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Slide RuleR What’s That???

Slide RuleR What’s That???. Tim Jehl – Math Dude. Contents. The Fundamental Problem Development of Logarithms Basic Properties of Logarithms History of the Slide Rule Building a slide rule with lumber, a ruler and a marker More History Scales found on Slide Rules.

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Slide RuleR What’s That???

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  1. Slide RuleRWhat’s That??? Tim Jehl – Math Dude

  2. Contents • The Fundamental Problem • Development of Logarithms • Basic Properties of Logarithms • History of the Slide Rule • Building a slide rule with lumber, a ruler and a marker • More History • Scales found on Slide Rules

  3. The fundamental problem • Adding is relatively easy. • 542 + 233 +187 = ? • Most students (and a couple of us adults) can solve this problem in a timely manner. • Multiplication is a bit more difficult • 542 x 233 x 187 = ? • This could take a bit…

  4. The fundamental problem (continued) • In ancient times (like when I went to high school), we had to do problems like this without the benefit of a calculator because, well, they didn’t exist. • The Slide Ruler was developed as a mechanical aid to assist in a variety of calculations

  5. Development of Logarithms • Logarithms were invented by the Scottish mathematician and theologian John Napier and first published in 1614. • Looking for a way of quickly solving multiplication and division problems using the much faster methods of addition and subtraction. • Napier's way invented a group of "artificial" numbers as a direct substitute for real ones, called logarithms (which is Greek for "ratio-number", apparently). • Logarithms are consistent, related values which substitute for real numbers. • They were originally developed for base e. • In1617, Henry Briggs adapted Napier's original "natural" logs to the base 10 format.

  6. Basic Properties of Logarithms • Product rule: logbAC = logbA + logbC • Ex: log464 = log44 + log416 = log4(4•16) • Quotient rule: logb(A/C) = logbA − logbC • Exlog3 27/9 = log327 - log39 = 3 – 2 = 1 • Power rule: logbAC = C(logbA) • Ex: log39² = 2log39

  7. A Logarithm Table

  8. Multiplying 9 x 8 • Look up logarithms of the factors • ln 9 = 2.197225 • ln 8 = 2.079442 • Add logarithms together • 2.197225 + 2.079442 = 4.276667 • Find the number who’s anti-log matches • ln 72 = 4.276666

  9. History of the slide ruler • In 1620, English astronomer Edmund Gunter drew a 2 foot long line with the whole numbers spaced at intervals proportionate to their respective log values. • A short time later, Reverend William Oughtredplaced two Gunter's scales directly opposite each other, and demonstrated that you could do calculations by simply sliding them back and forth.

  10. Building a slide rule with lumber, a ruler and a marker • Materials • A couple of 4-foot lengths of hardwood • Pine won’t do… it warps and won’t hold it’s shape properly • A ruler, square and permanent marker • Used for measuring lengths and marking the wood • A set of common log tables (or a handy calculator) • Time • About 30 minutes if you know what you’re doing. • All night if it’s your first try at it • Notes • The more accurate the measurements, the more accurate the instrument • Constructing a fixture to hold the slide would be nice, but might require carpentry skills

  11. Linear Scale • Let’s do mechanical addition • Mark a length of board for some fixed distance • For some bizarre reason, the marks in this demo were 89.6 cm apart • Divide the distance evenly into tenths, and attempt to mark accurately. • Total length times decimal value is the linear length to mark on your board • See table to the right

  12. Adding on the Linear Scale • Using the cleverly pre-fabricated matching board, I can add two numbers together by adding their lengths • The length of 0.2 is 17.92 • The length of 0.4 is 35.84 • The sum of those lengths is 53.76 • I can now look where these add together • The value 53.76 is the length of 0.6 (= 0.2 + 0.4) • Demo

  13. Logarithmic Scale • Let’s do mechanical multiplication • Mark a length of board for some fixed distance • For some bizarre reason, the marks in this demo were 89.6 cm apart • Divide the distance based on the logarithms from 1 to 10 • Total length times decimal value is the linear length to mark on your board • See table on right

  14. Adding on the Logarithmic Scale • Using the cleverly pre-fabricated matching board, I can add two numbers together by adding their lengths • The length of 2 is 26.97 • The length of 4 is 53.94 • The sum of those lengths is 80.91 • I can now look where these add together • The value 80.92 is the length of 8 (= 2 x 4) • Demo

  15. More History • Calculators did not appear until the mid-1970’s. • This is the sort of device your grand-parents used • This was what the Apollo mission astronauts used to do their calculations while orbiting the moon. Picket Model N600-EShttp://www.antiquark.com/sliderule/sim/virtual-slide-rule.html • Not all slide rules are straight

  16. Scales on Slide Rules (1)

  17. Scales on Slide Rules (2)

  18. Interesting Sites • Slide Rule Museum • http://www.sliderulemuseum.com/ • A digital repository for all things slide rule and other math artifacts • What can you do with a slide rule? • http://www.math.utah.edu/~pa/sliderules/ • Just what the name imples • Derek’s Virtual Slide Rule Gallery • http://www.antiquark.com/sliderule/sim/index.html • Software demos for a variety of slide rulers.

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