So... What exactly does population do?

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It's been awhile since I have played a Civ game and I am sitting here playing through the CivV tutorial and I need a question answered: What is the benefit of having a large happy city population? I'm not getting this one part.
 
You can have more people to work tiles, and I think you may can just set some to work in the city.Not certain of that, though(I don't use micromanagement much).But I think having a higher population increases production, and possibly wealth too.
 
1 population equals 1 beaker of science, then altered by effects from library, university, etc. So if you wanna move through the tech tree, focus on food tiles to get your population up.
 
More population means more tiles worked (so more food, hammers and gold), but also +1 scince per pop (1.5 with a library, 2 with a public school). When you get big university/observatory/National college/research lab modifiers, it makes the bulk of where all your science comes from.

Tall civs can also afford to run more specialists (and are more likely to have the buildings to house them thanks to increased hammers). With the freedom tree this is also true in a proportional sense, comparing to a load of small cities with the order tree. Specialists boost your science or culture which are awesome, and can also add a further boost to hammers or gold if needed. Importantly, they're also your main source to generating great people, who are awesome.

A large city is also more likely to have the hammers to build world wonders, and a tall civ has a much easier time of building national wonders. Because of the game mechanics, a tall civ is also much better at getting social policies and doing a cultural victory.

Remember, though that each population costs 1 hapiness, and so civs with large pop cities can only have a few cities, and so not work as much land as wide civs (i.e. small cities, but lots of them).
 
In the third tutorial it told me to make 4 improvements to my resources as one of my goals. Wasn't sure how to do that

Build a worker (different from your city population, you need to actually build it in your city as production) and move it on top of the resource. then on the bottom left it will reccomend improving the resource. The worker will get to work, and 5-11 turns later you'll get your resource (the worker will stop working, and the tile will look different). You might need to get the relevant technology before you can do this.
 
Stability? Does that mean there's something similar to Rhye's and Fall for CiV? I couldn't find anything like that on steam, but I really liked this mod.

Nope, I think stability was a reference to the defense of the city going up with the population.

Besides that, each citizen means another tile you can work, which means more hammers, more gold, and more apples, although each citizen eats 2 of those apples per turn.

More population also means more science generation, no matter which tiles or specialties the citizens are assigned to.
 
And city local hapiness is is capped at city population.
So to get maximal use of local hapiness you need population equal to it. Unluckily, I don't know whether there is some way how to see local hapiness available to city.
Known sources of local happiness: Happiness (Colloseum etc.) line buildings, not yet, but in next patch notes "Follower and Pantheon beliefs provide Local Happiness; Founder beliefs provide Global Happiness".
For now, pagoda/cathedral/mosque DO provide local happiness, other things I don't know (didn't tryed).
 
Imagine you have 26 population in a city.
You have at least eight grassland or flood plains tiles farmed next to a river, and you have already researched Civil Service (4 food per tile)
You then have 8 population working those tiles (4*8=32).
Each pop uses 2 food so 32-8*2=16
You have 16 surplus food, that means, you can support another 8 population, which can be working mines that yield 4 production, so you would now have 32 production, plus modifiers.
So basically, population generates production and science mostly, wealth if you want
 
Imagine you have 26 population in a city.
You have at least eight grassland or flood plains tiles farmed next to a river, and you have already researched Civil Service (4 food per tile)
You then have 8 population working those tiles (4*8=32).
Each pop uses 2 food so 32-8*2=16
You have 16 surplus food, that means, you can support another 8 population, which can be working mines that yield 4 production, so you would now have 32 production, plus modifiers.
So basically, population generates production and science mostly, wealth if you want

Well, those numbers don't add up to 26 pop, but your point is still right on -- excess food generated by farmers (together with food from sources that require no working citizens, like water mills, granaries, maritime city state friends, food-generating wonders like Hanging Gardens, etc.) free up other citizens to work tiles that generate little or no food, like mines, lumbermills, tiles with planted GPs, etc., and specialist slots. And remember that regardless of what those 26 (or 16 folks) are doing (farming, mining, whatever), they are producing 1 beaker apiece (base bpt, before libraries, observatories, unis, schools, labs, and other adders and multipliers). More pop = more bpt.
 
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