1 part sugar
1 part water
Combine both ingredients in a pot
Bring to a boil
Add to any beverage you want to sweeten
It really is simple to make. Keeping it, however, is a different matter.
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One thing you might run into is that the sugar crystallizes within a few days, creating a grainy, sticky mass more like sand mixed with honey than free-flowing simple syrup. Like this photo.
It still tastes the same and you shouldn't throw it away (see below), but it won't dissolve in an icy drink.
Solving The Solution
The crystallization is triggered by undissolved sugar crystals that latch onto dissolved ones and pull them out of solution. There are two primary approaches to preventing it. The first is to do everything you can to ensure all the sugar granules are dissolved. Stir it as little as possible, and use a wet pastry brush to wipe down the inside of the pan or put a lid on the pan so the steam washes the undissolved sugar off the sides. This will not give you 100% results, though.
The second, and easiest way to prevent crystallization is to use an interferer. Cream of tartar and lemon juice both contain acids that will prevent sugar from crystallizing. So, add a squeeze of lemon to your simple syrup after you remove it from the heat. Problem solved. Oh, and if you do end up with crystallized simple syrup, just heat it again to melt it and add some lemon juice. It'll be fine.