Why does blue and red make purple in paint but not with light?

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Actually blue and red don't make purple.

You can test this out yourself by mixing any blue and red paint. The result will--at best--be a muddy purple, certainly nothing like the purple we all know and love. The color wheels and color mixing methods most of us learned in school are incorrect. Red, blue, and yellow are not primary colors. The true primary colors are Magenta, Cyan, and yellow. That's why your printer takes those colors. If the primary colors were red, blue and yellow, those would be the colors your printer uses. Magenta, Cyan, and Yellow are called the subtractive primaries, because you start with white, and subtract portions of the light spectrum, until you reach black. So for instance, you start with a white piece of paper (the full spectrum is being reflected), and add magenta paint. You have now subtracted the green portion of the spectrum, which is absorbed by the magenta paint, and reflected the remainder of the spectrum, which we perceive as the color magenta.

The secondary colors in subtractive color mixing are red, blue, and green.

Magenta + cyan = blue
Cyan + yellow = green
magenta + yellow = red
all three = black

Now before I explain why light is different, you should know that there is no such thing as perfect primaries in paint, ink, dye, and other subtractive pigment based colors. What works in theory, does not work as well in practice. The biggest problem is that there are no primary pigments that can actually absorb all the spectrum, so you can never get dark black and good shades of colors. So that's why printers also have a black ink cartridge. Also, it's hard to find a good magenta, and cyan is not labeled as such in most artist colors.

To understand light, it's important to know that your eyes only have three color receptors, called cones. We have red, blue, and green cones. So it can be said that the entire color spectrum can be broken up into three even segments of red, blue, and green light. All colors we see are made by stimulating the three cones in our retinas to varying degrees.

Now light color mixing is the additive process. You start from dark (no light) and gradually add portions of the spectrum. The additive primary colors are Red, blue, and green. These are the colors you have in your computer monitor, and your TV. And they are also used in theatrical stage lighting. So in this system, you start with a dark room (or a computer screen), and then you turn on a red light. When you do this, you are seeing 1/3 of the light spectrum. Turn on a green light, and you are seeing 2/3 of the spectrum, Turn on a blue light, and you have white light. This is why it's called the additive process.

In the additive system:
Red + blue = magenta
Blue + green = cyan
green + red = yellow.
all three = white light.

By the way, just like your printer has black ink to help with creating deep black and shades of colors, your eyes have rods, which only "see" dark and light. The rods are like the black cartridge in your computer.
It's a complicated subject.
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