Editing Slide rule (section)

=== 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%.