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Appendix III. "No Year Zero" and Timespan Calculations
Why is there no Year Zero? We all know the BCAD system was instituted to honor Christ, and the year numbers are somehow geared to His birth. But unless we know the exact details of what is involved with the "gearing" of these yearnumbers, we cannot hope to do any math operations using them. To begin, let us forget all about the BCAD system, and focus on something very basic which we can all understand.
Man's birth initiates a numbering process Imagine that a boy named Fred is born. During his first year of life, Freddie is getting older. He is awaiting his first birthday. When that day comes, he is called "one year old". He is also called "one year old" for the entire year to come. From then on, all through his life, Freddie will be associated with the number of his most recent birthday. 1 year old: most recent birthday was # 1Going back to the matter of his first birthday: This happy event marks the beginning of his second year on earth. So, all the time he is called "1 year old", he is living in year # 2. This is because Fred like all of us mortal humans is tied to the lower of the two possible numbers: "45 years old", though living in Year # 46This convention does represent a mathematical truth:  one who is halfway through Year #46 is agreeably 45+1/2 years oldThe key point is, we associate a person with the number of complete years they have seen. Christ's birth initiates a numbering process The passage of time in general, which we also mark with an annually increasing number, is apt to be confused with our own, individual agereckoning process (outlined above). Why? Because we all know this tracking of the years is somehow a measure of accumulated time since the birth of the one key person, central to all history: Jesus Christ. When we decided to reckon our years by the Son of God, it was evidently the intent to track His era as one would an Old Testament ruler: By the nth year of his reign: Ezra 5:13 But in the first year of Cyrus the king of Babylon the same king Cyrus made a decree to build this house of God.The first year of Christ's life (as accurately as could be determined) was made Year # 1 of History. In mirror fashion, the first year prior to His birth earned the name "Year # 1 before Christ". BC  AD  = before Christ  = anno domini (latin, "year of the Lord")All of this appropriately and fittingly honored the Savior; yet a key mathematical disadvantage arises: There is no Year Zero To anyone who has studied math, the BCAD span of time may resemble a numberline: <++++++++++++++> 8 7 6 5 4 3 2 1 0 1 2 3 4 5On such a linegraph of numbers, calculation of distances from negativetopositive numbers is simple: distance d = A  B (absolute value of the difference) Example: Find the distance from 7 to 4 Solution: Let A = 7 and B = 4 ==> d = (7)  4 = 11 = 11 OR  Switch A & B. Solution: Let A = 4 and B = 7   Order is not ==> d = 4  (7)  important. = 4 + 7 = 11 = 11All of this is simple, and familiar to many. Yet the same math fails in the BCAD system when crossing from BCtoAD, or viceversa: For there is no "year zero" in the middle. Since a year has been left out, we expect our normal math to be off by one year. What follows is a detailed look at the math involved, and a set of formulas for calculating a BCtoAD span of time.
Calculation of yearspans
The span of time between a [BC moment] and an [AD moment] is composed of two portions: BC D AD D A A T T E E      Time spent on BC side  +  Time spent on AD side  ======================= =======================Each "portion" can itself be composed of two components: partial and complete BC D A T E   .. partial  +  complete _____ BC year   BC years   ==== ============'============'============    _______________________ + _______________________    ..  ______ complete  +  partial  AD years   AD year ============'============'============ ========  AD D A T EWe will demonstrate BCAD timespan calculations in the following order: BC AD   1. Complex timespans: date_X to date_Y 2. Anniversary timespans: date_X to date_X 3. Simple yearspan: year_P to year_Q
Complex timespans: date_X to date_Y
As diagrammed above, a BCtoAD timespan can consist of four parts: i. partial BC year Example: 91 2 BC to 61 2 AD i. 4 months (all of Sep, Oct, Nov, Dec) ii. 1 complete BC year (1 BC) ( 2  1 = 1 ...per ii. above) iii. 1 complete AD year (1 AD) ( 2  1 = 1 ...per iii. above) iv. 5 months (all of Jan, Feb, Mar, Apr, May) + __________________ = 2 years, 9 months Constructing a formula for complex timespans
i. ii. iii. iv.     BC_partial + (BC_yearNumber  1) + (AD_yearNumber  1) + AD_partial Reorder components: = (BC_yearNumber  1) + (AD_yearNumber  1) + BC_partial + AD_partial Move numbers to the right: = BC_yearNumber + AD_yearNumber + BC_partial + AD_partial  (2 years)
Anniversary timespans: date_X to date_X
If [startDate_BC] and [endDate_AD] share the same mm/dd values, we have a simplified version of the complex timespan above. From date_X in one year, to date_X in another year (i.e., matching mm/dd), is a complete number of years. Proving this will help simplify our anniversary timespan formula. Proof: Anniversary timespans consist of: 1.  a partial year at the start  2. Whole years, if any; short spans may have 1 or 0    1st whole year (Jan  Dec) of chosen timespan   2nd whole year (Jan  Dec) of chosen timespan   ... ...   ... ...   last whole year (Jan  Dec) of chosen timespan  3.  partial year at end We are seeking to verify the existence of complete years. Line 2, composed entirely of complete year(s), can be removed from the "hunt" without ill effect. There remains only to show that lines 1 & 3, when combined, will also add up to complete year(s).
Fact: Dates mmdd_BC and mmdd_AD, being identical (as they are anniversaries), will each bisect their respective years at the same spot. Choose any date: say, Aug 1 & Aug 1: Start year, BC: X================== "Line 1" Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec End year, AD: ============================X "Line 3" Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecAccordingly, we can label the resulting segments: ( BC_left ) ( BC_right ) X================== Start year, BC: Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec ( AD_left ) ( AD_right ) ============================X End year, AD: Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Fact 1: BC_left = AD_left and BC_right = AD_right Fact 2: BC_left + BC_right = 1 year Substitute BC_left in Fact 2 with its equivalent, AD_left (per Fact 1) Therefore: AD_left + BC_right = 1 yearBy this we have shown that in BCtoAD anniversary timespans, BC_partial + AD_partial = 1 yearThis fact will simplify our anniversary timespan formula. Rather than determining BC_partial & AD_partial and adding them to the equation, we will merely add their combined value of 1. Constructing a formula for anniversary yearspans Begin with the structure of the complex yearspan : i. ii. iii. iv. BC_partial + (BC_yearNumber  1) + (AD_yearNumber  1) + AD_partialKnowing from the above proof that for any anniversary timespan , ( BC_partial + AD_partial = 1 year ), we insert 1 for parts i & iv: ii. iii. i. + iv. (BC_yearNumber  1) + (AD_yearNumber  1) + 1 = BC_yearNumber + AD_yearNumber  1  1 + 1 = BC_yearNumber + AD_yearNumber  2 + 1 = BC_yearNumber + AD_yearNumber  1
Simple yearspans: year_P to year_Q
Yearspan Definition: For two nonidentical years P and Q, the yearspan is the count of whole years, equal to the number of spaces on a numberline of years, required to shift year P to a position squarely overlapping year Q (not merely touching year Q ). Determine BC years In the preceding cases of complex and anniversary timespans, the measurement of Time Before Christ required counting two parts: Whole years; and a Partial year. 1. First, whole BC years were counted, beginning at the BCAD juncture and working backwards into BC time, halting right before touching any part of startyear P: thus yielding a wholeyear count of (P  1), not (P). This gave the whole years.Example: Find the BC portion of a timespan beginning 10177 BC 1. 77 years  1 year = 76 years Yet such a twopart process is not required for simple yearspans from "year_P to year_Q". As P is a full year, it is counted with the wholeyears in Step 1; this leaves no partial year to be counted in Step 2. The total BC years is simply the value of the BC year named in the yearspan. Example: Find the BC portion of a yearspan beginning with the year 400 BC 1. 400 yearsThe BC portion of the simple yearspan (year_Ptoyear_Q) is: P Determine AD years Per the yearspan definition above, the span is determined by counting how many years the startyear needs to shift until reaching overlap with the endyear.At the completion of counting the BC years, the imagined year being shifted on the numberline has fully crossed into the AD area; overlapping the first AD year (1 AD). January 1 of this sliding imaginary year overlays January 1, 1 AD. To complete the shifting process toward the AD endpoint, 111 AD must overlay 11[year_Q] AD. To achieve a shift from (year 1)to(year Q) requires a shift of (Q1) years (proven below). This will be the AD portion of the simple yearspan. Proof: Yearspans confined to AD time (or confined to BC time ) can be measured using normal numberlinemath; as the BCAD spectrum, when graphed, is identical with a numberline in all points but the zeroless transition. Therefore transactions that avoid this middleground can be graphed on a "timeline", exactly as they would be on a numberline. The required shift in years to move from 1toQ is given by the numberline distance formula:
distance d = A  B or d = B  A When applying absolute value to a difference such as (x  y), either order of the operands produces the same result: x  y = y  x Find the distance in AD_years from 1 AD to QAD = 1  Q = Q  1  Using this order of operands  avoids negativenumber math. = Q  1 Example: Find the distance from 1 AD to 2011 AD Solution: Let A = 1 and B = 2011 ==> d = 1  2011 = 2010 = 2010 OR Solution: Let A = 2011 and B = 1  Switching A & B to  show that order is  not important. ==> d = 2011  1 = 2010 = 2010The AD portion of the simple yearspan (year_Ptoyear_Q) is (Q  1) Adding BC and AD Thus a simple yearspan of year_P (BC) to year_Q (AD) can be stated as: BC + AD ___ ______ P + (Q  1) = P + Q  1
Formula summary
John O'Leary / Biblecalculator 