Tuesday, 18 June 2019

Rarity of full-period prime denominators in rational approximations of irrational numbers

Ever since, a few months ago, I began to discover that – in addition to the very well-known approximation of 22/7 for ๐œ‹ – other irrational numbers had frequently used rational approximations, I have been struck by one fact: that full-period primes occur very rarely as denominators in such fractions, especially in the most useful rational approximations for hand calculations.

In order to test this hypothesis I have compiled a representative selection of irrational numbers in the table below, useful rational approximations for these numbers, and the periods of these rational approximations. For most numbers the most common rational approximation has been used; in certain cases like the square root of 6 and ๐œ‹, I have given more than one rational approximation, with that with the larger denominator naturally more accurate.
Number Decimal expansion Rational Approximation Period Prime factorisation of denominator Character and type of prime factors
√2 1.4142135623730950488016887242 99/70 6 2•5•7 Composite
Full-period and terminating factors
√3 1.7320508075688772935274463415 97/56 7 2•2•2•7 Composite
Full-period and terminating factors
√5 2.2360679774997896964091736687 161/72 1 2•2•2•3•3 Composite
Short-period (unique) and terminating factors
√6 2.4494897427831780981972840747 49/20 0 2•2•5 Composite
Terminating decimal
218/89 44 89 Half-period prime
√7 2.6457513110645905905016157536 127/48 1 2•2•2•2•3 Composite
Short-period (unique) and terminating factors
√10 3.1622776601683793319988935444 117/37 3 37 Short-period (unique) prime
√11 3.3166247903553998491149327366 199/60 1 2•2•3•5 Composite
Short-period (unique) and terminating factors
∛2 1.2599210498948731647672106072 63/50 0 2•5•5 Composite
Terminating decimal
∛3 1.4422495703074083823216383107 75/52 6 2•2•13 Composite
Half-period prime factor
∛4 1.5874010519681994747517056392 100/63 6 3•3•7 Composite
Half-period and short-period (unique) factors
227/143 6 11•13
∛5 1.7099759466766969893531088725 171/100 0 2•2•5•5 Composite
Terminating decimal
∛6 1.8171205928321396588912117563 467/257 256 257 Full-period prime
(accurate to 1-in-33,629,323!)
∜2 1.1892071150027210667174999705 44/37 3 37 Short-period (unique) prime
∜3 1.3160740129524924608192189017 25/19 18 19 Full-period prime
229/174 28 2•3•29 Composite
Full-period, short-period and terminating factors
21/5 1.1486983549970350067986269467 85/74 3 2•37 Composite
Short-period (unique) and terminating factors
21/12 1.0594630943592952645618252949 89/84 6 2•2•3•7 Composite
Full-period, short-period and terminating factors
๐œ‹ 3.1415926535897932384626433832 22/7 6 7 Full-period prime
355/113 112 113 Full-period prime
e 2.7182818284590452353602874713 193/71 35 71 Half-period prime
ee 15.154262241479264189760430272 197/13 6 13 Half-period prime
2849/188 46 2•2•47 Composite
Full-period prime factor
e๐œ‹ 23.140692632779269005729086367 1481/64 0 2•2•2•2•2•2 Terminating decimal
ln 2 0.6931471805599453094172321214 61/88 2 2•2•2•11 Composite
Short-period (unique) prime factor
log10 2 0.3010299956639811952137388947 59/196 42 2•2•7•7 Composite
Terminating and squared full-period prime factor
If we study this table, we see that, for whatever reason, there seem to be very few full-period prime denominators. The two well-known approximations for ๐œ‹, one approximation for the fourth root of three, and one remarkable approximation for the cube root of six are the only exceptions. [The cube root of six – which has minor notability as the geometric mean of 1, 2 and 3 – I did not originally intend to include but decided to do so because the approximation 467/257 is so amazingly accurate, being superior to the famous Milรผ approximation for ๐œ‹].

Why this should be so is an interesting question. It is possibly because the way in which the continued fractions used to find such approximations as 467/257 for ∛6 would add factors in the finding of “common denominators” needed for addition of fractions, although I have not checked this yet.

A final proof the “March on Canberra” is a quarter-century and counting overdue

According to a new paper in the journal Nature, regardless of what the rest of the world does, record-breaking temperature rises are already inevitable until 2040.

At the same time, the Sydney Morning Herald is noting a zero-emissions plan for Britain – whose parity emissions per capita are minimally four times those of Australia – as Australia approves the polluting Adani coal mine. There is – and was even before last month’s surprise election – a certainty Australia will expand fossil fuels whilst the EU moves to zero-net-emissions.

Many (including my brother) naรฏvely believe that Australia will eventually be condemned as a pariah state for expanding fossil fuels. Nevertheless, this viewpoint overlooks demographic reality. Australia already has substantially higher total fertility than those nations most advanced in decarbonisation. Recent trends towards lowest-low fertility in Finland (from 1.9 to 1.5 children since 2010) and other European nations whose fertility was the least low during the 2000s suggests that Morrison’s policies will widen this gap.

The fact is that – as I have emphasised for two decades – Australia must ecologically have by far the lowest emissions per capita in the world. This demand places human energy consumption upon its natural biological “footing”. Environment, Capitalism and Socialism demonstrated three decades ago that the money existed to finance a rapid transition to a carbon-free Australia as early as 2005 or 2010 – were major polluters taxed severely enough.

As Dimitri Lafleur has partially shown, a carbon-free Australia would remake the world economy by:
  1. radically limiting energy and materials use on a global scale, especially in desert nations with naturally low-energy ecologies and zero hydropower potential
  2. shifting “developing” economies towards renewable energy once they do not have cheap fossil fuels from Australia and the oil states
  3. shifting energy-intensive industries towards those (Enriched and Tropical) nations with large resources in hydropower
  4. shifting agriculture towards the high-latitude nations with youngest and most fertile soils
    1. this would occur because land clearing is a major source (around 20 percent) of greenhouse emissions in Australia
    2. also, Australian soils are thirty thousand times older and more weathered than soils of most other Quaternary landmasses
    3. young, high-latitude areas are also least affected by runaway climate change shown as certain by Nature
  5. shifting away from planned obsolescence towards long-lasting consumer goods that use fewer resources over the long term
What needed to be done back in the 1990s was for the globe to recognise that – regardless of its relatively small aggregate emissions that have led even environmentalists to neglect it – a rigid, zero-compromise, zero-emissions target for Australia no later than 2010 would have:
  1. largely solved global greenhouse gas emissions by radically altering global development patterns
  2. paid for the ecological crisis out of the pockets of those people – alongside the Arab Gulf royal families – with greatest duty and ability to pay
  3. achieved this in a manner in agreement with Earth’s natural ecology (smallest per-capita energy consumption and emissions in arid desert nations)
  4. in an Enriched World then and now crippled by excessive environmental regulations, which stand likely to achieve negligible global gains while Australia mines and uses more and more coal, created major economic opportunities including:
    • phase-out of economically crippling Enriched World farm subsidies as Australia’s unsustainable pastoral and broad-acre farmland would be converted to ecotourism
    • revitalising such industries as aluminum and titanium smelting when coal use in phased out in Australia and other nations lacking hydropower potential
Given the experiences of the past quarter-century and especially last month’s election, there exists zero possibility that Australia will ever elect a more environmentally responsive government. Thus, other countries are burdened with the critical task of clamping down on the worst environmental performer – a task entirely ignored but unless achieved even a total carbon phase-out in the EU and East Asia will achieve little in the long term.

Instead of decentralised global environmental protests (as seen in recent weeks) what was needed in 1994 and stands three decades overdue was and is a global focus upon the centre of power in the worst environmental performer: protests demanding uncompromising, rapid decarbonisation of Australia, or a “March on Canberra”. Whilst the effects would not be immediate, would be costly to the rest of the world and would need to be sustained over years and even decades, they possess potential to actually deal with the planet’s worst polluter rather than permit Australia indefinite emissions increases negating large-scale decarbonisation abroad.