Difference between revisions of "Logarithmic Scale and the Slide Rule"

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[[Image:Henry briggs.png|right|230px|border]]
 
[[Image:Henry briggs.png|right|230px|border]]
  
[http://en.wikipedia.org/wiki/John_Napier John Napier]'s invention of logarithms was really out of necessity. With the increasing amount of data and advancement in the natural science, mathematician and scientists were constantly occupied by huge stack of calculations which not only took up immense amount of time but also impeded further progress. This is especially true for the study of astronomy. Astronomers usually had to observe and collect data for many hours at night and then perform laboriously repetitive calculations (which was very prone to mistakes and errors) during the day time. This presented an imperative need for a new way of calculations that was fast and reliable. John Napier quietly stepped up for the challenge and excited the community of scientists and mathematician in 1617 with his publication of '''''Mirifici Logarithmorum Canonis Descriptio'''''in which he introduced the Logarithms and came up with the logarithms table. The history and development is a fascinating topic and hence I have dedicated an individual page [[The Logarithms, Its Discovery and Development]] to it. In it there are also more detailed and comprehensive information on logarithmic scale. Now, the important things about logarithms are some of its very useful properties some of which are known by even 9th graders, namely <math>log(xy) = log(x) + log(y) </math> and <math>log \frac{x}{y} = log(x) - log(y)</math>. So the that means if we have an operation in the linear scale such as <math>x \times y</math>, we can transform it into <math>x+y</math> on the logarithmic scale. [http://en.wikipedia.org/wiki/Henry_Briggs_(mathematician) Henry Briggs], England's most eminent mathematician of the time, traveled to meet Napier and introduced himself: "My Lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto astronomy...I wonder why nobody else found it out before, when, now being known, it appears so easy."  
+
[http://en.wikipedia.org/wiki/John_Napier John Napier]'s invention of logarithms was really out of necessity. With the increasing amount of data and advancement in the natural science, mathematician and scientists were constantly occupied by huge stack of calculations which not only took up immense amount of time but also impeded further progress. This is especially true for the study of astronomy. Astronomers usually had to observe and collect data for many hours at night and then perform laboriously repetitive calculations (which was very prone to mistakes and errors) during the day time. This presented an imperative need for a new way of calculations that was fast and reliable. John Napier quietly stepped up for the challenge and excited the community of scientists and mathematician in 1617 with his publication of '''''Mirifici Logarithmorum Canonis Descriptio''''' in which he introduced the Logarithms and came up with the logarithms table. The history and development is a fascinating topic and hence I have dedicated an individual page [[The Logarithms, Its Discovery and Development]] to it. In it there are also more detailed and comprehensive information on logarithmic scale. Now, the important things about logarithms are some of its very useful properties some of which are known by even 9th graders, namely <math>log(xy) = log(x) + log(y) </math> and <math>log \frac{x}{y} = log(x) - log(y)</math>. So the that means if we have an operation in the linear scale such as <math>x \times y</math>, we can transform it into <math>x+y</math> on the logarithmic scale. [http://en.wikipedia.org/wiki/Henry_Briggs_(mathematician) Henry Briggs], England's most eminent mathematician of the time, traveled to meet Napier and introduced himself: "My Lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto astronomy...I wonder why nobody else found it out before, when, now being known, it appears so easy."  
  
  
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==='''Taking Square Root'''===
 
==='''Taking Square Root'''===
  
Taking square root is the reverse process of squaring. Hence we are choosing a number on the B scale and read a the answer on the C scale. However, there is a trick to this process. B scale is divided into two halves. The left half is to find the square root of numbers with odd numbers of digits, i.e. 1, 345, 48529, etc. The right half is used to find the square root of numbers with even numbers of digits. But why is that you may ask. It is easy to see. Note that B scale is divided into two identical parts, with left side running from 1 to 10 and the right side running from 10 to 100. Since the full length of the B scale is denoted as 100, then the left side can denote <math>x</math> with <math>1 \leqslant x < 10</math>, or <math>1 \times 100 \leqslant x < 10 \times 100</math>, or <math>1 \times 10000 \leqslant x < 10 \times 10000</math>, etc and the right side can denote <math>y</math> with <math>10 < y \leqslant 100</math>, or <math>10 \times 100 < y \leqslant 100 \times 100</math>, or <math>10 \times 10000 < y \leqslant 100 \times 10000</math>. That is to say that we need the left side to find numbers with odd numbers of digits and the right to find numbers with odd number of digits.
+
Taking square root is the reverse process of squaring. Hence we are choosing a number on the B scale and read a the answer on the C scale. However, there is a trick to this process. B scale is divided into two halves. The left half is to find the square root of numbers with odd numbers of digits, i.e. 1, 345, 48529, etc. The right half is used to find the square root of numbers with even numbers of digits. But why is that you may ask. It is easy to see. Note that B scale is divided into two identical parts, with left side running from 1 to 10 and the right side running from 10 to 100. Since the full length of the B scale is denoted as 100, then the left side can denote <math>x</math> with <math>1 < x < 10</math>, or <math>1 \times 10^2 < x < 10 \times 10^2</math>, or <math>1 \times 10^4 < x < 10 \times 10^4</math>, etc and the right side can denote <math>y</math> with <math>10 < y < 100</math>, or <math>10 \times 10^2 < y , 100 \times 10^2</math>, or <math>10 \times 10^4 < y < 100 \times 10^4</math>. That is to say that we need the left side to find numbers with odd numbers of digits and the right to find numbers with odd number of digits.
  
 
Example: <math>\sqrt {4850}</math>.
 
Example: <math>\sqrt {4850}</math>.
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==='''Taking Cube Root'''===
 
==='''Taking Cube Root'''===
The K scale is separated into three identical section. Left section runs from 1 to 10; the middle section runs from 10 to 100; the right section runs from 100 to 1000. The same argument follows from the Taking Square Root section. Since the full length of the K scale is denoted as 1000, then the left section can denote <math>x</math> with <math>1 < x < 10</math>, or <math>1 \times 1000 < x < 10 \times 1000</math>, or <math>1 \times 1000000 < x < 10 \times 1000000</math>, etc, the middle section can denote <math>y</math> with <math>10 < y < 100</math>, or <math>10 \times 1000 < y < 100 \times 1000</math>, or <math>10 \times 1000000 < y < 10 \times 1000000</math>, etc and the right side can denote <math>z</math> with <math>100 < z < 1000</math>, or <math>100 \times 1000 < z < 1000 \times 1000</math>, or <math>100 \times 1000000 < z < 1000 \times 1000000</math>. Therefore, we use the left section to find numbers with 1, 4, 7...digits, the middle section to find numbers with 2, 5, 8...digits and the right section to find numbers with 3, 6, 9...digits.  
+
The K scale is separated into three identical section. Left section runs from 1 to 10; the middle section runs from 10 to 100; the right section runs from 100 to 1000. The same argument follows from the Taking Square Root section. Since the full length of the K scale is denoted as 1000, then the left section can denote <math>x</math> with <math>1 < x < 10</math>, or <math>1 \times 10^3 < x < 10 \times 10^3</math>, or <math>1 \times 10^6 < x < 10 \times 10^6</math>, etc, the middle section can denote <math>y</math> with <math>10 < y < 100</math>, or <math>10 \times 10^3 < y < 100 \times 10^3</math>, or <math>10 \times 10^6 < y < 100 \times 10^6</math>, etc and the right side can denote <math>z</math> with <math>100 < z < 1000</math>, or <math>100 \times 10^3 < z < 1000 \times 10^3</math>, or <math>100 \times 10^6 < z < 1000 \times 10^6</math>. Therefore, we use the left section to find numbers with 1, 4, 7...digits, the middle section to find numbers with 2, 5, 8...digits and the right section to find numbers with 3, 6, 9...digits.  
  
  

Revision as of 14:03, 29 June 2010

Inprogress.png
150 Extra Engineers
150 extra engineers thumb.jpg
Field: Algebra
Image Created By: IBM
Website: Global Nerdy --- Tech Evangelist Joey deVilla on software development, tech news and other nerdy stuff

150 Extra Engineers

This was a picture of an IBM advertisement back in 1953.


Basic Description

This was a picture of an IBM advertisement back in 1953 during an age when "computer" referred to human who did calculations and computation solely. The advertisement boasted that


"An IBM Electronic Calculator speeds through thousands of intricate computations so quickly that on many complex problems it's just like having 150 Extra Engineers. No longer must valuable engineering personnel...now in critical shortage...spend priceless creative time at routine repetitive figuring. Thousands of IBM Electronic Business Machines...vital to our nation's defense...are at work for science, industry and armed forces, in laboratories, factories and offices, helping to meet urgent demands of greater production."


Doesn't one just want to go back to that age when engineers were in "critical shortage" and "If you had a degree, you had a job. If you didn't have a job it's because you didn't want one." Oh well...


Notice that in the picture, in addition to the lack of women engineers back in those days, the male engineers almost all have receding hairlines (possibly due to constant overwork and the disproportional allocations of oxygen and other nutrients between brain cells and scalp skin). It was also typical of the male engineer to be defined by a standard uniform: "white shirt, narrow tie, pocket protector and slide rule." I need to do footnote here. But I don't know how. Yeah! Pocket protector! I did not know what it is until I saw the picture in the Scientific American Article, When Slide Rule Ruled, by Cliff Stoll, who is a brilliant engineer, physicist and educator, and whose TED talk is both hilarious and inspiring. However, it is neither Cliff Stoll, nor the pocket protector that I want to talk about. (It is just a thing that, despite making the engineers look very geeky and very attractive in author's opinion, protected their shirt pockets from being worn out so quickly due to the numerous engineering essentials they had in there.) Rather, it is the ruler looking stuff that is the engineer's hand that I want to talk about. The Slide Rule.


Born into the digital and automatic age, hardly anyone of our generation ever gets to know anything that is analog or manual. Not only that, we hate to have anything to do with stuff that is analog. We have been cultured to base our life and happiness on gadgets that allow us to access the world and all the information with a finger tip. We don't realize that in fact, things that are analog laid the foundation for our modern society and ushered us into the digital age. Slide Rule is one of "those analog things". In the pre-computer age, it was one of those ingenious tools that enabled engineers, mathematicians and physicists to do calculations, and because of it, we have witnessed the erections of skyscrapers, harnessing of hydroelectric power, building of subways, advancement in the aeronautics, beyond that, it helped us won the WWII, sent astronauts into the space (as a matter of fact, a Picket 600-T Dual base Log Log slide rule was carried by the Aopollo crew to the space and moon should the on-board computers fail) and eventually, hastened its own demise.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Algebra

Logarithm and logarithmic Scale

This section is not to preach you on logarithms since you are we [...]

Logarithm and logarithmic Scale

This section is not to preach you on logarithms since you are well familiar with its properties. Rather, I will simply introduce logarithmic scale which is less mentioned and understood but widely used and essential to the construction of a slide rule.

We are familiar with the linear scale, namely the equally divided scale, from the number 0 to whichever number as shown in scale A. We then use the identity x=log_22^x to relabel scale A to produce scale B. Then what we do to transform the linear scale to logarithmic scale is simply to take away the log_2, turning scale B to scale C. Calculating the numbers on scale C, we end up with scale D.

Sliderule001.png


Now, notice that the given numbers are equally divided but the differences between consecutive numbers are not the same. Our goal is to produce a logarithmic scale that has all the integers from 1 to 8. For example, if we want the number 3 on scale D, there will be a number x on scale A that corresponds to the number 3 on scale D. We have <template>AlignEquals




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About the Creator of this Image

International Business Machines (IBM) is a multinational computer, technology and IT consulting corporation headquartered in Armonk, North Castle, New York, United States. IBM is the world's fourth largest technology company and the second most valuable by global brand (after Coca-Cola). IBM is one of the few information technology companies with a continuous history dating back to the 19th century. IBM manufactures and sells computer hardware and software (with a focus on the latter), and offers infrastructure services, hosting services, and consulting services in areas ranging from mainframe computers to nanotechnology.


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