpi is not within the hardware. Instead lpi varies however the graphic software and printer driver choose to use the hardware. It is a software issue, to create the halftones above.
There is much uniformity in commercial practice, in that those images are typically sent to 2400 dpi imagesetters to generate screens for publication, and these are very single purpose with known standard lpi requirements. Many magazines use 133 or 150 lpi, for which scanning at 133 lpi x 1.5 = 200 dpi is fine for many cases, but many editors will habitually ask for 300 dpi. 150 to 200 dpi images are enough for printing in newspapers at 85 or 100 lpi.
But at home, we are at the mercy of many different software packages, and we all have a different printer and driver too. This makes it pretty difficult to use the (lpi x 1.5) formula, simply because we do not know it. You can however sometimes see this lpi value in your printer driver or image program options. For example, PhotoImpact indicates 85 lpi for my HP 600 dpi laser printer. That doesn't mean the printer is 85 lpi, it only means that PhotoImpact intends to generate 85 lpi images for it, which is the right ballpark for 600 dpi. But therefore, 85 x 1.5 = 128 dpi would be a good safe resolution to use, and I typically shoot for 150 dpi grayscale images for it because 150 dpi scans well.
That 85 lpi suggests 600 dpi / 85 lpi = 7 cells per inch, therefore a 7x7 halftone grid, with the resulting possibility of 50 shades of gray, which is bearable on a laser printer. The printer specs say 600 dpi and 128 shades of gray, but it cannot do both numbers at any one setting. Typically a middle of the road compromise is used. Using higher lpi for more resolution detail limits the smaller cells to fewer tones of gray, which is detail too. Perceived resolution depends on both factors.
The overall significance of this halftone grid is that the printer must use several of its dots to simulate tones of gray to represent each one pixel in the image. This greatly reduces the printer's real image resolution capability to a fraction of the printer's advertised dpi. Printer ink dots and image pixels are simply very different things. One gray image pixel requires many printer ink dots.