Math Is Fun Forum

Why does a slide rule work?

Normal scale align values proportional to the distance to the origin. Pick up a common axis, any axis, you will find the actual distance
between 0 and 3 is one half of that between 0 and 6 and twice as much as that between 0 and 1.5, in which actual distance means the
absolute distance between two points in the real world, usually measured in cm or inch.

Now look for the "L" scale on any slide rule. Notice the mark of 0, 0.1, 0.2, 0.3,...,1.0 and the distances between any neighboring two.
Yes, equal. The "L" scale behaves like any normal scale or any normal axis- the value increase is just proportional to the actual distance
passed. So they are equivalent in nature- just redifine the distance of 1.5 inches as 1 or something like that.

Then, please recheck the "D" scale and "L" scale together. Notice the "0" on L matches the "1" on D, and "1" on L matches the "10"
or "l" on D. The L scale has a "lgx" on the right and the D scale has a "x" on the right. They show any value on L is the logarithm of
the matched value on D at the base 10. Or put in simple symbols, L=lg(D). Inversely, D=10^L. D is actually 10 raised to power L.

D is usually called as "logarithm scale". It is true that the logarithm of D, L, is linear. But a value on D is exponetial to the actual
distance from the origin, or index, or begining. So perhaps it is wiser to call it as "exponetial". On a normal linear scale, equal
actual distance represents equal difference between the two values on the two ends. However, on a exponetially increasing
scale like D, equal distance represents equal times between the two instead. When L2-L1=L4-L3 on the L, 10[sup]L2[/sup]/10[sup]L1[/sup]=10[sup]L4[/sup]/10[sup]L3[/sup]
on the D. Look and pick up 1, 2, 4 and 8 on the D and see the distances separated by them and it is true. Moving a finger from
1 to 2, then to 4, then to 8, it feels like each time it doubles. This is indeed expontially increasing.

In addition, if one distance is twice as much as another, the multiplier of two values for the former is the square of the latter.
three times, cube ... Every distance represents a multiplier of the value pair, and the ratio between two distance represents
how much the multiplier is raised. The proof is either from the property of equal length, equal multiplier
a,b,c,d |ab|=|bc|=|cd|, b=ka, c=kb, d=kc,  so d/a=k³
when |ad|=3|xy| , |xy|=|ab|, thus y/x=k while d/a=k³.

Or from the L scale
L2-L1=b(L4-L3),
10[sup]L2[/sup]/10[sup]L1[/sup]=10[sup](L4-L3)b[/sup]={10[sup]L4[/sup]/10[sup]L3[/sup]}[sup]b[/sup]

The property above is crutial when  tracking the increase of sth like stock price logarithmatical scale(exponetial scale).
However, it's not a necessity to understand the slide rule.

When the middle bar slides to the right.
Simulated Pickette's N-600ES LL Speed Rule
supposedly 1 on C matches 2.5 on D.
find somewhere C1 on C and the actual distance between 1 and C1 is exactly the same as that between 2.5 and D1,
right under C1. Since C and D are in fact the same scale. C1/1= D1/2.5 thus D1=2.5(C1).

People in power is not only the master of themselves, but also the masters of others.

And once he has planted hatred by his brutal power, he is aware the best way of his life is to stay in power.

Also, a dictator can get almost anything in the world. Materially he gains.

How about OpenOffice.org? It's like the Microsoft Office, but it is less inferior to that to be frank.

Identity, well I got my slide rule from my father. For collection purpose I asked him to find the only remaining slide rules in his office and he suceeded.

On buying slide rules, I recommend you to search the internet.
Here are the sites I've found: (click)
Site 1
Site 2

I bet if you bring a slide rule to a physics exam, the old professor will not accuse you of laziness but
instead will praise your maths talent! If s/he ever used a slide rule once.

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Fortunately I have finally found out this online simulated slide rule based on a REAL model!
Simulated Pickette's N-600ES LL Speed Rule
And the very comprehansive detailed full manual for it!!
The Manual

Here is a slide rule with  measure conversions on its back
And here is the list of all the simulated slide rules provided

        The advantages of using a slide rule

First, it is a very educational instrument. Pressing the buttons is foolproof but it deprives
you of the opportunity to learn maths. Calculating with a pencil and a paper may become
exhausting and many times disabled(2.3^1.5?). The slide rule is the means between the
two-you gotta think, but you don't need to think too much on details. To use a slide rule
well, you need to master the properties of logs, expontials, ratios, sin-cos relations, and
inverse of functions, etc. Even if you don't, under curiosity you will discover them one by
one. What the silde rule can train you is the basic properties and concepts of
intermediate-maths (compared to basics like arithmetic and advanced like calculus) and
the need to master these concepts Never Dies out. Moreover, the way it trains you is
enjoyable because the training process is implicit. You don't mean to practice those
properties intentionly like doing some specialized lgAB exercises. Instead, you get pleasure
from applying those properties each time you solve a pratical calculating problem. So you
learn the concepts with fun.

Secondly, it is good for your psychological health to use a brain-needing slide rule in
place of a pressing-buttons calculator. Scientists have discoverd that using the brain
keeps one from losing intelligence through aging. So wanna be smart? Discard the
calculator unless necessary and use a slide rule everyday! Moreover, the slide rule is only
a tool which needs the human's participation while the calculator is a highly-automatic
machine which needs only pressing buttons. Evolutionarily the calculator seems more
advanced, but only in machinary perspective. Take transportation means for example,
people used to walk, then they use bicycles, then motor-cars, and then automatic cars in
the future. The bicycles and the cars are the most successful tools for human for the first
save strenth and the latter merely eliminate the need for physical strength. However, the
car deprives people from physical exercise the way the calculator deprive people of mental
exercise, causing physical unfit. Moerover, the more advanced auto-cars get a psychological and
marketing obstacle according to the engineers working on it. They say that true,
an automatic car can avoid accidents because the computer drives, but humen prefer to be
in control-they just want to decide their own route! Feeling in control is a crutial part of
feeling good. So let the machine rules thing ain't always good for our happiness, as the
film Matrix describes. A slide rule definately can provide you with more sense of
control than a calculator, and a calculator provide you with more than a laser-beam detector
of a counter in super market! Although the productivity may increase due to more application of
machines, the individual's well-being can gain more from simplier tools.

The last additional advantage of using a slide rule is that it can be employed in
circumstances that don't require precision that much. Checking is a good occasion. And
sometimes maths formulas don't rule the real world completely. For example, in biology
field-as long as the calculation always falls short of the reality with a considerable error,
what's the point to get many digits? After all, pressing buttons causes finger sour, doesn't
it?

A special model for measure conversion:

The middle bar is apparently double informationed and the user has to take it off, flip and snug back if s/he needs a conversion on the other side.

Take the back side for example, notice inches-centimetres match. Note 1inch≈2.54cm.  Imagine matching the inch arrow to 1 and you can see that 2.54 is pointed by the centimetres arrow.

Yes, the centimetres arrow is always 2.54 times ahead of the inch arrow. Theoritically it is log2.54 far right of the inch arrow. (Imagine a L scale exists)

This rule, I guess, can beat up computers and calculators in rough business. All you need to do is read, slide it, then read again, without any power lines or misc buttons to activate certain functions.

Go to this page and click on the left pink directions to see different brands of slide ruls

Read this page carefully before you read on:
The Principle of Slide Rule Multiplication

A Complex Slide Rule with more scales provides functions beyond multiplication and division.

Look at the slide rule above. The C and D scales are the usual scale for multiplication.
the LL3 scale is the exponetial of C-that is, e^x setting the scale on C as x
the LL2 scale is then e^10x
the LL/3 scale e^-x
the LL/2 scale e^-0.1x

How to calculate 1.5^1.2:

1.5^1.2=S
ln(1.5^1.2)=lnS
1.2ln1.5=lnS

S=exp(1.2ln1.5)

So here comes the trick!
define a scale on L3 as L, and that on D as x.
move the cursor to match 1.5 on L3, L=1.5 and apparently x=ln1.5

Now we need to multiply ln1.5 on D by 1.2.
using the previous knowlege, move the middle bar so that the 1.0 on C matches the hairline now, the ln1.5 on D.
move the cursor to let the hairline match 1.2 on C, and now the hairline indicates (ln1.5)1.2 on D.

Also e^[(ln1.5)1.2] on L3! The final result!

This simple problem requires only
1 moving the cursor and match
2 moving the middle bar to match
3 moving the cursor again to match
altogether three movings and matches, skilled engineers do it very quickly, 3 seconds.
Not bad compared to electronic calculators, er?

Moreover it is independent of battery.
And it can help you better understand logs and exponetials, multiplications and divisions, as well as inversions, etc.

Other functions can be simply added on using more scales like L3, L2. And the back of the rule is also made full use of.

Click this sentence to see the table illustrating how many scales one slide rule can have
Here is the museum showing the glorious past of Slide Rules
Some funs even made his own slide rule

You can buy one or make one of your own!!

Anyone knowing the slide rule??

In fact, the exponetial form is not a necessity. Whereas the trignometric form for complex numbers is.

The ancestor has three children, two of them have a grand-children each and the rest one has none.

Zhylliolom's factorization is very interesting- from infinite entries to infinite factors!

I know it's from the rule for 1-3x+2x[sup]2[/sup]= (1-2x)(1-x) sort of thing.

But still I am surprised to find that this rule applies to a polynomial of Infinite degree as well!!!