Accuracy of Rounded Fractions
The use of place notation is a huge boon for mathematics, making calculation and expression much easier in many cases than that of vulgar fractions, which had previously been the only way of referencing values less than a whole unit. However, while vulgar fractions can always exactly express their real value (such as 1/7), place-value fractions (which we will henceforth call "inline" fractions) sometimes cannot (such as the inline expression of 1/7, 0;186X35...).
Still, the convenience of inline fractions is such that we often wish to use them despite their inherent inaccuracy in such cases. Therefore, we round them; that is, we select a point in the inline expansion of the fraction which we will deem an acceptable level of inaccuracy. One common such level, used in the trigonometric and logarithmic tables common before digital calculators, was four digits; but any number of digits can be selected. The acceptable degree of inaccuracy is typically gauged in precisely this way: number of digits. The amount by which that number of digits is actually varying from the true value is rarely considered.
But not all roundings to the same number of digits reflects the same variation from the true value. Let's take a relatively simple, terminating fraction as an example: 0;0X5. We decide that three digits is too long, and we want to round it to two, giving us 0;0X. (Remember, this is dozenal; we round up at 6, the half, not at 5 as in decimal.) We have another fraction, 0;0X2, which we also want to round to two digits; this gives us 0;0X, as well. Clearly, the second rounding is more accurate; it varies from the true value by only 0;002, while the first varies from the true value by 0;005. So simply being rounded to the same number of digits doesn't indicate how close the rounded value is to the true value, except within certain fairly broad limits.
With non-terminating fractions, of course, the accuracy calculations are more difficult; we must round our inaccuracy values themselves to make them manageable. However, the basic concept is the same.
So let's examine some of the primary transcendental numbers for the relative accuracies of their roundings in both dozenal and decimal, and see what we arrive at. π, of course, we all know well as the ratio of the diameter of a circle to its circumference. e, Euler's number, is less well known as the base of natural logarithms. φ is the "mean and extreme ratio," that ratio of a line segment such that the ratio of the larger part to the smaller is equal to that of the whole to the larger part. And the last is the square root of two. Errors are in perbiquas and are themselves rounded.
A few things to note about the numbers we've explored here:
- Decimal is more inaccurate than dozenal for the same number of digits almost every single time. The only exceptions are the two-digit roundings for π, √2, and e; but in √2's case, the difference is miniscule, and even in these cases the dozenal three- and four-digit roundings are significantly more accurate than the decimal.
- In the case of φ, the dozenal two-digit rounding is more accurate than the decimal four-digit rounding. Decimal does not beat out dozenal's two-digit accuracy for φ until it reaches five digits; and the dozenal five-digit rounding is still more accurate than that.
- Often, the dozenal rounding is not only more accurate,
but much more accurate.
- The three-digit rounding of π, for example, is more than twice as accurate in dozenal than in decimal.
- The four-digit rounding of π is an entire order of magnitude more accurate in dozenal than in decimal.
- The three-digit rounding e is nearly a third more accurate in dozenal than in decimal.
- The roundings of φ have already been reviewed above.
All in all, dozenal clearly comes out on top in these calculations.
Other Irrational Fractions
Let's now examine some other difficult fractions; not transcendental numbers this time, but simply difficult ones. These are not the same in all bases; so, for example, 1/3 in decimal will be compared with 1/5 in dozenal, since 1/3 is trivially simple in dozenal (0;4) while 1/5 is trivially simple in decimal (0.2). On the other hand, 1/7 is a six-digit-period repeating fraction in both bases, so it will be compared to itself.
Surprisingly, for 1/7, decimal is slightly more accurate for a two-digit rounding. Two thousandths as opposed to nearly seven tricia explains that discrepancy. But otherwise, dozenal holds its own quite well even in sevenths; the three-digit rounding is slightly more accurate in dozenal than in decimal, while the four-digit rounding is nearly twice as accurate.
Decimalists will always point to the fifth as the Achilles heel of dozenal; but this chart gives that claim the lie. Not only is the third arguably a more important fraction anyway, but dozenal handles fifths better than decimal handles thirds. Observe our roundings here. The two-digit rounding of the dozenal fifth is more than twice as accurate as the two-digit rounding of the decimal third; the three- and four-digit roundings continue to blow decimal's out of the water.
Even in the most difficult numbers; even in the numbers that are some of dozenal's few weaknesses; even in these cases, dozenal is the best base.