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In American English "colour" is written "color" and "grey" is written "gray".
The 7 colours of the rainbow
The seven colours of the rainbow are the "visible" part of the electro-magnetic spectrum - they are visible to (or can be seen by) the human eye. A rainbow is what we sometimes see in the sky when the sun is shining and it is raining somewhere at the same time.
This mnemonic can help you remember the colours of the rainbow and their order:
The prefix "infra-" means "below" or "under". In the electro-magnetic spectrum, infra-red is "below" the colour red (and not visible). The prefix "ultra-" means "beyond". Ultra-violet is "beyond" the colour violet (and not visible).
Black and white
Technically, black is not a colour (it is the absence of all colours), and white is not a colour (it is a combination of all colours). Grey also is not a colour since it is a mixture of black and white. But in normal language we usually describe black, white and grey as colours.
Three more very common colours
*Silver and gold are shiny "metallic" colours.