# Renard series

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Renard series are a system of preferred numbers dividing an interval from 1 to 10 into 5, 10, 20, or 40 steps. [1] This set of preferred numbers was proposed in the 1870s by French army engineer Colonel Charles Renard.[2] 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 approximately constant (before rounding), namely the 5th, 10th, 20th, or 40th root of 10 (approximately 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. One application of the Renard series of numbers is fuse amperages.

The most basic R5 series consists of these five rounded numbers, which are powers of the fifth root of 10, rounded to two digits. Note that the Renard numbers are not always rounded to the closest three-digit number to the theoretical geometric sequence:

``` 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. These are rounded to a multiple of 0.05. Where an even finer grading is needed, the R20, R40, and R80 series can be applied. The R20 series is usually rounded to a multiple of 0.05, and the R40 and R80 values interpolate between the R20 values, rather than being powers of the 80th root of 10 rounded correctly. In the table below, the additional R80 values are written to the right of the R40 values in the column named "R80 add'l". The R40 numbers 3.00 and 6.00 are higher than they "should" be by interpolation, in order to give rounder numbers.

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. In the table below, rounded values that differ from their less rounded counterparts are shown in bold.

least rounded
R5 R10 R20 R40 R80 add'l
1.00 1.00 1.00 1.00 1.03
1.06 1.09
1.12 1.12 1.15
1.18 1.22
1.25 1.25 1.25 1.28
1.32 1.36
1.40 1.40 1.45
1.50 1.55
1.60 1.60 1.60 1.60 1.65
1.70 1.75
1.80 1.80 1.85
1.90 1.95
2.00 2.00 2.00 2.06
2.12 2.18
2.24 2.24 2.30
2.36 2.43
2.50 2.50 2.50 2.50 2.58
2.65 2.72
2.80 2.80 2.90
3.00 3.07
3.15 3.15 3.15 3.25
3.35 3.45
3.55 3.55 3.65
3.75 3.87
4.00 4.00 4.00 4.00 4.12
4.25 4.37
4.50 4.50 4.62
4.75 4.87
5.00 5.00 5.00 5.15
5.30 5.45
5.60 5.60 5.75
6.00 6.15
6.30 6.30 6.30 6.30 6.50
6.70 6.90
7.10 7.10 7.30
7.50 7.75
8.00 8.00 8.00 8.25
8.50 8.75
9.00 9.00 9.25
9.50 9.75
10.0 10.0 10.0 10.0 10.3
medium rounded
R'10 R'20 R'40
1.00 1.00 1.00
1.05
1.12 1.12
1.20
1.25 1.25 1.25
1.30
1.40 1.40
1.50
1.60 1.60 1.60
1.70
1.80 1.80
1.90
2.00 2.00 2.00
2.10
2.20 2.20
2.40
2.50 2.50 2.50
2.60
2.80 2.80
3.00
3.20 3.20 3.20
3.40
3.60 3.60
3.80
4.00 4.00 4.00
4.20
4.50 4.50
4.80
5.00 5.00 5.00
5.30
5.60 5.60
6.00
6.30 6.30 6.30
6.70
7.10 7.10
7.50
8.00 8.00 8.00
8.50
9.00 9.00
9.50
10.0 10.0 10.0
most rounded
R′′5 R′′10 R′′20
1.0 1.0 1.0
1.1
1.2 1.2
1.4
1.5 1.5 1.6
1.8
2.0 2.0
2.2
2.5 2.5 2.5
2.8
3.0 3.0
3.5
4.0 4.0 4.0
4.5
5.0 5.0
5.5
6.0 6.0 6.0
7.0
8.0 8.0
9.0
10 10 10

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 millimetres. But one would need to use an appropriate number base to avoid ending up with two incompatible sets of nicely spaced dimensions, if for instance they were applied with both inches and feet. In the case of inches and feet a root of 12 would be desirable. i.e. n √12 or 121n where n is the desired number of divisions within the major step size of twelve. Similarly a base of two, eight, or sixteen would fit nicely with the binary units commonly found in computer science.

Each of the Renard sequences can be reduced to a subset by taking every nth value in a series, which is designated by adding the number n after a slash.[2] For example, “R10′′/3 (1...1000)” designates a series consisting of every third value in the R′′10 series from 1 to 1000, that is, 1, 2, 4, 8, 15, 30, 60, 120, 250, 500, 1000.