the number 18 stamped on type 44 wheels

You CANNOT have 18" wheels and 17" tires, this is impossible.

18 must be a mold mark that is significant to manufacturing or something like that, but is meaningless to the rest of the universe.

You ought to be able to identify the BMW Part Number, and the rim size (which will look like 7.5x17JJ, or similar). Any other markings are not important to you.

Your tires are 225/45x17. These numbers mean ...

225 This is the width of the tread (not exactly, but close enough for this discussion) in milimeters. You can see other cars at the traffic light around you and notice that this number will be smaller on small cars and higher on performance cars. A KIA might have a tire where this number is in the range of about 185 and a Corvette might have this number up around 285, for example.

45 This is called the Aspect Ratio. It is a number that represents the size of the sidewall as a percentage of the tread width (the first number). In your case, the sidewall is 45% of 225. If you wanted to put 18 or 19 inch rims on your car, you could not simply use another 225/45x because this would result in an overall tire diameter that would not fit inside the fenders. In order to maintain the same width of tire on a larger rim, you need to reduce the sidewall height to keep the same overall diameter. In your case, your 225/45x17 would be equivelent to a 225/40x18 or a 225/35x19. (I'm not suggesting that this alteration is a good idea, I'm only illustrating how one needs to reduce the sidewall height to allow for a larger rim size.)

17 This is the diameter of your rim. The tires MUST be the same diameter as the rims.

The rim also has a width that is typically 7.5 inches, but can be both wider or narrower depending on the marketing/engineering decisions. The width of the rim will govern the width of the tire. It is difficult to explain any hard and fast rules for the relationship of rim width and tire width, but as a personal matter, I like my tires to be about 1" wider than my rims for a standard fitment of most passenger cars. My BMW 3 Series has 7.5" rims and

225 tires (I have the same tires you have), and if one converts mm to inches, you will find that this is what you have. 225mm is just a bit over 8.5 inches.