John Napier: The Rationalization of Arithmetic

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John Napier The Rationalization of Arithmetic

• Lyndsi Monjon
• October 30, 2007

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Who is John Napier?

• Born in 1550
• Died in 1617
• What he considered his major work for humanity
• His campaign against Roman Catholic supremacy in
  Scotland and later in England
• Fanatical Scottish Presbyterian
• Follower of John Knox

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More about Napier

• Most important contributions not to religion but
  to Mathematics
• Many regret him not making number-study his main
  work
• Wanted to free the minds of those bedeviled by
  numbers
• Wanted to abolish arithmetic altogether and
  replace it with a rational system so simple that
  even a child could do it

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Just a little bit more!

• Probably the first if the public benefactors who
  took up the study of math
• Babbage was the greatest

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Books By John Napier

• A Plaine Discoverie of the Whole Revelation of
  Saint John
• Took 27 years to write
• Identified the pope as the antichrist
• Miraculous Canon of Logarithms
• Calculations took 20 years for this book
• Rabdologiae
• In Latin
• Bones or divining rods

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Napiers Bones

• Also known as Divining Rods
• Take most of the pain our of multiplying and
  dividing
• Can also be used to find roots and powers
• All you need to know
• the Arabic numbers 0 to 9
• What Arabic notation means
• How to add and subtract
• Can replace tables

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How to make you own Bones!

• Can be made using strips of papers or pieces of
  wood
• Need 10 strips about 5 inches long and half an
  inch wide
• Draw lines and write numbers on the strips
• Each strip represents a line on the
  multiplication table
• End Rod has the Roman numerals I to IX written
  on it

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In-Class Activity

• Going to use Bones to multiply and divide some
  problems
• Use the strips provided by me and make your own
  bones
• Solve the following problems using the bones

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Multiplication with Bones

• Multiply 643 by 249
• Begin by removing the rods for 6,4,3
• Also remove the end rod which serves as a marker
• Lay the rods side by side

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More Multiplication

• Copy down the results for multiplying 643 by 9
  units, then by 4 tens and then by 2 hundreds
• Now add all the results
• What do we get?
• 160,107

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Division With Bones

• Done in much the same way
• Difference
• Look at the bones for results
• Dont start from the marker , we finish there
• Do all the same steps in reverse

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Division Example

• Divide 160,107
• Look for a number equal to or less than 160,107
• Subtract this number from 160,107
• Now look for a number equal to or less than the
  difference
• And so on.

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Advantages of method

• Learn to dispense with number tables
• We think about numbers in a new way, not taking
  them for granted
• We cease to fuss about carries, these being
  done quite automatically

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Why were Bones helpful?

• May not seem like a tremendous advance
• Better than mental math
• Example on Page 169

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The Invention of Logarithms

• Realized that all s could be thought of as
  being one continuous series
• Same rules applies throughout whole numbers,
  fractions, and mixed numbers
• Napier discovered the relation between two
  well-known series of numbers
• The arithmetic series and the geometric series

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Logarithms Cont.

• Discovered that one series could be written in
  terms of the other
• A geometric series could be written as a
  arithmetic series and vice-versa
• Use 2 as an example
• Two times two time two is eight
• Two to the power of three is eight
• Logarithm eight to the base two is three

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Calculating Logarithms

• Tried many ways of calculating powers to bases
• Would calculate numbers such as 2 to the power
  1000, then count the number of places in the
  answer to get 3011 places, subtract 1 to get
  3010and that is the logarithm of 2 that is
  correct to four decimal places (see page 171)

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More Calculations

• Usually worked to 7 decimal places but soon
  decided there was no future in this procedure
• Set out to find other methods
• Arithmetic and geometric mean

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Starting Principles

• Start with any two numbers and their known
  logarithms
• For a new and unknown log, work out the
  arithmetic mean of the two earlier logarithms
• For the unknown number, work out the geometric
  mean of the numbers

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Arithmetic Mean

• The arithmetic mean of any two numbers is found
  by adding them and dividing the answer by 2
• For example 4 and 16
• 41620
• 20/2 10

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Geometric mean

• Used for scientific purposes
• How to find multiply the two numbers and take
  the square roots of the results
• For Example 4 and 16
• 4x1664
• Square root of 648

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A few more facts!

• Napiers work was immediately accepted
• Useful to astronomers, ships captains,
  scientists, and engineers
• His logs were used until the inventions of
  electric calculators and electronic computers

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More facts..

• Until late 1960s all secondary school children
  would have been familiar with their books about
  tables
• Calculators and computers owe much to Napier and
  his Bones

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Resources

• Websites
• http//www.nls.uk/scientists/images/results/napier
  .gif
• http//www.forpd.ucf.edu/newsletter/question.mark.
  jpg
• http//www.in.gov/dcs/booksforyouth/images/booksta
  ck.jpg
• http//www.jjhc.info/images/napierjohn1617.jpg
• http//gwydir.demon.co.uk/jo/numbers/machine/bones
  .gif
• http//www.gutenberg.org/files/20196/20196-h/image
  s/napier.jpg
• http//www.cam.k12.il.us/ms/8th/jones/Multiplicati
  on.jpg
• http//www.cam.k12.il.us/ms/8th/jones/Division.jpg
  
• Books
• The Story of Numbers

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Any Questions???