I've heard a lot of people say this for a long time, but I don't think that's the case. On the Minecraft Wiki, it says that Emerald Ore attempts to spawn 11 times per chunk in groups of 1, while Diamond Ore attempts to spawn 1 time per chunk in groups of up to 8. That would mean that Emerald is more common to find, wouldn't it?

And just because Emerald is exclusive to Mountains doesn't mean it's rare either, especially since Mountain biomes arent hard to find. If you dug around in the Mountains, you'd probably run into more Emerald veins, but find more Diamonds just because they Spawn in bigger groups.

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you'd probably run into more emerald veins, but find more diamonds just because they spawn in a bigger groups

You answered your own question, Emerald veins are more common than diamonds, (assuming you're mining in the right biome) but emeralds themselves aren't (if we ignore the arguably broken trading mechanics)

I just never really saw Emeralds as rare since I'd find it so often, even if it was in a very small vein. Ignoring real life population statistics, it would be like a whole bunch of Lions scattered across an infinite savanna next to a whole bunch of even more scattered herds of gazelles. You'd run into more lions, but in the long run you'd see a higher quantity of gazelles when you did find them

They really are more rare than diamonds in game. While they are slightly more plentiful than Diamonds in the areas that you can mine them, it’s perhaps one more on average.

One extra emerald ore per chunk doesn’t make up for the number of biomes that they simply don’t spawn in while diamonds can still be mined. I’ve never had massively expansive mountain ranges but I have had massive oceans.

Emerald ore yes, AFAIK it generates at the same rate or similar to diamonds but only in select biomes whereas diamond ore generates everywhere.

Emeralds? no, you can get them from villagers easily or amass absolutely loads with a raid farm.

There are less mountain biomes than other biomes, so yes they are more rare. However you can autofarm emeralds you cant autofarm diamonds.

emeralds are a renewable resource. diamonds are not. there is a finite number of diamonds in an area and when you mine them all they are gone forever. emeralds can be acquired via trading. you can trade renewable (farmable) resources for emeralds. thus, emeralds are infinite in quantity and renewable.

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you want emeralds? i'll stop by my manual crop farm and harvest some wheat and potatoes. when i'm done with that i usually walk away with 22 BLOCKS of emeralds.