Dyes and color additives are regulated by organizations such as the FDA. In order to distinguish between the many different color additives, it was important to add numbers to distinguish them from each other. There isn’t only one kind of red dye, so numbers and descriptive names are used to distinguish between them.
You can find the list of approved uses of dyes [here](https://www.fda.gov/industry/color-additive-inventories/color-additive-status-list) for the FDA.
There are hundreds of food additives, including dozens and dozens of color additives. They can’t just say “yellow”, because there are many yellow compounds that can be used in food. So they are given numbers by the FDA so they can be distinguished.
See a list of FD&C color additives here, many of which are now restricted:
https://www.fda.gov/industry/color-additive-inventories/color-additive-status-list#list1
It’s for identification. The FDA has a list of certified colorings that are allowed to be used in food. Countries other than the US use different numbering systems.
FD&C Blue No.1 has had many names. It’s full chemical name is disodium;2-[[4-[ethyl-[(3-sulfonatophenyl)methyl]amino]phenyl]-[4-[ethyl-[(3-sulfonatophenyl)methyl]azaniumylidene]cyclohexa-2,5-dien-1-ylidene]methyl]benzenesulfonate. So you can see why it might need to be identified by a number.
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