Need a Math Fiend to Help Me!

[HenryChavez]

I am working on a program to use the zone system. Basically, the program documents the aperture, shutter speed and gives the EV number. You choose what zone you want to put the reading at and the program gives you the new aperture and shutter speed based on what zone you selected. Here is my question; does anyone know what the calculation/formula is to take an existing shutter speed or aperture and obtain the difference based on the number of stops? For example, f/8 @ 1/8 and I want to increase exposure by 2 stops. So I want, in this case to increase the aperture by 2 stops. I know the answer is f/2, but I want to know the equation/calculation to get to this answer. Anyone have any ideas? I did a search and could not find anything.

Kindly,

Henry

[JoeVanCleave]

I'm no math fiend, but I believe the f-stop sequence is based on SQRT2, which is about 1.414. So one stop below f/8 is 8/1.414=f/5.6. And the next stop below f/5.6 is 5.6/1.414=f/4, etc.

~Joe

[HenryChavez]

I'm no math fiend, but I believe the f-stop sequence is based on SQRT2, which is about 1.414. So one stop below f/8 is 8/1.414=f/5.6. And the next stop below f/5.6 is 5.6/1.414=f/4, etc.

~Joe

Thanks Joe, I'd seen the number, but didn't know it was the square root of 2. This is the missing piece. Thanks!

Best,
HC

[DaveBell]

Joe of course is correct, Henry. The reasoning might help with grasping the whole concept:
f ratio is equal to the focal length divided by the diameter of the aperture.
But the amount of light that gets through is proportional to the *area* of the aperture, which is of course proportional to the square of the diameter.
So, doubling the diameter increases the light by four times (or decreases the exposure time by four.)
To get exactly double the light, you need 2 times the area, so sqrt(2) times the diameter, which reduces the f ratio by sqrt(2).

Pedantically,
Dave

[HenryChavez]

Joe of course is correct, Henry. The reasoning might help with grasping the whole concept:
f ratio is equal to the focal length divided by the diameter of the aperture.
But the amount of light that gets through is proportional to the *area* of the aperture, which is of course proportional to the square of the diameter.
So, doubling the diameter increases the light by four times (or decreases the exposure time by four.)
To get exactly double the light, you need 2 times the area, so sqrt(2) times the diameter, which reduces the f ratio by sqrt(2).

Pedantically,
Dave

Thanks Dave. Very cool to know.

HC

[earlj]

it's easy to generate an f number table in Excel: first column - integer numbers from 1 to whatever (1,2,3, ...) 2nd column: =(SQRT(2))^A1, =(SQRT(2))^B1, =(SQRT(2))^C1, ...
Each f number is the square root of two raised to the power of an integer number. This relationship also gives you an easy calculation for the number of stops between two known f numbers, using the logarithm with a base of the square root of two. So, let's say your pinhole camera is f295, and your light meter reads to f22. To calculate the number of stops between those two f numbers, subtract the log of each using a base of the square root of two: =LOG(295,SQRT(2))-LOG(22,SQRT(2)) The result is 7.49, so there are 7 and 1/2 stops between f22 and f295.

[HenryChavez]

it's easy to generate an f number table in Excel: first column - integer numbers from 1 to whatever (1,2,3, ...) 2nd column: =(SQRT(2))^A1, =(SQRT(2))^B1, =(SQRT(2))^C1, ...
Each f number is the square root of two raised to the power of an integer number. This relationship also gives you an easy calculation for the number of stops between two known f numbers, using the logarithm with a base of the square root of two. So, let's say your pinhole camera is f295, and your light meter reads to f22. To calculate the number of stops between those two f numbers, subtract the log of each using a base of the square root of two: =LOG(295,SQRT(2))-LOG(22,SQRT(2)) The result is 7.49, so there are 7 and 1/2 stops between f22 and f295.

Thanks Earl, that is what I am using. This tells me I am going in the right direction.

Best,
Henry

[Ned.Lewis]

HC: Here's some more, if it seems like too much then just ignore.... you already have good answers!

Joe's answer is completely practical.

Dave's answer explains why the square root of 2 comes into play.

Earl's answer uses logarithms to further simplify the problem. Logarithms are used to turn problems with powers and multiplication into arithmetic problems, and the f-stop scale is naturally a logarithmic one. Earl's answer simplifies the calculation of a difference of f-stops, and it simplifies the understanding of that difference if you are comfortable and intuitive with logarithms.

If you only have a calculator and not a program that can perform logarithms in any base, then you can change the base of a logarithm by dividing as follows ( lower case "log" is the log base 10 function on your calculator )

LOG( x, sqrt(2) ) = log( x )/log(sqrt(2))

This comes in handy when doing quick calculations designing pinhole or lensed cameras! ( Like estimating how many "stops" falloff will there be at the corner of your film in a pinhole camera... )

I don't know that we do such a great job teaching mathematics. The story of Napier and Briggs and the invention of logarithms is fascinating and it was a revolution in science. Before Napier, people solved difficult multiplicative problems with tables of sines and other tricks, but logarithms freed scientists from time-consuming tedious calculations. More importantly maybe, it led to thinking and intuition about powers arithmetically, which is often very natural just like it is with f-stops. There are some hints that Kepler got to his 3rd law by "logarithmic thinking" he certainly knew the work of his contemporaries Napier and Briggs. Today's scientists use that thinking unconsciously having been steeped in it through school. It was a major stepping stone along the way to modern science, with amazing consequences, yet the story is hardly known.

[HenryChavez]

HC: Here's some more, if it seems like too much then just ignore.... you already have good answers!

Joe's answer is completely practical.

Dave's answer explains why the square root of 2 comes into play.

Earl's answer uses logarithms to further simplify the problem. Logarithms are used to turn problems with powers and multiplication into arithmetic problems, and the f-stop scale is naturally a logarithmic one. Earl's answer simplifies the calculation of a difference of f-stops, and it simplifies the understanding of that difference if you are comfortable and intuitive with logarithms.

If you only have a calculator and not a program that can perform logarithms in any base, then you can change the base of a logarithm by dividing as follows ( lower case "log" is the log base 10 function on your calculator )

LOG( x, sqrt(2) ) = log( x )/log(sqrt(2))

This comes in handy when doing quick calculations designing pinhole or lensed cameras! ( Like estimating how many "stops" falloff will there be at the corner of your film in a pinhole camera... )

I don't know that we do such a great job teaching mathematics. The story of Napier and Briggs and the invention of logarithms is fascinating and it was a revolution in science. Before Napier, people solved difficult multiplicative problems with tables of sines and other tricks, but logarithms freed scientists from time-consuming tedious calculations. More importantly maybe, it led to thinking and intuition about powers arithmetically, which is often very natural just like it is with f-stops. There are some hints that Kepler got to his 3rd law by "logarithmic thinking" he certainly knew the work of his contemporaries Napier and Briggs. Today's scientists use that thinking unconsciously having been steeped in it through school. It was a major stepping stone along the way to modern science, with amazing consequences, yet the story is hardly known.

Ned,
Seriously fascinating. I was a poor student with math as a youngster, only as a result of my arrogance. As an adult I have found a great need for it and have had to teach myself as a necessity. The necessity coming from my hobbies. This is a prime example of needing math. I do so enjoy learning when I play. I had to research and understand log, albeit rudimentary, to work on this program. Your insight is greatly appreciated and enjoyed.

Best,
Henry

[HenryChavez]

Well, I have a working copy up and running of the program I was writing. I'm still debugging it, but it seems to work fairly well. I'm thinking of adding ISO, but I haven't decided yet. I'll see if I find it useful before I release it to the library. I had fun working on it and I really appreciate all the help I got here. It was ALL VERY useful.

[JoeVanCleave]

A bit off-thread, but this reminds me of my long-standing interest in the abacus, upon which addition and subtraction are relatively easy, while multiplication and division are easy but tedious. Square roots are also possible, using a variation of the paper and pencil method. But I discovered, several years ago, that multiplication and division are quicker using logs. I made a small laminated card with a simple log table, which you can use with the abacus to multiply and divide numbers by adding or subtracting their logarithm. I'll post an image of one of my handmade abaci.

~Joe

[Ned.Lewis]

Hi Joe and HC,

This made me laugh out loud. First we have a picture of a phone app, followed by a comment about an abacus. I think many of us here must appreciate things that are fundamental and somehow "close to reality". The kinds of photography we do fits this as well. I spend my days programming, managing quantities of data that are difficult to comprehend, administering modern servers that have astonishing capabilities, and I do a lot of mathematical statistics. To stay sane and counterbalance, I don't own a cell phone, I write and do math with a pencil and paper, and I like more "hands-on" kinds of photography. These are not contradictory, but complimentary. I might joke about being a Luddite, but everything is connected in it's way.

Anyway, it's been a long time since I read about this, but there is something called "Napier's bones" that Napier invented as a calculating aid. You could probably look it up and it might be something you would find interesting. It was another "calculation aid" using logarithms and a sort of precursor to the slide rule. I have a vague memory that there was also something called "Napier's Abacus", but I don't know what it was.

Logarithm tables were incredibly valuable. When I graduated from high school, we were offered a beautiful bound set of tables in several volumes, as something valuable to carry forward into scientific careers. My dad has a beautiful slide rule that was a cherished and well-worn possession when he was an engineering student in the 1950's. Today we forget the simple amazement of being able to get logarithms so easily. The slide rule helped eliminate using those bulky tables!

Those tables were made at an incredible price. Groups of monks worked on them as a sort of "service to humanity" and some particular values are very difficult to calculate and can take days or weeks, and then days more to check and verify. Many lifetimes of work went into making those tables. Now anyone with a calculator or "smart phone" or any kind of computer can compute them by pressing a button. This might disconnect us from what logarithms actually are and the intuition that comes from that.

Some of the work I've been doing recently has had me going back and reading work from the early days of probability and statistics. Just yesterday I was reading a paper about methods to compute a function I work with called the incomplete beta function. I learned that Karl Pearson, who is one of the great founders of modern statistics, spent ten years of his life compiling tables of the values of this function. I admire that dedication and also the selflessness of that effort. There is also a different hidden sort of value here, like the value of a handmade abacus. I don't know what that would be called exactly, but I have a hunch it's related to the value of homemade photographs in the way it's connected to the real world in a tangible way.

End of rambling thoughts....:)

[JoeVanCleave]

Great thoughts, Ned. And continuing with the tangent, here's my first handmade abacus. The side pieces of the frame are of Brazilian hardwood, while the upper and lower dowels are black walnut. The rods and cross bar are solid brass. This frame design is my own idea, diverging quite a bit from the traditional Asian abacus frame construction.

But the beads are indeed Japanese soroban beads, which I had to purchase from Tomei Soroban at 10 cents per bead!

The thin wire is used to hang the abacus for display.

~Joe

http://farm8.staticflickr.com/7399/10433687183_0e5c5ae467_c.jpg

[JoeVanCleave]

Continuing with the tangent, this second version uses the same frame design as the first but is much larger, a tabletop design for use by adult-sized fingers. Being as I could not find the biconic style of soroban bead in larger sizes, and my lathe skills aren't up to the task, I opted instead to use round wooden balls from Hobby Lobby, predrilled with holes.

The wood used in the frame is the same as the smaller version, with the addition of a back piece for strength. The rods are solid brass while the cross bar is a square brass rod, with tiny screws denoting the thousands commas and decimal point.

All told, this is a very pleasant abacus for daily use, especially satisfying knowing that I made it by hand.

As in the previous image, the copper wire is for hanging on the wall.

~Joe

http://farm4.staticflickr.com/3728/10433587374_d7752fe761_c.jpg

[HenryChavez]

Great thoughts, Ned. And continuing with the tangent, here's my first handmade abacus. The side pieces of the frame are of Brazilian hardwood, while the upper and lower dowels are black walnut. The rods and cross bar are solid brass. This frame design is my own idea, diverging quite a bit from the traditional Asian abacus frame construction.

But the beads are indeed Japanese soroban beads, which I had to purchase from Tomei Soroban at 10 cents per bead!

The thin wire is used to hang the abacus for display.

~Joe

http://farm8.staticflickr.com/7399/10433687183_0e5c5ae467_c.jpg

Joe,
This is lovely. I like your idea regarding the side rails, it looks like it would be easier to hold with these instead of the traditional box design. Now I wan to learn to use it! Dang it! I already have too many things that occupy my little brain. Well, just one more thing won't hurt. :-) Thank you for sharing this with us.

Best,

[HenryChavez]

Continuing with the tangent, this second version uses the same frame design as the first but is much larger, a tabletop design for use by adult-sized fingers. Being as I could not find the biconic style of soroban bead in larger sizes, and my lathe skills aren't up to the task, I opted instead to use round wooden balls from Hobby Lobby, predrilled with holes.

The wood used in the frame is the same as the smaller version, with the addition of a back piece for strength. The rods are solid brass while the cross bar is a square brass rod, with tiny screws denoting the thousands commas and decimal point.

All told, this is a very pleasant abacus for daily use, especially satisfying knowing that I made it by hand.

As in the previous image, the copper wire is for hanging on the wall.

~Joe

http://farm4.staticflickr.com/3728/10433587374_d7752fe761_c.jpg
Joe,
This is one is nice too, although I must say I do like the soroban beads, this one is also very nice. How big is the one with the soroban beads, can you show something for scale?

Best,

[HenryChavez]

Hi Joe and HC,

This made me laugh out loud. First we have a picture of a phone app, followed by a comment about an abacus. I think many of us here must appreciate things that are fundamental and somehow "close to reality". The kinds of photography we do fits this as well. I spend my days programming, managing quantities of data that are difficult to comprehend, administering modern servers that have astonishing capabilities, and I do a lot of mathematical statistics. To stay sane and counterbalance, I don't own a cell phone, I write and do math with a pencil and paper, and I like more "hands-on" kinds of photography. These are not contradictory, but complimentary. I might joke about being a Luddite, but everything is connected in it's way.

Anyway, it's been a long time since I read about this, but there is something called "Napier's bones" that Napier invented as a calculating aid. You could probably look it up and it might be something you would find interesting. It was another "calculation aid" using logarithms and a sort of precursor to the slide rule. I have a vague memory that there was also something called "Napier's Abacus", but I don't know what it was.

Logarithm tables were incredibly valuable. When I graduated from high school, we were offered a beautiful bound set of tables in several volumes, as something valuable to carry forward into scientific careers. My dad has a beautiful slide rule that was a cherished and well-worn possession when he was an engineering student in the 1950's. Today we forget the simple amazement of being able to get logarithms so easily. The slide rule helped eliminate using those bulky tables!

Those tables were made at an incredible price. Groups of monks worked on them as a sort of "service to humanity" and some particular values are very difficult to calculate and can take days or weeks, and then days more to check and verify. Many lifetimes of work went into making those tables. Now anyone with a calculator or "smart phone" or any kind of computer can compute them by pressing a button. This might disconnect us from what logarithms actually are and the intuition that comes from that.

Some of the work I've been doing recently has had me going back and reading work from the early days of probability and statistics. Just yesterday I was reading a paper about methods to compute a function I work with called the incomplete beta function. I learned that Karl Pearson, who is one of the great founders of modern statistics, spent ten years of his life compiling tables of the values of this function. I admire that dedication and also the selflessness of that effort. There is also a different hidden sort of value here, like the value of a handmade abacus. I don't know what that would be called exactly, but I have a hunch it's related to the value of homemade photographs in the way it's connected to the real world in a tangible way.

End of rambling thoughts....:)

Ned,
Thank you so much for your contribution to this odd, but interesting thread. I vehemently agree with your comments. It is all connected. Consider the artist who uses a digital or even film camera. He is capable of creating beautiful images and may not even consider how his tool works or what led up to the extreme ability that he has to instantly capture images with his specific tool. He could live out his entire life producing wonderful works that are prized by many people and people yet to come, but yet not know one thing about the tool. I can image if the knowledge of which you speak of where to ever disappear, people would look upon these works with wonder as we today look at the structures of long lost civilizations. IMHO there are very few people who care to know why and how and that makes them a precious commodity. . .to steal from. ;-)

I have a tremendous thirst for how things work and that is why I am here, wandering about, talking to all these intelligent folks and running about like a kid in a candy store. I often consider film in the same light. I imagine the first people that used their intellect to further the photographic process with awe. The math involved crushes my little brain to even consider. What with focus, light waves and what not, it is a wonder to me that it ever came to pass, but it did. So this is the place to be for me. I run about looking at what everyone has with amazement and reading about how they did it and why. Then I find these wonderful associations that put me onto paths of new interests that are no less fascinating and exciting to see. I am curious to read about Napier, I've seen the name before. Very interesting indeed.

We all, not only stand on the shoulders of giants, we also get to tug at their waist coats and ask stupid questions without being yelled at! :-) Thank you.

Best,
Hank of all trades, master of none.

[JoeVanCleave]

Joe,
This is one is nice too, although I must say I do like the soroban beads, this one is also very nice. How big is the one with the soroban beads, can you show something for scale?

Best,

Henry,

The small abacus with biconic soroban-style beads is about 4" wide, and so those beads are best suited for child-sized fingers. Meanwhile the larger abacus uses 3/4" diameter round beads, so it's much larger.

You might be interested in a blog article I wrote recently, about finding an adult-sized Japanese soroban at a thrift store here in Abq. Here's the link. (http://joevancleave.blogspot.com/2013/10/a-double-find-day.html?m=0)

To keep from detouring this thread any further, I can recommend some abacus links if you're curious, just PM me.

~Joe

[DaCh]

Or you could try
1517
;)