lesson
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If you open a drawer of electronic parts, you will quickly notice something strange: you can easily buy a 470Β Ξ© or 680Β Ξ© resistor, but a neat, round 500Β Ξ© resistor is nowhere to be found.
Why would component makers choose such bizarre numbers instead of simple steps like 100, 200, 300, and 400?
The Problem with Linear Spacing
Resistors have a tolerance, which is the maximum percentage error between a component's labeled value and its actual measured resistance.
If you space values linearly (adding a fixed 100Β Ξ© each step), jumping from 100Β Ξ© to 200Β Ξ© is a massive 100% increase, but jumping from 1,000Β Ξ© to 1,100Β Ξ© is only a 10% increase.
πInteractive diagram
Linear steps create massive gaps at low values and wasteful crowding at high values. How can we make every single step represent the exact same relative percentage jump?
The Geometric Principle
In the 1870s, French military engineer Charles Renard faced this exact problem while standardizing cordage sizes for military airships, inventing what we now call preferred numbers.
Instead of adding a fixed amount, a geometric progression multiplies each value by a constant ratio r to find the next value.
In electronics, we divide each decadeβa range spanning a factor of 10, such as 1Β Ξ© to 10Β Ξ© or 10Β kΞ© to 100Β kΞ©βinto n equal logarithmic steps.
Starting at 1 and multiplying by the ratio r exactly n times must bring us to 10, which gives us the governing formula:
rn=10βΉr=101/n=n10β