True, but only one of them out of an Asteroid Field cluster with several Asteroids needs to be inside your ZOC in order to assign the Asteroid Miner to it, and then the entire Field will be in your ZOC.
Do not bother building Asteroid Miners for single Asteroid fields early in the game, concentrate on Fields with multiple Asteroids. Try to colonize worlds that have multiple Asteroids near by.

Asteroids do not get consumed.  If you mine an asteroid in a cluster (adjacent to other asteroids) the entire cluster is mined (so it is better to go after big clusters first).  Only one asteroid in a cluster need be in your ZoC for the entire cluster to be mined (but you must target the single asteroid in your zone).

If you zoom out to icon levels, mined asteroids appear in your civ color.

Each hex in the cluster provides the mining bonus. So one hex of asteroids gives a base bonus of .1 minerals; if there are 3 hexes of asteroids in a cluster then you are getting .3 minerals.

Happy Monday !

Yes, a cluster of asteroids (hexes with asteroids adjacent to each other) are all mined with a single miner.  So a cluster of 7 gives a base of 0.7, while a single asteroid gives 0.1.  

It is important to remember that this is reduced by (1) the distance of the asteroids to their closest planet, and (2) the decay from the planet with the asteroid bonus to its Core planet (unless, of course, the asteroids feed a Core).  So closer asteroids are better than far asteroids, as well as the cluster size.

As long as the Asteroids touch each, One Asteroid Miner will start mining for all of them. Only one Asteroid in the Field needs to be inside your Zone of Influence/Control, to start mining all of them.

Gaps, planets, Durantium resource locations, etc. can break an Asteroid Field into multiple sections, each of which require a separate Miner.

When an Asteroid Miner costs 40 Shipping production to build, and you only get 0.1 mining yield per turn per Asteroid mined, even with 3 Asteroids mined because of one Asteroid Miner, it takes a long time to pay off the investment, e.g. even 3 Asteroids being mined will take 133  (ONE HUNDRED THRITY THREE) Turns turns to pay off the investment, in Ship Production (with no efficiency gains taken into account).

Four Hundred turns to pay off 1 Asteroid being mined (with no gains taken into account), for just Ship Production.

However, take these estimates with a big grain of Salt, first Core worlds also get a benefit to their Social (i.e. surface) Production from the Asteroid, Second Social Production can get a lot of % gain bonuses, 3rd Ship Production can also get Military bonuses. So Asteroid Mining gives increases to both Social and Ship Production, not just the Ship Production used to build them.

So Asteroid Fields with several, preferably 3 or more Asteroids, are worth mining even with an Asteroid Miner that costs 40 (with enough efficiency gains). But Single isolated Asteroids should be ignored for a long time. With only 2 Asteroids in a Field, I would give other build projects a higher priority for a long time.

Just be aware of what you are paying out (40 per Asteroid Miner) and what your return is (0.1 per Asteroid per turn) and take into account efficiency gains, and how long it takes, when deciding on building Asteroid Miners, and When to build them.

Asteroids being Mined do have one additional benefit, they help extend your Zone of Influence (or Control) (each one generates Influence), which in turn discourage AI colony and construction ships, as well threaten to flip control of Colonies they build  nearby, once your Zone of Influence grows to include them.

There are ways to increase yields from Asteroids, but they are expensive and should only be done after you have a LOT of existing Asteroids being mined.

There is an Ideology that helps by 0.1 per Asteroid, but Culture Points are hard to come by, and you have to reach a high enough level in the right area in Ideology. You need a 7 Creativity to take the Resourceful choice in the Creativity branch of Ideology. So that is not something that you will be doing soon.

There are also 2 Techs, Collector Drones and Asteroid Factories that increase mining yields by 0.1 each, but they are even more expensive. First you have to research Drone Production Tech then you have to research those Techs, which require that you spend one Drone to unlock Collector Drones Tech and an additional 2 Drones to unlock Asteroid Factories Tech so you can research them. These Drones cost 250 in Ship Production (EACH) to build.

So take into account all of the costs and returns when deciding on building Asteroid Miners and where to send them. As well as when to try to increase those yields.