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Many of us will readily recognize the following sequence of numbers:
1, 2, 4, 8, 16, 32, 64, 128, 256, 512.
Save for the first digit (1), each number is twice that of the number to it's left. So you could say that, as you read from left to right, the numbers gets doubled. Conversely, if you were to read the numbers right to left, the numbers get halved.
If you've been involved with photography for a significant length of time, you will also recognize some familiar numbers;.. and you usually call them f-stops, or f-numbers.
But those numbers, when stamped into a lens barrel, have some additional numbers in them.
2, 2.8, 4, 5.6, 8, 11, 16, 22, and maybe 32.
And for some reason, you've been told that this sequence, just like the list above, is either doubling or halving depending on which direction you are reading them.
OK, so what gives?
I'll start out with a word of warning;.. if math gives you a headache, you may want to stop here and go do something else. What I'm going to attempt to do is explain F-stops in a bit more detail. And that involves math. No, not rocket-science math or quantum physics. But math nonetheless.
Starting with this: 8 x 2 = 11.
OK, I cheated. f8 x 2 = f11.
Make sense now? I didn't think so. Nor does f8 / 2 = f5.6, right?
To understand those f-numbers, lets take a look at an ordinary camera lens. In this case, I'll use my 105mm Nikkor Micro, which is pictured above. If you want to increase the exposure by one stop;, you've probably been told to use a smaller number. I'll use going from f8 to f5.6 for this example. F5.6 lets twice as much light in as f8. But even your kids and grandkids know that 8 / 2 = 4, not 5.6!
The reason behind the odd photographic sequence is because f-numbers are actually the result of a ratio. Remember the term quotient from your school days? That's the name given to the results of a simple division problem. F-numbers are really quotients!
OK, so where do the other two numbers come from (recall dividend or numerator and divisor or denominator? If so, you’ve probably got a head-ache now!)? The first number (dividend) is the focal length of your lens. The second number (divisor) is the diameter of the aperture inside your lens.
So let's say you have a 200mm lens. You turn the aperture ring until the aperture blades create a circle inside the lens that measures 25mm in diameter. The result is the equation 200 / 25 = 8. You now have the lens set to f8. If you were to open the aperture until it measures 50mm in diameter, you have the equation 200 / 50 = 4;.. meaning the lens is set to f4.
In essence, you are letting in four times as much light at f4 as you were with the lens set to f8. F5.6 would be twice as much light as f8.
So now you're probably wondering why f5.6 lets in twice as much light as f8. Why isn't it f4 instead? Well, sorry to say, it's time for some more math; and it's for the same reason 50mm is four times as large as 25mm. Confused? Don't feel bad. It's a concept that takes some people time to wrap their heads around.
Let's take that 25mm opening. How do you calculate the area of that opening? Time to think back to school;.. remember the formula? pi x r². I'll use the everyday value of 3.14 for;. r is for radius, which is half the diameter. So a 25mm circle will have an area of 3.14 x 12.5² = 3.14 x 156.25 = 490.625 mm².
A 50mm opening would be 3.14 x 25² = 3.14 x 625 = 1962.5 mm². A 50mm opening, while being twice the diameter as a 25mm opening, has 4 times the area! (1962.5 / 490.625 = 4) So a 50mm opening will let in 4 times as much light!
In order to get twice as much light, you would need a 35.35mm diameter opening (3.14 x 17.68 x 17.68 = 3.14 x 981.51 mm²; and 490.625 x 2 is roughly 981.51;..remember I'm using rounded-off numbers for simplicity!). And where would this size opening fall into the f-stop number? 200 / 35.35 = 5.6.
Does 5.6 sound familiar?
This is why f5.6 lets in twice as much light as f8;.. the area is twice the size; not the number (quotient!) itself!
OK, so why not just use the areas created by the aperture blades instead of this seemingly long-winded way? Well, if you told someone you took a shot at ISO 100, 1/640th of a second, and an aperture set to 25 mm²;.. you may be providing accurate information, but you're not providing complete information.
To explain why, I placed two lenses up on a table and set them both to f8. On the left is a 105mm, on the right is a 28mm. Remember, they're both set to f8! Notice how much larger the opening is on the 105mm on the left compared to the 28mm on the right?
If I set the 105mm lens' aperture to, let's say 10mm diameter, I would have an f-number of roughly f10.5. But if I set the 28mm to the same 10mm diameter, it would be set to f2.8! To get f10.5 on a 28mm lens, the aperture would need a diameter of 2.67 mm.
So even though the areas created by the aperture blades in lenses of different focal lengths are different; optically they create the same f-number;. And ultimately the same exposure! That's why f-numbers are used instead of the areas created by the aperture blades;.. it makes it just that much easier for us to work with. Otherwise, converting aperture areas from one lens to create an identical exposure in another lens would REALLY give you a headache!
So what the manufacturers do when they put seemingly archaic numbers like 2.8, 5.6 and 11 on the lenses is really just taking the math out of the equation for us! So f/8 on one lens gives us the same exposure aperture on any other lens! Whether it's a $50 point-and-shoot, a vintage 8x10 view camera or a 5-digit Hassy dream setup.... f/8 is f/8.
1, 2, 4, 8, 16, 32, 64, 128, 256, 512.
Save for the first digit (1), each number is twice that of the number to it's left. So you could say that, as you read from left to right, the numbers gets doubled. Conversely, if you were to read the numbers right to left, the numbers get halved.
If you've been involved with photography for a significant length of time, you will also recognize some familiar numbers;.. and you usually call them f-stops, or f-numbers.
But those numbers, when stamped into a lens barrel, have some additional numbers in them.
2, 2.8, 4, 5.6, 8, 11, 16, 22, and maybe 32.
And for some reason, you've been told that this sequence, just like the list above, is either doubling or halving depending on which direction you are reading them.
OK, so what gives?
I'll start out with a word of warning;.. if math gives you a headache, you may want to stop here and go do something else. What I'm going to attempt to do is explain F-stops in a bit more detail. And that involves math. No, not rocket-science math or quantum physics. But math nonetheless.
Starting with this: 8 x 2 = 11.
OK, I cheated. f8 x 2 = f11.
Make sense now? I didn't think so. Nor does f8 / 2 = f5.6, right?
To understand those f-numbers, lets take a look at an ordinary camera lens. In this case, I'll use my 105mm Nikkor Micro, which is pictured above. If you want to increase the exposure by one stop;, you've probably been told to use a smaller number. I'll use going from f8 to f5.6 for this example. F5.6 lets twice as much light in as f8. But even your kids and grandkids know that 8 / 2 = 4, not 5.6!
The reason behind the odd photographic sequence is because f-numbers are actually the result of a ratio. Remember the term quotient from your school days? That's the name given to the results of a simple division problem. F-numbers are really quotients!
OK, so where do the other two numbers come from (recall dividend or numerator and divisor or denominator? If so, you’ve probably got a head-ache now!)? The first number (dividend) is the focal length of your lens. The second number (divisor) is the diameter of the aperture inside your lens.
So let's say you have a 200mm lens. You turn the aperture ring until the aperture blades create a circle inside the lens that measures 25mm in diameter. The result is the equation 200 / 25 = 8. You now have the lens set to f8. If you were to open the aperture until it measures 50mm in diameter, you have the equation 200 / 50 = 4;.. meaning the lens is set to f4.
In essence, you are letting in four times as much light at f4 as you were with the lens set to f8. F5.6 would be twice as much light as f8.
So now you're probably wondering why f5.6 lets in twice as much light as f8. Why isn't it f4 instead? Well, sorry to say, it's time for some more math; and it's for the same reason 50mm is four times as large as 25mm. Confused? Don't feel bad. It's a concept that takes some people time to wrap their heads around.
Let's take that 25mm opening. How do you calculate the area of that opening? Time to think back to school;.. remember the formula? pi x r². I'll use the everyday value of 3.14 for;. r is for radius, which is half the diameter. So a 25mm circle will have an area of 3.14 x 12.5² = 3.14 x 156.25 = 490.625 mm².
A 50mm opening would be 3.14 x 25² = 3.14 x 625 = 1962.5 mm². A 50mm opening, while being twice the diameter as a 25mm opening, has 4 times the area! (1962.5 / 490.625 = 4) So a 50mm opening will let in 4 times as much light!
In order to get twice as much light, you would need a 35.35mm diameter opening (3.14 x 17.68 x 17.68 = 3.14 x 981.51 mm²; and 490.625 x 2 is roughly 981.51;..remember I'm using rounded-off numbers for simplicity!). And where would this size opening fall into the f-stop number? 200 / 35.35 = 5.6.
Does 5.6 sound familiar?
This is why f5.6 lets in twice as much light as f8;.. the area is twice the size; not the number (quotient!) itself!
OK, so why not just use the areas created by the aperture blades instead of this seemingly long-winded way? Well, if you told someone you took a shot at ISO 100, 1/640th of a second, and an aperture set to 25 mm²;.. you may be providing accurate information, but you're not providing complete information.
To explain why, I placed two lenses up on a table and set them both to f8. On the left is a 105mm, on the right is a 28mm. Remember, they're both set to f8! Notice how much larger the opening is on the 105mm on the left compared to the 28mm on the right?
If I set the 105mm lens' aperture to, let's say 10mm diameter, I would have an f-number of roughly f10.5. But if I set the 28mm to the same 10mm diameter, it would be set to f2.8! To get f10.5 on a 28mm lens, the aperture would need a diameter of 2.67 mm.
So even though the areas created by the aperture blades in lenses of different focal lengths are different; optically they create the same f-number;. And ultimately the same exposure! That's why f-numbers are used instead of the areas created by the aperture blades;.. it makes it just that much easier for us to work with. Otherwise, converting aperture areas from one lens to create an identical exposure in another lens would REALLY give you a headache!
So what the manufacturers do when they put seemingly archaic numbers like 2.8, 5.6 and 11 on the lenses is really just taking the math out of the equation for us! So f/8 on one lens gives us the same exposure aperture on any other lens! Whether it's a $50 point-and-shoot, a vintage 8x10 view camera or a 5-digit Hassy dream setup.... f/8 is f/8.
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