F-Stop Math... Please?

<p>Please help me understand the math behind f-stops. For example, if I'm at f1.8, and I want to add two stops, what's the formula? Perhaps it's highly obvious, but honestly folks, I'm missing it! Thanks.</p>

<p>Randy</p>

<p>I don't recall the formula in a way I could write it down, but I did sketch it all out once and put it on my website where I could refer to it:<br>

http://www.thepeaches.com/photography/fractional_stops.htm<br>

Your answer (f/1.8 + 2 stops) = f/3.5</p>

<p>It's far easier just to memorize the values:</p>

<p>http://en.wikipedia.org/wiki/F-number</p>

 

<p> or if you round it to f/2, two stops down is f/4. </p>

<p>Two stops is an easy one -- double it.</p>

<p>To add one stop, multiply by the square root of two (about 1.4142, but we typically approximate it to 1.4.)<br /> To add two stops, mutiply by the square root of two twice, which is two.</p>

<p><em>Edit:</em> It's worth noting that ƒ/1.8 is an approximation -- it's a third stop below ƒ/2, which is actually about 1.78. Two stops up from there is about 3.56, which some round to ƒ/3.5 (in the normal photographic rounding scheme), or ƒ/3.6 (by proper mathematical rounding).</p>

Did somebody say formula? See http://www103.pair.com/parv/cgi-bin/fstop.cgi .

<p>every two stops the f number doubles, the in betweeners you'll have to memorize.</p>

<blockquote>

<p>For example, if I'm at f1.8, and I want to add two stops, what's the formula? </p>

 

</blockquote>

<p>Well, there is the math and then there is simply doing it in the field. Is your camera set to 1/3rd or 1/2 f-stops? Most are set to one-third. So this mean each click of the dial is one-third a stop. 3-clicks = 1/stop. So, from f/1.8 we simply dial in 6-clicks for two stops. This will take you to f/3.5. Done. If the camera is set to 1/2 stops it is 2-clicks of the dial for each full stop, or four clicks to go from f/1.8 to f/3.5. </p>

<p>Take 1.0 and 1.4, then double to 2.0 and 2.8, then double to 4.0 and 5.6, then double to 8.0 and 11(.2 but we drop the decimal at this point), then double to 16 and 22, then double to 32 and 44, 66 and 88.... If it's easier to remember you could start at something that your lenses actually have (assuming you don't have f/1.0 or f/1.4 lenses). 4.0 and 5.6 are pretty common. This of course isn't the "formula", but it's how I keep things straight.</p>

<p>If you're starting at a half or third stop and want to go 2 stops from there it's just doubling the number. Eventually you'll likely just memorize the values that you have available to you.</p>

<p>The full (and more than you wanted to know) discussion is still, I think, at http://en.wikipedia.org/wiki/F-number.<br /> However, just scroll down to the "Standard full-stop f-number scale" section learn what the full stops are. (e.g.,<br /> f/# 0.5 0.7 1.0 1.4 2 2.8 4 5.6 8 11 16 22 32 45 )<br /> However, you will almost always have a camera handy when you're doing this, so just note that each marked f/stop on a lens (at least above f/2 or so, depending on the lens) is one full stop. To keep the same exposure just slow down one marked speed for each marked aperture you stop down. (This may not apply to really old pre WWII cameras with weird speed intervals.)</p>

<p>The camera lens performs much like a funnel in that it gathers light and projects a miniature image onto the light sensitive surface of film or chip. Now the diameter of the lens is one key fact. In other words the larger the diameter of the lens, the more light gathered. Consider astronomical telescopes have large diameters so they can gather more light. As an example<br />Mt. Palomar in southern California is 200 inches in diameter. As to the camera, the greater the diameter of the lens, the more light gathered, the greater the light energy that can play on the film or chip.</p>

<p>The diameter of the lens is only one ingredient. Equally affecting the amount of light that can play on the film or chip is the focal length. The focal length is a measure of lens to image distance when the camera is looking at and focuses upon a far distant object. The greater this distance (focal length) the more the image will be magnified. With increasing magnification come light loss. In other words if we increase the focal length we get a more feeble image<br />playing on the film or chip. Doubling the focal length results in a 4 fold loss of light playing on the film or chip.</p>

<p>The facts are, focal length and lens diameter are intertwined. So entangled are these two details, we need a ratio to help untwine. We divide the focal length by the lens diameter to obtain a ratio that is universal. In other words, the ratio obtained is pure and devoid of dimension. Consider that a lens 400mm in focal length with a diameter of 100mm works out to 400 ÷ 100 = 4. If a lens has a focal length of 50mm and a diameter of 12.5mm this works out to 50 ÷ 12.5 = 4. Both will deliver the same image brightness. By convention we call this focal ratio value f/number or f/# or in this case f/4. Note that any lens, regardless of dimensions functioning at f/4, delivers the same light energy to the film or chip. This is the universal nature of the f/number system. We need such a system to take chaos away. The focal ratio allows us to tell each other how to set our cameras regardless of the dimensions<br />involved.</p>

<p>Now likely the maximum diameter available on our lens will deliver too much light and the result will be an over exposure. We need to have the ability to adjust the working diameter. A mechanism modeled after the human eye's ability to adjust pupil size is what we need. This is the Iris, named for the Greek god of the rainbow because this is colored portion of the human eye. The camera iris is a series of overlapping blades that makes a circular opening just behind the front element of the camera lens. The opening is called the aperture.</p>

<p>Because of the way film responds to light, it was deemed that adjustment to light energy allowed to play on the film or chip be adjustable in 2x increments. This is a doubling or halving of the amount of light allowed to play on the film or chip. To accomplish we alter the size of the aperture opening. Since this opening is a circle we must fall back on the geometry of the circle. Fact --- to double the surface area of a circle we multiply its diameter by the<br />square root of 2, which is 1.4142. We need only take this value to two decimal<br />places so 1.41 is just fine. <br /> Starting with a circle of one: The sequence is:<br /> 1 - 1.4 - 2 - 2.8 - 4 - 5.6 - 8 - 11 - 16 - 22 - 32 - 64<br /> Note each number going right is its neighbor to the left multiplied<br />by 1.41 and rounded.<br /> Each number going left is its neighbor on its right divided<br />by 1.41 and rounded.<br /> These values are often called "stops" because they stop some light and pass some light based on their hole size.<br /> For most needs, these values in 2x increments are just fine. If we need finer, we can make a set in 1/2 stop increment based on the multiplier 1.189, the fourth root of 2.<br /> If we need even finer we can go to 1/3 stop increments based on the sixth root of 2, it is 1.122.<br /> <br />I call all of this stuff gobbledygook but it is the heart of photo science.</p>

<p>If you remember the basic f/stop formulae you quarter the size of the aperture for two stops and divide it into the focal length of the lens. F/stop can be written as f/a where f is focal length and a is the diameter of the aperture. Definition .... stop=aperture<br>

Next question ... How do you find out the diameter of the aperture inside the lens? Sorry I cannot answer that one :-) IT is all theory and probably the suggestion to go to f/2 and close two stops to f/4 and then open up a third is the simpler way to work it out.<br>

Also thank your lucky stars you are not working with European lens of fifty plus years ago where instead of f/5.6 f/8 f/11 you could find lens marked f/6.3 f/9 f/12.5 :-)</p>

<p>The f-stop is the ratio of focal length to aperture diameter. A smaller number means that the aperture is larger relative to the focal length, hence more light.</p>

<p>However, the amount of light let in is a function of the area that is open, not the diameter. Double the area, double the light. The area is pi r squared. So, if you double the area, you are increasing the diameter (or the radius) by the square root of 2, which is approximately 1.4.</p>

<p>So a one-stop additional opening, which doubles the light, changes by a factor of 1.4. E.g., going from f/2.8 to f/2.0 (approximately) doubles the light.</p>

<p>Hence, changing the f-stop by a factor of 2 increases or decreases the light by a factor of 4.</p>

<p>The f-stop sequence is based on the area of the circle (aperture) through which the light passes.</p>

<p>Each time you change that area by a factor of two, you double or halve the amount of light.</p>

<p>The f numbers are based on the diameter (or radius if you prefer) of that circle, so they progress in steps equal to the square root of two, which is roughly 1.4. So one stop smaller than f/2 is f/2.8, and one stop less than f/4 is f/5.6.</p>

<p>The f-number is the denominator of a fraction in which the lens focal length is the numerator, so they're written as f/n or f:n.</p>

<p>As the f-number increases the area of the circle decreases so you get less light.</p>

<p>- Leigh</p>

 

<p>1.8 x 2 = f/3.6 is the two stops smaller aperture from f/1.8.</p>

<p>see: http://www.scantips.com/lights/fstop.html</p>

<p>Now changing the subject (maybe):</p>

<p>You are the captain of Cavalry “A” Troop. One hundred men with 100 horses marching through the American South Western Desert on patrol. Water is a problem but you expect rain. You order the troops to bivouac for the night and to dig a circular pit 8 feet in diameter. They do so and line this depression in the sand with canvas tent fabric. It rains as expected. The pit begins to collect rainwater. Due to your West Point training, you know that an 8-foot diameter pit will collect and hold enough water for your troop's needs. Unexpectedly a lookout spots “B” Troop approaching -- another 100 men with horses. You order your men to expand the diameter of the circular so it will collect water sufficient to accumulate 200 men and horses.</p>

<p>How big must the revised pit be to double the amount of collected rainwater?<br /> Answer: You multiply the pit diameter (8 feet) by 1.4142. This value is the square root of 2. The answer is 11.3 (rounded its 11 feet). So you order the pit expanded to 11 feet in diameter. Surprise, this new value causes the pit to accumulate twice as much water as before. Why? The surface area (catch basin) now has double the surface area; thus it can capture twice the amount of rain.</p>

<p>The lens opening or aperture is also a circular geometric figure. The area of any circle (thus its ability to collect light) is doubled if you multiply its diameter by 1.4 (1.4142 rounded). Using this factor a number set emerges:<br /> 1 – 1.4 – 2 – 2.8 – 4 - 5.6 – 8 – 11 – 16 – 22 – 32 – 45 – 64<br /> Note each number to the right is its neighbor on the left multiplied by 1.4 and then rounded. Each number to the left is its neighbor on the right divided by 1.4 and then rounded.<br /><br /><br /> To stop down one f/stop we multiply by 1.4.<br /> To open up one f/stop we divide by 1.4 or If you hate to divide, you can multiply by 0.7.</p>

<p>Two f/stops closed down multiply by 2.<br /> Two f/stops opened up divide by 2 or multiply by 0.5,</p>

<p>Three f/stops closed down multiply by 2.8<br /> Two f/stops opened up divide by 2.8 or multiply by 0.35</p>

<p>Four f/stops opened down multiply by 4.<br /> Four f/stops opened up divide by 4 or multiply by 0.25</p>

<p>More gobbledygook or is this photo science?</p>

<blockquote>

<p>if I'm at f1.8, and I want <strong>to add </strong>two stops</p>

</blockquote>

<p>Use two shutter speeds slower. Adding two stops means adding more light. f/1.8 is probably the widest aperture available. So, it is unlikely that you will be able to add two stops by operating the aperture ring. Add two stops by adding more light by picking two doublings of the shutter speed, slower. </p>

<p>If you are f/1.8 and 1/125, and you want to add two stops, then use f/1.8 and 1/30. </p>

 

<<Two stops is an easy one -- double it.>>

 

 

 

Be careful here. Double means multiply by 2.

 

 

f/1.8 x 2 = 2f/1.8 = f/0.9

 

 

Probably not what you want. If you said double the denominator (bottom number), then you would be correct.

 

 

f/(1.8 x 2) = f/3.6

 

 

Which is closer to what your want.

 

 

 

Be careful again, though. What did the OP mean by adding two stops? Two stops MORE light or two stops LESS

light? Two stops LESS light is f/3.6. Two stops MORE light is f/0.9, but I don't think such a lens exists.

<p>You know, I love these threads when they pop up, as they do every so often.</p>

<p>The development as it goes along, with each person trying to clarify or make more accurate the posts before (at least the immediate one before) is a sort of example of how human knowledge develops.</p>

<p>The "engineers" give complicated formulae for precise solutions of the problem.<br>

The "humanists" apply complicated metaphors involving buckets and funnels, or some analogy.<br>

By the time everybody has had their shot, the OP has long since wandered (and maybe wondered, too) off. ;)</p>

<p> </p>

<p>Been away - just catching up... Wow. Thank you all for a really fine thread, explanations, teaching, application, etc. I love this forum!</p>

<p>Randy -</p>

<p>@JDM - nice way to sum things up. Still here! Thanks -</p>

<p>R. Nelson, nothing personal, just a generalization.. ;)</p>

<p>Randy, here are a couple of key ideas.</p>

<p><strong>1. Fractions get smaller as the bottom number gets bigger.</strong></p>

<p>For any given focal length (f), f/8 is a smaller number than f/4. This also applies to shutter speeds that are expressed as a fraction of a second. 1/125th of a second is SHORTER than 1/60th of a second. Aperture math differs slightly from shutter math. I'll discuss how and why in a minute.</p>

<p><strong>2. Think about circles.</strong></p>

<p>When you see something like f/4, it's describing the diameter of a circle. If f = 100 mm, f/4 = 25 mm or about one inch. The aperture is a circular hole that's approximately one inch in diameter.</p>

<p><strong>3. Shutter speeds are linear; apertures are squared.</strong></p>

<p>If you double the shutter speed (multiply by 2), you let in two times as much light.</p>

<p>1/30 second gives the sensor or film twice as much light as 1/60 second.</p>

<p>If you double the f-stop, you let in FOUR TIMES as much light.</p>

<p>f/4 gives the sensor or film four times as much light as f/8. The reason is that the AREA of the circle is what lets in the light, and the area is the SQUARE of the radius. The radius is half of the diameter.</p>

<p>Example: f = 80 mm</p>

<p>f/4 = a diameter of 20 mm, or a radius of 10 mm<br>

AREA = 2 x pi x radius squared = (approximately) 2 x 3 x (10x10) = 600 square millimeters</p>

<p>f/8 = a diameter of 10 mm, or a radius of 5 mm<br>

AREA = 2 x pi x radius squared = (approximately) 2 x 3 x (5x5) = 150 square millimeters</p>

<p>600 square millimeters is four times the area of 150 square millimeters. Four times the area projects four times the light at the same focal length.</p>