The job of expressing quantities between 0 and 1 usually falls to the percent (%) sign.  And when it’s used in this context, the % sign takes on the same value as 0.01.

The parts per million (ppm) convention works similarly to express quantities far smaller than 1 %.  In this context, 1 ppm equals 1 x 10^-6 (i.e. one divided by 1 million).  The notion extends to parts per billion (ppb) and parts per trillion (ppt) wherever even smaller quantities are necessary. 

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The dimensionless parts-per-quantity approach can express extremely small numbers without resorting to awkwardly large numbers of zeros right of the decimal point (or exponents, for that matter).

Resistors for example have a resistance (R) that will vary with a change in temperature (T-T_ref), but only slightly.  Consequentially the temperature coefficient of a resistor (α_ppm) may be small enough to approximate using ppm/°C…

R\ \cong\ R_{ref}\left(1+\alpha_{ppm}\frac{T-T_{ref}}{{10}^6}\right)

…where R_ref is the resistance at a reference temperature T_ref (such as 20 °C).

There are 2 reasons why this blog avoids the parts-per-quantity convention whenever possible.

Reason #1: For Clarity

From a technical writing standpoint, relatively little insight can be gained from the ppm notation compared to other expressions.  An alternate way to specify the above resistor’s temperature coefficient or any other parts-per-quantity form would be to include the unit of measure as part of the statement.

For example, a resistor with a temperature coefficient of 1 ppm/°C could also be expressed as:

1 (Ω/10⁶ Ω)/°C

Multiplying the inside of the parenthesis by 10^-6/10^-6 will get the exponential term out of the denominator…

10^-6 (Ω/Ω)/°C

Finally, the micro (µ) metric prefix can replace the number 10^-6…

1 (µΩ/Ω)/°C

The presence of µΩ/Ω immediately signals that the expression relates to electrical resistance rather than the many other units ppm could possibly refer to.

Likewise, the idea may be extended to ppb and ppt…

1 ppm –> 1 (µΩ/Ω)

1 ppb –> 1 (nΩ/Ω)

1 ppt –> 1 (pΩ/Ω)

Reason #2: Language Independence

There is also a language conflict between the words “billion,” “trillion” and the quantities they are meant to express.  This is because “billion” can mean either 10⁹ or 10¹², depending on whether the publication uses the short scale decimal system or the long scale system.

Table 1: Short Decimal Scale Versus Long Decimal Scale

In North America (where Unboxing Tomorrow is based), the short scale system is almost a given.  Other countries, historic documents, or publications from the United Kingdom prior to the 1970’s are a different matter.  This trouble is compounded by the naming conflict between parts-per-thousand (ppt) and parts-per-thousand (also ppt).

Unless the context is completely unambiguous, this blog will normally follow the National Institute for Standards and Technology (NIST) recommended unit/unit notation. 

References

Thumbnail Image Credit: Bart Everson (CC BY 2.0)

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