Editing Slide rule (section)

==== 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.