I know similar questions have been asked before, however, it is not really getting me the results I want. So I have been trying to learn about how binoculars work, which by my understanding, the first number is power with second being magnification. Anyways, I basically want to be able to see a dime in fine detail from lets say 100 yards, to the point where I can see the head on the front of the dime. My old binoculars was 8x25 specs and was just horrible as I couldn't even see a dime from that far. So I went to Walmart and the guy there recommended a $100 binocular for this, and it did not get in close enough, it looked clear but seems like it is not close enough. It was however, miles better than my old binoculars. The $100 binoculars that I got was the Nikon Aculon A211 10x42. So I was wondering if maybe 16x50 would be able to do it as amazon list that as $120? I also heard that some binoculars lie about there specs, so what binoculars exist that would be reasonable like $100-$200 that I can see a dime like I was standing a foot away from it even though I am 100 yards out and are not fake specs? Would the same brand binoculars except a spec rating of 16x50 do it? Thanks in advance.
If you want to see something from 100 yards away as if it were 1 foot away, you can work out what magnification you need. 100 yards is 300 feet. To make 300 feet seem like 1 foot, you need 300x magnification. That's the first number. (The second number is the objective lens diameter in mm.) 300x magnification basically does not exist in handheld binoculars, so you will have to compromise.
You can find a bunch of good information at a page such as How to Choose Binoculars (REI web site):
What do the numbers (specs) on binoculars mean? Model numbers on binoculars essentially tell you their strength (magnification power) and size (objective lens diameter). In 8x42 binoculars, for example, “8” is the magnification power and “42” is the diameter (in millimeters) of the objective lenses (the lenses closest to the object you’re viewing).
I found a quite nice explanation of all the considerations for binocular choice here. It gets a little technical near the end though.
There's a bit to unpack here - you are a little off about the second number of the binoculars - the first number is the magnification. This number tells you how many times closer an object will appear if you look at it from any given distance. The higher this number is, the harder it will be to focus on objects that are closer to you, and the more light you will need to see objects that are further away... which is where the second number comes into play.
The second number is the objective diameter in millimeters. The bigger this number, the wider the lens and the more light it will let in (generally, there are other considerations too), giving you a clearer view of a small object at a greater distance. The other factors are the type of lens you have - how good they are and what they are made out play a big role in colour and how well the light gets transmitted, as well as how much distortion you will see in any image. Generally a cheap lens means it won't work as well as a more expensive one of the same size - but not always
For magnifying things - what you are looking at is the radian - this is the ratio of how big it is across, to how far away it is:
A dime is
17.91 mm across - lets say
20 mm for convenience.
100 yards is about
90 m or
90000 mm 20/90000 = 0.00022 radians. At
1 foot (300 mm) a dime is
20/300 = 0.0667 radians
So how much magnification would you need to get a dime to appear the same as one at a foot away with the naked eye?
0.0667/0.00022 = 300 (as also noted by Greg Hewgill). So to see a dime at 100 yards as if it were 1 foot away from you, you would need to magnify it by 300x.
You will not find a 300x pair of binoculars anywhere - this is because to have that high a magnification, the lens would need to be massive in both senses of the word - big in diameter to pick up enough light, and heavy as a consequence. Lenses of this size require supports like tripods and will be in the zoom-telephoto lens size that you see sports photographers with.
If you want to get good reliable lenses - make sure you do your research; cheap is not usually an option. If you are interested in things other than binoculars, check out spotting scopes (these ones are the Zeiss ones - excellent optics) Spotting scopes come with higher mag than binoculars and can be mounted with a phone or camera (model dependent) for photos, but you do need a tripod or something to rest them on.