# Kinda like a slide rule

Discussion in 'Math' started by someonesdad, Jan 27, 2011.

Jul 7, 2009
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I wrote this a number of years ago, then forgot about it. It will be familiar to the geezers, but you young folks may want to try it out before tossing it in the garbage can.

The relevant file is scale.pdf and it will let you do many of the things you could do with a slide rule. If you have some dividers, it will let you do anything you could do with a slide rule. Now, if you think that's a relic of the past, you're right. But you should also know that it's valuable when you want to get two or three significant answers to a problem, especially when no calculator is handy.

The other file is some documentation.

• ###### scale_27Jan2011.zip
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2. ### Wendy Moderator

Mar 24, 2008
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I remember. I prefer a good scientific. Seems to me they have gotten carried away with some models though, memory should be a single keystroke, instead of three.

3. ### magnet18 Senior Member

Dec 22, 2010
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Maye if I lose every calculator in my house, and at school, and my computer...
too many little lines for me, but thanks for sharing
ill stick with my Ti-89 Titanium for now, and its 600 page manual
i generally like using old stuff, but calculators are an exception

4. ### thatoneguy AAC Fanatic!

Feb 19, 2009
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I was grabbing gas just the other day and all I could find in the car was a cheapy calc with dead batteries.

Well, other than my Droid.

I guesss one may look silly figuring out their MPG on a slide rule though. I still use mine since the "back side" has some convenient SI->IMP scales.

Does it do Hex to binary conversions? I lost my Hex slide rule.

5. ### studiot AAC Fanatic!

Nov 9, 2007
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Slide rules, calculators, pdas, computers, functions on your phone...

Fewer and fewer people these days have a 'feel' for what the correct answer to a numerical calculation should be, to the nearest deciomal point.

This doesn't really matter for mpg in your scorpion tank, but say you are a medical practitioner drawing up a drug dose.

go well

Jul 7, 2009
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The miles per gallon calculation is simple on the x cubed scale (A3). Suppose you want to know what the mpg is for 220 miles driven with 14 gallons of gas. Take a piece of paper and mark on the edge the location of the 14 and 220 points without moving the paper. Then slide one mark to the 1 position and read the miles per gallon at the other mark. This is especially fast with a pair of dividers (navigators have used that method for a long time).

It should be obvious that the method extends to any ratios a/b = c/d where you know any three of the values.

7. ### studiot AAC Fanatic!

Nov 9, 2007
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$\frac{3}{7}{\left( {\frac{{331}}{{58}}} \right)^2}\pi$

This is a simple example of what I mean. Suppose you are faced with the above calculation.

I look and say

58 is nearly 60 and 300/60 = 5 so 331/58 is 5 and a bit.

5 squared is 25 so 5 and a bit squared is say 30. But 4 times 7 is 28 so say 28 and a bit.

Cancel the 7 into the 28 goes 4

3 pi is 10 and 4 and a bit times 10 is a little over 40.

My calculator says 43.8

$\frac{3}{7}{\left( {\frac{{331}}{{58}}} \right)^2}\pi \approx 40 = 43.8$

Such a feel would have prevented a recent injection of 400μg of a drug instead of 40.

go well

Jul 7, 2009
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Studiot, your example is excellent and that method of approximate calculation that we had to do when doing things with slide rules, logs, or by hand is a tool that's still oh-so-relevant today. In various things I've written over the years, I try to show folks why this method of approximation is so valuable, regardless of whether you're doing the calculation by hand, calculator, or computer, because it can prevent you from making serious errors.

For other readers, what Studiot did is an approximate calculation to estimate the order of magnitude of the answer. The basic technique is to do the calculation with the terms rounded to one significant figure because that will lead to doing integer arithmetic, which we should all be able to do without help (if not, you need help ). Such a tool when combined with intelligent guesstimates of physical values lead to Fermi calculations, but of course people were doing such things long before Fermi's name got attached to it.

Here's another arithmetical technique that can be useful and probably isn't used as much. Suppose we want the decimal value of 96/103. Just by inspection, we know it will be around 0.93 or so (in your mind, you think if the denominator was 100, the decimal would be 0.96, but it's 3 units over 100, so we adjust the decimal fraction down 3 units). Write it as

$\frac{96}{103} = \frac{100 - 4}{100 + 3} = \frac{1 - 0.04}{1 + 0.03}$

Next, use one of the most-used approximations in technical work

$\frac{1}{1 \pm \delta} \approx 1 \mp \delta$

where $\delta << 1$. This gives

$\frac{96}{103} \approx (1 - 0.04)(1 - 0.03)$

Using simple algebra, this is 1 - 0.04 - 0.03, where we ignore the (0.04)(0.03) term because it is much smaller than the other terms. This gives us 0.93, which is off the correct answer by about 2 parts in 1000.

Such little tricks are handy when you don't have a calculator. For example, I've been up on a roof cutting plywood and need a quick calculation and such tools come to my aid.

If you want more examples of this, consult E. B. Wilson's "Introduction to Scientific Research", early 1950's (my copy is in the other room and I'm too lazy to go get it to make sure I got the reference right ).

Nov 9, 2007
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10. ### studiot AAC Fanatic!

Nov 9, 2007
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say sin(55)

Well I know that sin(x) must lie between 0 and 1, and that most of the increase is at the beginning to try sin(55) = 3/4, (0.75 is not a bad value since since sin(55) = 0.81).

Here is an old mnemonic table to help get a better value. I have always found this table fascinating.

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Jul 7, 2009
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An old saying is "never do card tricks for the people you play poker with". When I start getting too cocky about my abilities to do approximate arithmetic, I'm pretty careful not to do them in front of another person with roughly my experience in this stuff. That's because I know he'd toss me a problem like

$\sqrt{\frac{1}{tan 1234567891}}$

and I know I wouldn't be coming up with a reasonable answer too quickly.

12. ### studiot AAC Fanatic!

Nov 9, 2007
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When playing poker it is not a good idea to play by imaginary rules either.

The result is an imaginary number.

It's fairly easy to show this, without a calculator.

manually divide 1234567891 by 60 and then by 6 to yield

3249355.253

Discard the whole numbers of revolutions

253/1000 times 360 works out at around 91 degrees thus the tangent is negative and very large.

the rest follows.

I once wrote a published paper entitled "The Use of the Fifth Quadrant" for calculating latitudes and departures with angles larger than 360 on a computer or calculator.

Jul 7, 2009
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I gave the problem in radians, as I know how to handle degrees pretty easily (same as you). My only point was that a trig function like the tangent ranges over the real numbers and any approximate rounding could get you into trouble (and especially because the output could be negative, meaning the square root would become complex. To do this problem by hand with no other resources, about the only approach I can see is to do the long division to calculate 1234567891 mod 2*pi. Oh, and you have to have a few digits of pi memorized too...

14. ### studiot AAC Fanatic!

Nov 9, 2007
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I looked, but nowhere did I see this specification, so I carried on using the format already specified and in use in the thread as would be convention.

Anyway such is missing the point. I admired your nomograms and added a point of my own.
C'est assez.

go well