Editing Slide rule (section)

==Operation==

=== Logarithmic scales ===
The following [[logarithmic identities]] transform the operations of multiplication and division to addition and subtraction, respectively:

<math display="block">\log(x \times y) = \log(x) + \log(y) \, ,</math><math display="block">\log(x/y) = \log(x) - \log(y) \, .</math> 

==== Multiplication ====
With two logarithmic scales, the act of positioning the top scale to start at the bottom scale's label for <math>x</math> corresponds to shifting the top logarithmic scale by a distance of <math>\log(x)</math>. This aligns each top scale's number <math>y</math> at offset <math>\log(y)</math> with the bottom scale's number at position <math>\log(x) + \log(y)</math>. Because <math>\log(x) + \log(y) = \log(x \times y)</math>, the mark on the bottom scale at that position corresponds to <math>x \times y</math>. With {{Math|1=x=2}} and {{Math|1=y=3}} for example, by positioning the top scale to start at the bottom scale's {{Math|2}}, the result of the multiplication {{Math|1=3×2=6}} can then be read on the bottom scale under the top scale's {{Math|3}}:

[[File:Slide rule example2 with labels.svg|550px]] 

While the above example lies within one decade, users must mentally account for additional zeroes when dealing with multiple decades. For example, the answer to {{Math|1=7×2=14}} is found by first positioning the top scale to start above the 2 of the bottom scale, and then reading the marking 1.4 off the bottom two-decade scale where {{Math|7}} is on the top scale:

[[File:Slide Rule Duplex.svg|A duplex slide rule set to multiply any 2 by any number up to 50.]]

But since the {{Math|7}} is above the ''second'' set of numbers that number ''must'' be multiplied by {{Math|10}}. Thus, even though the answer directly reads {{Math|1.4}}, the correct answer is {{Math|1=1.4×10 = 14}}.

For an example with even larger numbers, to multiply {{Math|88×20}}, the top scale is again positioned to start at the {{Math|2}} on the bottom scale. Since {{Math|2}} represents {{Math|20}}, all numbers in that scale are multiplied by {{Math|10}}. Thus, any answer in the ''second'' set of numbers is multiplied by {{Math|100}}. Since {{Math|8.8}} in the top scale represents {{Math|88}}, the answer must additionally be multiplied by {{Math|10}}. The answer directly reads {{Math|1.76}}. Multiply by {{Math|100}} and then by {{Math|10}} to get the actual answer: {{Math|1,760}}.

In general, the {{Math|1}} on the top is moved to a factor on the bottom, and the answer is read off the bottom where the other factor is on the top. This works because the distances from the {{Math|1}} mark are proportional to the logarithms of the marked values.

==== Division ====
The illustration below demonstrates the computation of {{Math|{{sfrac|5.5|2}}}}. The {{Math|2}} on the top scale is placed over the {{Math|5.5}} on the bottom scale. The resulting quotient, {{Math|2.75}}, can then be read below the top scale's {{Math|1}}:

[[File:slide rule example4.svg]]

There is more than one method for doing division, and the method presented here has the advantage that the final result cannot be off-scale, because one has a choice of using the {{Math|1}} at either end.

With more complex calculations involving multiple factors in the numerator and denominator of an expression, movement of the scales can be minimized by alternating divisions and multiplications. Thus {{Math|{{sfrac|5.5×3|2}}}} would be computed as {{Math|{{sfrac|5.5|2}}×3}} and the result, {{Math|8.25}}, can be read beneath the {{Math|3}} in the top scale in the figure above, without the need to register the intermediate result for {{Math|{{sfrac|5.5|2}}}}.

==== Solving Proportions ====

Because pairs of numbers that are aligned on the logarithmic scales form constant ratios, no matter how the scales are offset, slide rules can be used to generate equivalent fractions that solve proportion and percent problems.

For example, setting 7.5 on one scale over 10 on the other scale, the user can see that at the same time 1.5 is over 2, 2.25 is over 3, 3 is over 4, 3.75 is over 6, 4.5 is over 6, and 6 is over 8, among other pairs. For a real-life situation where 750 represents a whole 100%, these readings could be interpreted to suggest that 150 is 20%, 225 is 30%, 300 is 40%, 375 is 50%, 450 is 60%, and 600 is 80%.

===Other scales===
{{Main article|Slide rule scale#Scales}}

[[File:Pickett slide rule.jpg|frame|center|This slide rule is positioned to yield several values: From C scale to D scale (multiply by 2), from D scale to C scale (divide by 2), A and B scales (multiply and divide by 4), A and D scales (squares and square roots).]]
In addition to the logarithmic scales, some slide rules have other mathematical [[function (mathematics)|functions]] encoded on other auxiliary scales. The most popular are [[trigonometric function|trigonometric]], usually [[sine]] and [[tangent (trigonometric function)|tangent]], [[common logarithm]] (log{{small|10}}) (for taking the log of a value on a multiplier scale), [[natural logarithm]] (ln) and [[exponential function|exponential]] (''e<sup>x</sup>'') scales. Others feature scales for calculating [[hyperbolic functions]]. On linear rules, the scales and their labeling are highly standardized, with variation usually occurring only in terms of which scales are included and in what order.<ref>{{Cite web|last=Marcotte, Ph.D.|first=Eric|date=2002|title=Eric's Types of Slide Rules and their Scales|url=https://www.sliderule.ca/scales.htm|access-date=2021-07-13|website=www.sliderule.ca}}</ref>
{| class="wikitable"
|-
| C, D || '''single-decade logarithmic''' scales, single sections of the same length, used together for multiplication and division, and generally one of them is combined with another scale for other calculations
|-
| A, B || '''two-decade logarithmic''' scales, two sections each of which is half the length of the C and D scales, used for finding square roots and squares of numbers
|-
| K || '''three-decade logarithmic''' scale, three sections each of which is one third the length of the C and D scales, used for finding cube roots and cubes of numbers
|-
| CF, DF || '''folded''' versions of the C and D scales that start from '''pi''' ([[Pi|π]]) rather than from unity; these are convenient in two cases. First when the user guesses a product will be close to 10 and is not sure whether it will be slightly less or slightly more than 10, the folded scales avoid the possibility of going off the scale. Second, by making the start π rather than the square root of 10, multiplying or dividing by π (as is common in science and engineering formulas) is simplified.
|-
| CI, DI, CIF, DIF || '''inverted''' scales running from right to left, used to simplify [[Multiplicative inverse|reciprocal]] ({{Fraction|1|x}}) steps
|-
| S || used for finding '''sines''' and '''cosines''' on the C (or D) scale
|-
| T, T1, T2 || used for finding '''tangents''' and '''cotangents''' on the C and CI (or D and DI) scales
|-
| R1, R2 || '''square root scales''' – setting the cursor to any value <math>r</math> on R1 or R2, find <math>{\pi}r^2</math> ([[area of a circle]] of radius <math>r</math>) under the cursor on the DF scale
|-
| ST, SRT || used for '''sines''' and '''tangents''' of '''small angles''' and '''degree–radian''' conversion
|-
| Sh, Sh1, Sh2 || used for finding '''hyperbolic sines''' on the C (or D) scale
|-
| Ch || used for finding '''hyperbolic cosines''' on the C (or D) scale
|-
| Th || used for finding '''hyperbolic tangents''' on the C (or D) scale
|-
| L || '''linear scale''' used for addition, subtraction, and (along with the C and D scales) for finding base-10 logarithms and powers of 10
|-
| LL0N (or LL/N) and LLN || '''log-log folded''' <math>e^{-x}</math> and <math>e^x</math> scales, for working with logarithms of any base and arbitrary exponents. 4, 6, or 8 scales of this type are commonly seen.
|-
| Ln || '''linear scale''' used along with the C and D scales for finding natural (base <math>e</math>) logarithms and <math>e^x</math>
|-
|P
|'''Pythagorean''' scale of <math>\sqrt{1-x^2}</math> to (1) solve the [[Pythagorean theorem]] and (2) to accurately determine cosine for small angles (with the S scale)
|}

{| style="width:150px; font-size:90%; border:1px solid #ccc; padding:4px; background:#f9f9f9;"
|-
|
{| style="border-collapse: collapse;border-spacing:0;padding:2px;"
|-
| [[File:Slide rule scales front.jpg|300px]]
| [[File:Slide rule scales back.jpg|300px]]
|}
|-
| style="text-align: left" | The scales on the front and back of a [[Keuffel and Esser]] (K&E) 4181-3 slide rule
|}

====Roots and powers====
There are single-decade (C and D), double-decade (A and B), and triple-decade (K) scales. To compute <math>x^2</math>, for example, locate x on the D scale and read its square on the A scale. Inverting this process allows square roots to be found, and similarly for the powers 3, 1/3, 2/3, and 3/2. Care must be taken when the base, x, is found in more than one place on its scale. For instance, there are two nines on the A scale; to find the square root of nine, use the first one; the second one gives the square root of 90.

For <math>x^y</math> problems, use the LL scales. When several LL scales are present, use the one with ''x'' on it. First, align the leftmost 1 on the C scale with x on the LL scale. Then, find ''y'' on the C scale and go down to the LL scale with ''x'' on it. That scale will indicate the answer. If ''y'' is "off the scale", locate <math>x^{y/2}</math> and square it using the A and B scales as described above. Alternatively, use the rightmost 1 on the C scale, and read the answer off the next higher LL scale. For example, aligning the rightmost 1 on the C scale with 2 on the LL2 scale, 3 on the C scale lines up with 8 on the LL3 scale.

To extract a cube root using a slide rule with only C/D and A/B scales, align 1 on the B cursor with the base number on the A scale (taking care as always to distinguish between the lower and upper halves of the A scale). Slide the slide until the number on the D scale which is against 1 on the C cursor is the same as the number on the B cursor which is against the base number on the A scale. (Examples: A 8, B 2, C 1, D 2; A 27, B 3, C 1, D 3.)

====Roots of quadratic equations====
[[Quadratic equations]] of the form <math>ax^2 + bx + c = 0</math> can be solved by first reducing the equation to the form <math>x^2 - px + q = 0</math> (where <math>p = -b/a</math> and <math>q = c/a</math>), and then aligning the index ("1") of the C scale to the value <math>q</math> on the D scale. The cursor is then moved along the rule until a position is found where the numbers on the CI and D scales add up to <math>p</math>. These two values are the roots of the equation.

====Future value of money====
The LLN scales can be used to compute and compare the cost or return on a fixed rate loan or investment.
The simplest case is for continuously compounded interest. Example: Taking D as the interest rate in percent,
slide the index (the "1" at the right or left end of the scale) of C to the percent on D. The corresponding value on LL2 directly below the index will be the multiplier for 10 cycles of interest (typically years). The value on LL2 below 2 on the C scale will be the multiplier after 20 cycles, and so on.

====Trigonometry====
The S, T, and ST scales are used for trig functions and multiples of trig functions, for angles in degrees.

For angles from around 5.7 up to 90 degrees, sines are found by comparing the S scale with C (or D) scale. (On many closed-body rules the S scale relates to the A and B scales instead and covers angles from around 0.57 up to 90 degrees; what follows must be adjusted appropriately.) The S scale has a second set of angles (sometimes in a different color), which run in the opposite direction, and are used for cosines. Tangents are found by comparing the T scale with the C (or D) scale for angles less than 45 degrees. For angles greater than 45 degrees the CI scale is used. Common forms such as <math>k\sin x</math> can be read directly from ''x'' on the S scale to the result on the D scale, when the C scale index is set at&nbsp;''k''. For angles below 5.7 degrees, sines, tangents, and radians are approximately equal, and are found on the ST or SRT (sines, radians, and tangents) scale, or simply divided by 57.3 degrees/[[radian]]. Inverse trigonometric functions are found by reversing the process.

Many slide rules have S, T, and ST scales marked with degrees and minutes (e.g. some Keuffel and Esser models (Doric duplex 5" models, for example), late-model Teledyne-Post Mannheim-type rules). So-called ''decitrig'' models use decimal fractions of degrees instead.

====Logarithms and exponentials====
Base-10 logarithms and exponentials are found using the L scale, which is linear. Some slide rules have a Ln scale, which is for base e. Logarithms to any other base can be calculated by reversing the procedure for calculating powers of a number. For example, log2 values can be determined by lining up either leftmost or rightmost 1 on the C scale with 2 on the LL2 scale, finding the number whose logarithm is to be calculated on the corresponding LL scale, and reading the log2 value on the C scale.

====Addition and subtraction====
Addition and subtraction aren't typically performed on slide rules, but is possible using either of the following two techniques:<ref>{{cite web|url=https://www.antiquark.com/2005/01/slide-rule-tricks.html|title=AntiQuark: Slide Rule Tricks|work=antiquark.com}}</ref>

# Converting addition and subtraction to division (required for the C and D or comparable scales):
#* Exploits the [[Identity (mathematics)|identity]] that the quotient of two variables plus (or minus) one times the divisor equals their sum (or difference):<math display="block">\begin{align}
\left(\frac{x}{y} + 1\right) y &= x + y \, \text{ (addition)}, \\
\left(\frac{x}{y} - 1\right) y &= x - y \, \text{ (subtraction)}.
\end{align}</math>
#* This is similar to the addition/subtraction technique used for high-speed electronic circuits with a [[logarithmic number system]] in specialized computer applications like the [[Gravity Pipe]] (GRAPE) supercomputer and [[hidden Markov models]].
# Using a linear L scale (available on some models):
#* After sliding the cursor right (for addition) or left (for subtraction) and returning the slide to 0, the result can be read.

===Generalizations===
[[File:Cikk-skalak-r-en.gif|thumb|alt=|Quadratic and reciprocal scales]]
Using (almost) any strictly [[Monotonic function|monotonic scales]], other calculations can also be made with one movement.<ref>{{cite journal |last=Istvan |first=Szalkai |title=General Two-Variable Functions on the Slide Rule |url=http://www.oughtred.org/jos/pages/JOS_2018_Vol_27_1_TOC.jpg |journal=Journal of the Oughtred Society|volume=27|issue=1|pages=14–18 |year=2016 |arxiv=1612.03955}}</ref><ref>{{cite arXiv |last=Istvan |first=Szalkai |title=General Two-variable Functions on the Slide-rule |eprint=1612.03955 |class=math.HO|year=2016}}</ref> For example, reciprocal scales can be used for the equality <math>\frac{1}{x} + \frac{1}{y}=\frac{1}{z}</math> (calculating [[Series and parallel circuits#Parallel circuits|parallel resistances]], [[harmonic mean]], etc.), and quadratic scales can be used to solve <math>x^2 + y^2 = z^2 </math>.