1) On correcting the current practice for the rounding of numbers
For ’rounding purposes’, a unit can be represented by (0.5 ← x → 0.5). A unit is a discrete quantity; it can only proceed to the next unit (a clue is in the choice of the word ‘unit’ to describe it) when the number itself increases, if only by an infinitesimal amount (+δx). A digit is therefore a threshold, not a centre, illustrated by (n – δx, n + δx) in the limit. Rounding off can therefore only be performed within the unit itself; it cannot be rounded up past the next digit to the next unit. [Analogy: An apple that barely scrapes through a hole in an apple grading template does not qualify for its inclusion in the larger size group.]
To balance their interests with those of their customers, banks traditionally use the ‘odd/even’ fix. Using the term ‘centre’ here wrongly proposes that a single coordinate can simultaneously represent two mutually exclusive states. Under this system, halves following an odd digit round up (e.g. 4.135 → 4.14), while halves following an even digit round down (e.g. 3.125 → 3.12). While theoretically neutral, this practice heavily favours financial institutions in real-world markets. Truly equitable results require this precise boundary model paired with ‘Fair Momentum’ – a dynamic system that tracks running rounding errors to guarantee absolute statistical equilibrium.
2) On coordinating the dating system
2a) On correcting the flaws of the ‘BP’ and 1950 AD baseline
One of the most distant physical records of a solar eclipse ever committed to writing is found on a cuneiform clay tablet excavated at the ancient city of Ugarit. Modern astronomers and historians believe it captures a total solar eclipse that occurred on March 5, 1223 abs BC.
Aside from rare historical-astronomical cross-links like this, dates farther into the past than and including 1000 BC rely on scientific instrumentation or scarce historical cross-links; they are therefore by definition inherently approximate and once this is accepted the use of circa (c.) within this deep-time era becomes superfluous.
The majority of archaeological dates older than 1100 BC – the era marking the final collapse of the Mycenaean civilisation – end in at least two zeros, with the vast majority ending in exactly three zeros.
With such dates not having qualifiers (limits, etc.) attached, the only recourse on inspection is to interpret such dates mathematically, e.g. 1100 BC (‘0’s are the homes of ’rounding off’) equates to 1050-1150 or 1100±50 BC.
Note: The mathematical errors (±5, ±50, ±500) associated with the dates (xxx0; x,x00; xx,000), respectively, are generally the absolute minimum possible error ranges; in practice, the actual range (when not mentioned) is almost always much greater.
Geologists and archaeologists have made the boundary between the Pleistocene and Holocene epochs (11,700 BP) a fixed threshold with zero margin of error. Their purpose in doing this is unclear. Using it as a constant in calculations is of course acceptable, but as a derived date it remains an approximation.
Dates on either side of this threshold, such as 11,800 and 11,600 BP, are not absolute; they are variables with inherent margins of error. Such dates can therefore be interpreted as representing rounded figures that reflect inseparable measurements or multiple conflicting sources. Consequently, a date formatted as xx,000 BP can be logically interpreted as having implicit error limits of ±500 years. Similarly, dates formatted as xx,x00 BP can be interpreted as having implicit error limits of ±50 years.
Herein lies the systemic problem with using the constant 1950 AD for conversion into a BC date. This practice visually compresses and distorts the original measurement’s implicit error limits. For example, converting 11,800 BP (which carries an implicit ±50 year limit) into 9850 BC forces an artificial ’50’ into the notation.
A simple mathematical interpretation of a zero-ending decade date like 9850 BC thus understates the actual range of errors, falsely implying an over-precise ±5-year limit and breaking the visual integrity of the original century-scale rounding.
Converting an ancient date like 1100 BC (which has an implicit ±50 BC limit) into the current BP system distorts the figure into 3050 BP. Authors are then caught in a dilemma: leaving it as 3050 BP implies a false sense of specific precision, while following standard rounding rules creates a century-wide distortion of 3100 BP. The current system forces researchers to compromise their data.
While organisations operating within a closed framework should undoubtedly continue using 1950 AD and ‘BP’, the rest of the world should have a logical, more usable framework that works for them. For this, the constant 1950 AD should be replaced by the constant 2000 AD. Under this standard, adding the constant to a rounded millennium date clearly preserves the trailing zeros, ensuring the notation remains an honest reflection of the data’s true, unmeasurable limits (e.g. 20,000 BC scales perfectly to 22,000 BP, additionally, the BP is now valid until 2500 and for xx00 until the year 2050).
It should be noted that since the 11,700 BP boundary is an artificial, rounded threshold rather than the date of a single sudden event, adjusting it by 50 years for the year 2000 represents a negligible change to its definition (c.0.4 %).
In texts, the unit ‘kya’ or ‘ka’ should be used exclusively for the Palaeolithic era, running down to the threshold boundary of 11.7 kya / 9700 BC; the unit ‘BC’ reserved solely for the Holocene epoch, handling the narrative from that human turning point down to 1 BC; and by changing the constant to 2000 AD, the general use of ‘BP’ notation becomes a true statement of fact.
Note: The most consequential turning point in human history is not the transition from BC to AD. That boundary is merely a man-made inaccuracy based on the supposed birth of Jesus Christ, an event historians estimate actually happened between 6 BC and 4 BC, and serves only as an arbitrary reference point. There is simply no comparison between an arbitrary calendar error and the true planetary turning point of 9700 BC. This boundary marks the monumental environmental shift from the end of the Palaeolithic era (11.7 kya) to the beginning of the European Mesolithic and the Holocene (9700 BC), when the planet warmed and human civilisation began to evolve.
2b) On a more efficient use of ‘circa’ (c.)
While organisations operating within a closed framework should undoubtedly continue using 1950 AD and ‘BP’, the rest of the world should have a logical, more usable framework that works for them, (as established in Section 2a). The constant of 1950 AD is replaced entirely by 2000 AD. Under this rule, the unit ‘kya’ is used exclusively for the Palaeolithic era, running down to the threshold boundary of 11.7 kya / 9700 BC, while the unit ‘BC’ is reserved solely for the Holocene epoch, handling the narrative from that human turning point down to 1 BC.
With few exceptions, dates farther into the past than and including 1000 BC rely on scientific instrumentation or scarce historical cross-links; they are therefore by definition inherently approximate and once this is accepted the use of circa (c.) within this deep-time era becomes superfluous, (as established in Section 2a).
The exceptions to this rule are absolute dates verified via tree-ring dating/dendrochronology, approximate dates with refined limits (‘±xxxx BC’ and ‘xxx±xx cal BC’) – noting that if these specific qualifiers are omitted, the dates simply revert to standard approximations.
From the above, it can be seen that European Mesolithic dates from 9700 to 1000 BC are, with few exceptions, defined as approximate and do not require ‘circa’. Dates between 999 and 1 BC fall into two distinct groups: those ending in the digits 1 through 9 are derived from written histories that match modern astronomical computer models. They are absolute by default and therefore do not require ‘circa’. Conversely, dates closer to the present than 1000 BC that end in the digit ‘0’ serve historians as the designated resting places for approximations. For these dates, the use of ‘circa’ is mandatory, except when a zero-ending date is historically or astronomically exact – in which case it is explicitly qualified with the modifier ‘abs’ (e.g. 440 abs BC).
It should be noted that in the years between 999 and 1 BC, absolute dates ending in zero are highly unusual, totalling a tiny fraction of the 500 to 1,000 recorded absolute dates ending in one through nine.
It should also be noted that under this new structure the simple ‘circa’ is strictly reserved for two- or three-digit dates lacking exact error margins or formal calibration labels, e.g. King Arthur ruled c.500 AD.
Thus, dates from the deep past up to and including 1000 BC are approximate unless they have qualifiers, and dates 999-1 BC are absolute unless they have qualifiers.
It seems historians, in their pursuit of precision, have taken the wrong path with the dates of the Ancient Egyptian kings. Since these dates are assessments from three separate sources and are highly challenged which means that the presentation of ancient chronology is fundamentally relational, not absolute. For example, in ‘1492–1479 Thutmose II Aakheperenre’, the presence of the final digits 2 and 9, by inspection, suggests absolute dates. Since these dates are assessments from three separate sources and are highly challenged, it would appear they should include circa. However, doing so would compromise the use of the specific digits 2 and 9 above. An honest presentation would round these dates off, letting the recorded lengths of the kings’ reigns ‘float’ within the inherent ±5 range of a zero-ending decades (xxx0). Comparing these ’rounding off” dates other more reliable dates obtained of other cultures of the same period would reinforce or perhaps improve the comparison. These new lists would need to be headed by a note: ‘The following dates (c.xxx0) are approximate. The actual dates of the kings ‘float’ between the given dates are shown between brackets (…) after the kings’ names. Consequently, future dynastic lists should be prefaced by a definitive note: ‘The following dates (c.xxx0) are approximate. The actual dates of the kings ‘float’ within the brackets (…) displayed after the kings’ names.’ For example:
| 2060–1650 BC | MIDDLE KINGDOM PERIOD (2055–1650 BC) |
| 1990–1730 BC | 12th Dynasty Baseline (1985–1733 BC) |
| 1990–1960 BC | Amenemhat I Sehetepibre (1985–1956) |
| 1960–1910 BC | Senusret I Kheperkare (1956–1911) |
| 1910–1880 BC | Amenemhat II Nubkaure (1911–1877) |
| 1880–1870 BC | Senusret II Khakheperre (1877–1870) |
| 1870–1830 BC | Senusret III Khakaure (1870–1831) |
| 1830–1790 BC | Amenemhat III Nimaatre (1831–1786) |
| 1790–1780 BC | Amenemhat IV Maakherure (1786–1777) |
1780–1770 BC | Sobekneferu, Queen (1777–1773 |
Summary
The only introduction necessary would be the heading,’Unmarked dates from the deep past up to and including 1000 BC are approximate, and dates from 999 to 1 BC are absolute, which means that only the exceptions require qualifiers, namely ‘circa’ (c.), xxxx±xxx BC, xxxx±xxx cal BC, and the anomalies circa (c.) and xxxx abs BC’.
The unit ‘kya’ and the constant 2000 AD should replace ‘BP’ and 1950 AD respectively
In the conversion from ‘BC’ to ‘kya’, the qualifiers – or the lack of them – are carried through completely unchanged.
With very few exceptions, dates further back in the past than, and including, 1000 BC are approximate. Having been identified as such and isolated, the use of ‘circa’ (c.) here is unnecessary and therefore should be abandoned.
Conversely, dates from 999 BC to and including 1 BC are mostly absolute. Although chronological progress was ragged as dates moved towards the present, during and following the Middle Ages these absolute dates have become the larger group.
To prevent unverified zero-ending dates from looking identical to verified ones, iron-clad historical exceptions must use the scientific ‘abs’ standard. A premier example is the eclipse of 310 BC, when the Greek ruler Agathocles slipped his fleet out of the harbour of Syracuse. This is rendered as 310 abs BC.
| Era / Date Range | Example | Qualifiers (BC / AD) | Notes |
| Up to 11.7 kya | 15.0 kya | None | Approximate (implicit) |
| 9700-1000 BC | 3500 BC | None | Approximate (implicit) |
| 9700-1000 BC | 6000±200 | xxxx±xxx BC | Refined limits |
| 9700-1000 BC | 3100±100 | xxxx±xxx cal BC | Refined limits |
| 9700-1000 BC | 1223 abs BC | xxxx abs BC | Verified absolute exceptions |
| 999-1 BC (ends 1–9) | 431 BC | None | Absolute (implicit via history/ astronomy) |
| 999-1 BC (ends 0) | c.440 BC | Circa (c.) | Mandatory approximation marker |
| 999-1 BC (ends 0) | 310 abs BC | xx0 abs BC | Absolute exception ending in zero |
3) On tidying up the presentation
Standard printing rules frequently fail to satisfy the requirements of strict mathematical logic and efficient data processing.
The thousands separator comma should be omitted for all four-digit years in new entries. Dates should be formatted cleanly without punctuation (e.g. 1066, 1453, 1492, 2026). Traditional printing standard that inserts a space between ‘c.’ and the first digit is redundant; computers read the data perfectly without it.
Punctuation choices must also protect the layout from visual misinterpretation. The en-dash is far too prominent for what is essentially a chronological aside. Using a simple hyphen (-) between dates and a spaced hyphen ( – ) between distinct text blocks ensures that numeric ranges are never mistaken for compound or hyphenated words. Ultimately, spaces in text must carry a distinct semantic meaning. The four spaces in (c. 800 – c. 400 BC) serve no analytical purpose, and they should therefore be omitted, leaving (c.800-c.400 BC) as a single, unbroken text block that must move to the next line together rather than being split. The common shorthand format of (c.800-400 BC) is both grammatically and mathematically unsound, particularly because a second date can occasionally be absolute. For complete clarity, circa must always be explicitly stated for both parameters, as seen in (c.800-c.400 BC).
The current practice of writing, e.g. (c.700 BC-c.AD 50) is awkward and unnecessary. Better in Modern English is and more natural to read is (c.700 BC-c.50 AD), i.e. ‘from about 700 years before Christ until about 50 years after the death of Christ.’
In both the BC and AD eras, relative time markers like ‘earlier’ or ‘older’ trigger an immediate conflict between a forward-moving historical flow and an absolute numerical countdown. For instance, stating that an event occurred ‘prior to 1000 BC’ can mean either further back in time (e.g. 1200 BC) or closer to the present (e.g. 800 BC), depending on whether a reader visualises history chronologically or numerically. Similarly, a phrase like ‘prior to the 18th century’ causes readers to mistakenly target the 1800s (1801-1899) rather than the 1700s (1701 -1800) because the absolute digits override the historical coordinate. This mental shortcut consistently compromises chronological clarity. To eliminate this confusion, raw decade/century bounds (e.g. the 1700s) or explicit descriptive clauses such as ‘further back into the past‘ and ‘closer to the present’ should be used.
Succeeding the Pre-Roman Iron Age, the term ‘Roman Period’ should be used in place of ‘Roman Iron Age’. Once complex imperial administration and broad trade networks consolidated, the Roman epoch evolved far beyond raw material production; reducing a complex society to a single metal label oversimplifies its history. Consequently, material-based terms for subsequent eras – such as ‘Migration Iron Age’ or ‘Viking Iron Age’ – should be rejected, as these tags reduce massive human, societal, and seafaring movements to mere technological classifications. This is particularly misleading given that iron-working itself underwent no revolutionary change during these times.
The choice of transition date between the Pre-Roman Iron Age and the Roman Period in northern and northeastern Europe is fluid. Because the Romans never occupied these regions, this boundary can only be determined by when significant Roman trade likely began. The date AD 1, though used by some, is inaccurate because it implies an absolute, exact threshold. A more realistic boundary post-dates the Battle of Teutoburg Forest in AD 9, during which Germanic tribes temporarily blocked contact with northern Europe. Allowing for a period of consolidation, the date c.50 AD cited by several sources is much more reasonable. It is accepted, however, that actual margins of error are much greater than strict mathematical limits imply expressed here as c.50 AD±5, although possible circa (c.) variances are always to be excluded from general presentation.
4) On a mathematical solution to the centuries-old ‘Two-Year Problem’ in BC/AD dating
The period between 1 BC and 1 AD is one year. On a timeline, year intervals are bounded by tick marks placed at the start of BC years and at the end of AD years. Here, the total time lapse between 1 BC and AD 1 measures two years, generating a mathematical distortion across the timeline – the so-called ‘Two-Year Problem’. Traditionally, chronologists have corrected this by applying a manual arithmetic patch, such as (64 + 27 – 1 = 90), an equation that masks rather than represents the underlying spatial reality. Because the final second of the BC era must reach and pass through zero before the first second of the AD era can commence, the boundary should be treated as a strict mathematical axis ’00’ in space (xx-00-xx) rather than as an artificial, discrete ‘Year 0’ or ‘Year 1’ block.
On a graphic timeline, by labelling the ticks as mid-year points (1 July), the 1 BC and 1 AD tick marks sit exactly six months to the left and right respectively of this zero datum 00, naturally forming a single, continuous one-year span. This allows historical intervals to be calculated dynamically by treating time as a linear distance on a spatial scale. For example, the interval 64-00-27 can be calculated from the clean scalar addition of their respective spatial coordinates from the zero axis (63.5 + 26.5 = 90 years). This approach entirely removes the need for the clunky, counter-intuitive ‘minus one’ arithmetic step, represents the exact physical situation, fixes the so-called ‘Two-Year Problem’, and provides a systemic justification for the introduction of the datum ’00’. By doing so, we achieve the same seamless interval calculations utilized by astronomical year numbering and ISO 8601 protocols without needing to invent a fictional, empty ‘Year 0’ or ‘Year 0000’ block.
Note: The primary benefit of this system for general data presentation is that the fragmented, clunky ‘BC/AD’ notation is replaced by a smoother, continuous -00- axis, eliminating the need for politically or culturally contentious labels like BC, AD, CE, or BCE. It is important to note that the scalar addition formula yields a precise historical duration only when both anchor points are absolute calendar dates. However, applying either method to approximate prehistoric dates is entirely swallowed by the standard ± error range (e.g. c.64 + c.27 – 1 = c.90), where the inherent mathematical rounding-off error of ±5 or more renders a single-year discrepancy completely irrelevant.
Note: 1 July 64 BC to 1 July AD 27 = 63.5 + 26.5 = 90.0 years; 5 February 64 + November 25 = – (-0.4 – 63.5) + (+26.5 + 0.4) = 63.9 + 26.9 = 90.8 years.
Afterthought: Using (c.500) (00), (00) (c.500), (c.500-c.100)-(00), (00)-(c.100-c.500) – this actual format of continuity and presentation of the dates seems to work best – would solve the problem there is with using BC, BCE, AD, CE (these are all Christian dates) but sadly, this perhaps would be a leap too far!!!
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