Slide Rule Calculations by Example

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Introduction

This isn't really a tutorial, it's more of a self-guided demo. This page gives numeric examples of the basic calculations that a slide rule can do. Just follow the step-by-step instructions and you will be amazed by the power and versatility of the venerable slipstick. Just start up a virtual slide rule (opens in new window) and start calculating.



Multiplication

Simple Multiplication (uses C and D scales)

Example: calculate 2.3 × 3.4

'Wrap-Around' Multiplication (uses C and D scales)

Example: calculate 2.3 × 4.5

Folded-Scale Multiplication (uses C, D, CF and DF scales)

Example: calculate 2.3 × 4.5

Multiplication by π (uses D and DF scales)

Example: calculate 123 × π

Division

Simple Division (uses C and D scales)

Example: calculate 4.5 / 7.8

Reciprocal (uses C and CI scales)

Example: calculate the reciprocal of 7.8, or 1/7.8

Trigonometry

Sin(x) for angles between 5.7° and 90° (uses S and C scales)

Example: calculate sin(33°)

Cos(x) for angles between 5.7° and 90° (uses S and C scales)

Example: calculate cos(33°).

Tan(x) for angles between between 5.7° and 45° (uses T and C scales)

Example: calculate tan(33°).

Tan(x) for angles between between 45° and 84° (uses backward T and CI scale)

Example: calculate tan(63°).

Tan(x) for angles between between 45° and 84° (uses forward T and C scale)

Example: calculate tan(63°).

Sin(x) and tan(x) for angles between 0.6° and 5.7°  (using the ST and C scales)

In this range, the sin and tan functions are very close in value, so a single scale can be used to calculate both.

Example: calculate sin(1.5°)

Sin(x) and tan(x) for other small angles (using C and D scales)

For small angles, the sin or tan function can be approximated closely by the equation:
sin(x) = tan(x) = x / (180/π)  =  x / 57.3.
Knowing this, the calculation becomes a simple division. This technique can also be used on rules without an ST scale.

Example: calculate sin(0.3°)

Squares and Square Roots

Square (uses C and B scales)

Example: calculate 4.7 2

Square Root (uses C and B scales)

Example: calculate √4500

Cubes and Cube Roots

Cube  (uses D and K scales)

Example: calculate 4.73

Cube Root (uses D and K scales)

Example: calculate 3√4500

Log-Log Scales

Log-log scales are used to raise numbers to powers. Unlike many of the other scales, log-log scales can't be learned simply be memorizing a few rules. It is necessary to actually understand how they work. These examples are intended to gradually introduce you to the concepts of log-log scales, so you gain that understanding. Hopefully, the power of 10 examples don't bore you, as they lay the foundation for later examples.

Since there are many slight variations of log-log scales on different slide rules, I'll refer only to the scales found on the Pickett N3, Pickett N600 and Pickett N803 slide rules (among others). If you want to view a virtual N3, click here, if you want a virtual N600, click here (opens in a new window.)

Another interesting aspect of  LL scales is that the decimal point is "placed." That is, you don't have to figure out afterwards where the decimal point belongs in your result. The disadvantage to this is that LL scales are limited in the numbers they can calculate. Typically, the highest result you can get is about 20,000, and the lowest is 1/20,000 or 0.00005. One exception to this is the Picket N4 (virtual here), which goes up to 1010.

Raising a Number to Powers of 10 (N>1)

To raise a number to the power of 10, simply move the cursor to the number and look at the next highest LL scale. (These examples are for numbers greater than 1. )

Example: calculate 1.35 10      (uses LL2 and LL3 scales)
Example: calculate 1.04 100     (uses LL1 and LL3 scales)
Example: calculate 1.002 1000  (uses LL0 and LL3 scales)
Example: calculate sequential powers of ten of 1.002 (uses LL0 to LL3 scales)

Raising a Number to Powers of 10 (N<1)

The reciprocals of the LL scales are the -LL scales. They work the same way, but you have to make sure that you look for the answer on a -LL scale.

Example: calculate 0.75 10     (uses -LL2 and -LL3 scales)

Finding the 10th Root

As you've seen in the previous examples, to raise a number to the 10th power, you simply look at the adjacent number on the next highest LL scale. To find a tenth root, you look at the adjacent number on the next lowest LL scale. Remember also that finding the tenth root is the same as raising a number to the power of 0.1.

Example: calculate 10√5, or 5 0.1     (uses LL2 and LL3 scales)
Example: calculate 100√0.15, or 0.15 0.01   (uses -LL3 and -LL1 scale)

Arbitrary Powers (Staying on Same LL Scale)

Occasionally, depending on the numbers, it is possible to calculate a power without switching scales.

Example: calculate 9.1 2.3         (uses LL3 scale)
Example: calculate 230 0.45      (uses LL3 scale)
Example: calculate 0.78 3.4      (uses -LL2 scale)
Example: calculate 0.78 0.45      (uses -LL2 scale)

Arbitrary Powers (Switching LL Scales)

One of the rules of exponents is that (A B ) C is equal to A B x C. We can use this fact, along with our knowledge of powers of ten, to calculate arbitrary powers.

Example: calculate 1.9 2.5        (uses LL2 and LL3 scales)
Example: calculate 12 0.34      (uses LL3 and LL2 scales)
Example: calculate 0.99 560    (uses -LL1 and -LL3 scales)

Log-Log Approximations

In general, LL scales don't handle numbers extremely close to 1, such as 1.001 or 0.999. This is not a problem because there is an accurate approximation for numbers in this range. In general, if you have a very small number 'd', then:

    (1 + d) p = 1 + d p

Example: calculate 1.00012 34   (uses C and D scales)
Example: calculate 0.99943 21    (uses C and D scales)


Copyright (c) 2005 Derek Ross

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