Counting[ edit ] It is believed that a mathematical notation to represent counting was first developed at least 50, years ago [1] — early mathematical ideas such as finger counting [2] have also been represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The tally stick is a way of counting dating back to the Upper Paleolithic. Perhaps the oldest known mathematical texts are those of ancient Sumer.

What makes them special, however, is that they obstruct a precisely controlled fraction of incoming light, and do so uniformly. However, the viewfinder will still appear very dark, so photographers often compose their image prior to placing the filter in front of the lens.

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Unfortunately for consumers, manufacturers do not use a standardized classification system to describe the strength of their filters. Hoya designates its 9 f-stop grey filter by x based on its filter factor which is somehow lower than the absolute value of x Lets see an overview of the different descriptions concerning ND filters as a function of f-stop reduction and optical density but adding also a new parameter the filter factor.

What is the filter factor?

For example, a 3-stop ND filter ND 0. In the below table, you can clarify the properties of each filter. What happens if we stack ND filters? It is very common to stack ND filters to increase their light-reducing strength, but adding multiple filters to the end of a lens can cause vignetting.

Keep in mind that no matter how many ND filters are used, the densities of the filters were added and the filter factors need to be multiplied. For example, if you start with an ND filter with a density of 0. Basic steps to perform a Long exposure shot First you have to find out the unfiltered exposure time.

Measure this with your camera in aperture priority mode using your desired f-stop. There are several ways of doing this: Cover the camera with a dark cloth. Use a small cap which is designed for this purpose.

Is there a mathematical formula to calculate the exposure times?

Yes, there is a simple mathematical formula and it is described in the following figure with a simple example. Useful timetables I have compiled some useful tables where you can find the appropriate exposure time when you have already chosen the unfiltered time and the type of ND filter you use.

All the values are in seconds except it is declared otherwise. I believe this table below is maybe more useful since it deals with darker ND filters and longer exposure times.

How can we read these tables? Check first the unfiltered shutter speed you want this is the one you determine form the aperture priority settings. And that is your new exposure time. Lets say for example you read in your camera settings a shutter speed of 1 sec very low-light scene then see table below if you are using a ND 3.

I am afraid no. Just to finish with the tables i will give you a most useful table where we can have exposure times for stacking filters. Be aware and use the minimum number of filters to achieve your required number of f-stop reduction.

If you want to achieve a 16 f-stops reduction use a and a 6- filter and not combinations like -6, -6, Increasing the number of ND filters can cause serious vignetting problems.

You can work both ways with this chart and every chart in this post. Then looking at this table below you see that for an hour exposure you need 16 f-stops.

You have two stops filters that is two ND 3. Then after looking at this table you go for an hour of exposure time How accurate are these tables?

In theory these tables should give you accurate exposure times.Containing more than 6, entries, CRC Standard Mathematical Tables and Formulas, 33 rd Edition continues to provide essential formulas, tables, figures and detailed descriptions.

The newest edition of this popular series also features many diagrams, group tables, and integrals that are not available online.

e-books in Mathematical Statistics category Beginning Statistics by Douglas S. Shafer, Zhiyi Zhang - timberdesignmag.com, This book is meant to be a textbook for a standard one-semester introductory statistics course for general education students.

MathType is a powerful interactive equation editor for Windows and Macintosh that lets you create mathematical notation for word processing, web pages, desktop publishing, presentations, elearning, and for TeX, LaTeX, and MathML documents.

2 Edexcel AS/A level Mathematics Formulae List: M1–M5, S1–S4 Contents Page – Issue 1 – September 20 09 14 Mechanics M1 14 There are no formulae given for M1 in addition to those candidates are expected to know.

CONTENTS page Greek Alphabet 3 Indices and Logarithms 3 Trigonometric Identities 4 Complex Numbers 6 Hyperbolic Identities 6 Series 7 Derivatives 9 Integrals Using the Method', a mathematical formula used to calculate the annual adjustment of salaries, which reflects the evolution of national civil servants' purchasing power from a sample of eight member states, the Commission had calculated that the salaries should be increased by 1.

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