Notes on b-adic representations, radix, and their usefulness in providing a comparative to decimal

Decimal is not the point. What's the point? I have noticed that decimal arithmetic is introduced too soon before fraction operations are well in hand. Such an approach is illogical for it places the cart before the horse. The decimal system is a variation on Old Babylonian sexagesimal, both of which are particular cases of a much larger context called radix. The point is an example of a separatrix, which is a punctuation chosen to separate digits representing whole numbers from those representing fractions. Thus, the decimal system is much more than a period. Consider the decimal 3.47. Most read 3.47 as "three point four seven". However, teachers and students should take care to read it as "three and forty seven hundredths". For those that need some remediation, "three and four tenths and seven hundredths" is a better reading; the use of decimal grids and fraction models help improve intuition about why there are many different, but equally valid readings of a number containing decimal fractions. Drilling arithmetic algorithms for decimal numbers outside of the context of fraction operations renders the decimal system meaningless. The difficulties fraction operations pose must be overcome prior to the introduction of decimal arithmetic. Otherwise, discussions about decimal operations become nothingmore than magical inscriptions using meaningless symbols. I fear this is far too common a mistake and the most likely reason students struggle with algebra and beyond. They simply lack the proportional reasoning that comes from careful study of fraction operations.

There is nothing special about the decimal system. In the computational sector, three similar systems are in use: binary, octal, and hexadecimal. There are reasons that these three systems are more appropriate than decimal and some argument that systems other than decimal are better because they have fewer fractions. Old Babylonian sexagesimal is such a system. Beyond any reasons for or against decimal, understanding decimal is sorely lacking among working adults in America. It's not because we stick to British measurement. Instead, no similar system is offered to provide contrast to decimal and without comparison to competing systems there is little understanding.

A curiosity about repeating decimal periods... it has been observed that for a unit fraction with denominator p other than 2^m5ⁿ, its period will be a divisor of p - 1. Furthermore, its multiples can be quickly produced by tracking the remainder cycles which result from decimal fraction extraction from the first few multiples; 1/3 and 2/3 are rather trivial for their period is one. 1/7 is more interesting. We can calculate the decimal equivalents of its multiples in the following way. First, we observe that its period is either 1, 2, 3, or 6. It turns out that its period is 6. By tracking the remainder cycle as subscripts of the quotients they produce, we see that 1/7 = 0.1₃4₂2₆8₄5₅7₁. Since the remainders of division by 7 are 0 - 6, we see that the fundamental remainder cycle is (3 2 6 4 5 1), leading us to the digits in the period of 1/7. To construct the decimal equivalents of 2/7, 3/7, 4/7, 5/7, and 6/7 we need only locate the numerator in the fundamental remainder cycle and permute the digits by means of a left shift accordingly. Thus, since 4/7 corresponds to the remainder cycle (4 5 1 3 2 6), we get the decimal 0.571428...

Let's try this with 1/13. The period will be either 1, 2, 3, 4, 6, or 12. 1/13 = 0.0₁₀7₉6₁₂9₃2₄3₁. So, the first fundamental remainder cycle is (10 9 12 3 4 1); let's denote this by FRC1. Since 2 is not in that cycle, we must calculate 2/13 to get a second fundamental remainder cycle which contains it. 2/13 = 0.1₇5₅3₁₁8₆4₈6₂, giving us a second fundamental remainder cycle of (7 5 11 6 8 2); let's denote this by FRC2. Interestingly, we get 3/13, 4/13, 9/13, 10/13, and 12/13 from FRC1, the others are related to FRC2. More precisely, the periods are divided equally between the two FRCs. Another point of interest is the distance of the numerators associated with FRC1 from 13/2 seems to be symmetric: 5.5, 3.5, 2.5, 2.5, 3.5, 5.5. For FRC2, the spread of distances from 13/2 of the numerators 2, 5, 6, 7, 8, 11: 4.5, 1.5, 0.5, 0.5, 1.5, 4.5.

Do the FRCs of 1/7 and 1/13 shed any light on the period of 1/61 and its multiples?

Let's predict the results for 1/17. We will either have 1 FRC of length 16, 2 FRCs of length 8, 4 FRCs of length 4, 8 FRCs of length 2, or 16 FRCs of length 1. 1/17 = 0.0₁₀5₁₅8₁₄8₄2₆3₉5₅2₁₆9₇4₂1₃1₁₃7₁₁6₈4₁₂7₁

1/3 gives us 1 FRC of length 1, 1/7 gives us 1 FRC of length 6, 1/11 gives us 5 FRCs of length 2, 1/13 gives us 2 FRCs of length 6, and 1/17 gives us 1 FRC of length 16. What will 1/19 give us?

1/19 = 0.0₁₀5₅2₁₂6₆3₃1₁₁5₁₅7₁₇8₁₈9₉4₁₄7₇3₁₃6₁₆8₈4₄2₂1₁