Perhaps one of the most difficult concepts for photographers and other types of digital artists to wrap their minds around is the concept of "Resolution". No, I'm not talking about that steadfast determination to throw your money away on a gym membership come January 1st. I'm referring to the fundamental principle that must be abided by when viewing and printing your photos at a photo lab or on a consumer grade printer. Hopefully I can answer the question, "What resolution does my image need to be to print correctly?".
Let's start at the very beginning with the pixel. The pixel (short for Picture Element) is the digital building block of your image. A pixel is basically a square containing light and color information. There are 256 possible colors that a pixel can represent from 0; solid black, to 255; solid white. Your image is made up of thousands and even millions of these. In fact; the term "Megapixel" literally means 1 million pixels. The more of these babies your image contains, the sharper it will appear. Too few will result in your image looking "Pixellated", when you can actually see the individual pixels. Pretty basic stuff so far.
Now lets look at a term you've all seen many times but might not have been aware as to what it actually refers to. The term in question is, "PPI". It's a term thats often used interchangeably with "DPI" at the expense of accuracy. It's OK. We all do it. DPI just rolls off the tongue better. But the fact is, PPI refers to Pixels Per Inch whereas DPI refers to Dots Per Inch. Dot. Pixel. It's all the same right? Well, no. A pixel is, as I said earlier, a digital construct. A dot is what is used to reproduce a digital image (made up of pixels) as ink on paper. It may take a few dots to reproduce one pixel. That is why consumer printers like inkjet printers will have such a high amount of DPI available to them. It should be noted though, that when printing to an inkjet you do not need to size your image at an outrageous PPI. This type of printer just needs the extra DPI to accurately reproduce crisp images on paper because as I said, it sometimes takes multiple dots to reproduce one pixel. But I am getting ahead of myself.
Lets look closer at PPI. Pixels Per Inch refers to exactly that, an amount of pixels in a given inch. Most sources agree that the human eye can see detail somewhere in the 200 DPI range (yes, thats dots per inch). This has been the case when viewing printed photographs for many, many years. With a digital image, 240 PPI is generally high enough to reproduce a quality print at say, 4x6. You will also see 300 PPI used quite frequently. Going back to 240 PPI, this actually means that there are 240 pixels displayed for every inch. A 4x6 at 240 PPI is 960 pixels by 1440 pixels. Simply multiply 4 by 240, and 6 by 240 to get the total image size in pixels.
Now look at the example below. Three details, each one inch by one inch, from a 12x18 are represented at various resolutions. The first example, a, is at 10 PPI. At this resolution you can really see the individual pixels that make up the image. If you count from left to right and top to bottom inside the red stroke you'll see that there are 10 pixels in each inch-long row. The total image size in pixels of a 12x18 at 10 PPI is 120 px by 180 px. Again, multiply each dimension by the resolution, in this case, 10. Incidentally, the red stroke around each example is one pixel wide and the bright green square represents one pixel. As you can see, by increasing the resolution the pixels get smaller in order to fill up that inch. At 50 PPI (example b) more detail is clearly visible but you can still see the individual pixels. If you are able to, you can count 50 pixels across and down. In example c it becomes nearly impossible to see each pixel at 240 PPI. Example c would make a good print.
For more on this topic see part 2. 
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