Preferred Number - Renard Numbers

Renard Numbers

The French army engineer Col. Charles Renard proposed in the 1870s a set of preferred numbers for use with the metric system. His system was adopted in 1952 as international standard ISO 3. Renard's system of preferred numbers divides the interval from 1 to 10 into 5, 10, 20, or 40 steps. The factor between two consecutive numbers in a Renard series is constant (before rounding), namely the 5th, 10th, 20th, or 40th root of 10 (1.58, 1.26, 1.12, and 1.06, respectively), which leads to a geometric sequence. This way, the maximum relative error is minimized if an arbitrary number is replaced by the nearest Renard number multiplied by the appropriate power of 10.

The most basic R5 series consists of these five rounded numbers:

R5: 1.00 1.60 2.50 4.00 6.30

Example: If our design constraints tell us that the two screws in our gadget should be placed between 32 mm and 55 mm apart, we make it 40 mm, because 4 is in the R5 series of preferred numbers.

Example: If you want to produce a set of nails with lengths between roughly 15 and 300 mm, then the application of the R5 series would lead to a product repertoire of 16 mm, 25 mm, 40 mm, 63 mm, 100 mm, 160 mm, and 250 mm long nails.

If a finer resolution is needed, another five numbers are added to the series, one after each of the original R5 numbers, and we end up with the R10 series:

R10: 1.00 1.25 1.60 2.00 2.50 3.15 4.00 5.00 6.30 8.00

Where an even finer grading is needed, the R20, R40, and R80 series can be applied:

R20: 1.00 1.25 1.60 2.00 2.50 3.15 4.00 5.00 6.30 8.00 1.12 1.40 1.80 2.24 2.80 3.55 4.50 5.60 7.10 9.00 R40: 1.00 1.25 1.60 2.00 2.50 3.15 4.00 5.00 6.30 8.00 1.06 1.32 1.70 2.12 2.65 3.35 4.25 5.30 6.70 8.50 1.12 1.40 1.80 2.24 2.80 3.55 4.50 5.60 7.10 9.00 1.18 1.50 1.90 2.36 3.00 3.75 4.75 6.00 7.50 9.50 R80: 1.00 1.25 1.60 2.00 2.50 3.15 4.00 5.00 6.30 8.00 1.03 1.28 1.65 2.06 2.58 3.25 4.12 5.15 6.50 8.25 1.06 1.32 1.70 2.12 2.65 3.35 4.25 5.30 6.70 8.50 1.09 1.36 1.75 2.18 2.72 3.45 4.37 5.45 6.90 8.75 1.12 1.40 1.80 2.24 2.80 3.55 4.50 5.60 7.10 9.00 1.15 1.45 1.85 2.30 2.90 3.65 4.62 5.80 7.30 9.25 1.18 1.50 1.90 2.36 3.00 3.75 4.75 6.00 7.50 9.50 1.22 1.55 1.95 2.43 3.07 3.87 4.87 6.15 7.75 9.75

In some applications more rounded values are desirable, either because the numbers from the normal series would imply an unrealistically high accuracy, or because an integer value is needed (e.g., the number of teeth in a gear). For these needs, more rounded versions of the Renard series have been defined in ISO 3:

R5″: 1 1.5 2.5 4 6 R10′: 1 1.25 1.6 2 2.5 3.2 4 5 6.3 8 R10″: 1 1.2 1.5 2 2.5 3 4 5 6 8 R20′: 1 1.25 1.6 2 2.5 3.2 4 5 6.3 8 1.1 1.4 1.8 2.2 2.8 3.6 4.5 5.6 7.1 9 R20″: 1 1.2 1.5 2 2.5 3 4 5 6 8 1.1 1.4 1.8 2.2 2.8 3.5 4.5 5.5 7 9 R40′: 1 1.25 1.6 2 2.5 3.2 4 5 6.3 8 1.05 1.3 1.7 2.1 2.6 3.4 4.2 5.3 6.7 8.5 1.1 1.4 1.8 2.2 2.8 3.6 4.5 5.6 7.1 9 1.2 1.5 1.9 2.4 3 3.8 4.8 6 7.5 9.5

As the Renard numbers repeat after every 10-fold change of the scale, they are particularly well-suited for use with SI units. It makes no difference whether the Renard numbers are used with metres or kilometres. But one would end up with two incompatible sets of nicely spaced dimensions if they were applied, for instance, with both yards and miles.

Renard numbers are rounded results of the formula

,

where b is the selected series value (for example b = 40 for the R40 series), and i is the i-th element of this series (with i = 0 through i = b).