Calculating B.C. to A.D. or A.D. to B.C. Dates

Dates are calculated using a simple formula.  The basic formula is:

B.C. Date = A.D. Date - Number of Years - 1

In this formula, all B.C. Dates are negative numbers and A.D. Dates are positive numbers.  The Number of Years is how many years one wants to traverse across time to reach from an A.D. Date to a B.C. Date.  For example, one might want to know what the B.C. Date is if you start at 5 A.D. and go back 10 years.  Here is what the calculation would look like:

B.C. Date = 5 AD - 10 - 1 = 5 - 10 - 1 = -5 - 1 = -6 (the answer is -6 or 6 B.C.)

This formula above can be rearranged to yield several useful variations:

A.D. Date = B.C. Date + Number of Years + 1

Number of Years = A.D. Date - B.C. Date - 1

For difference of years for a B.C. to A.D. date

Let us consider for a moment how to derive and use this formula for dates.  The derivation is based on very simple arithmetic so should not be hard to understand.  Ordinary numbers can be calculated using a number line (for counting purposes) which has a zero, but for the B.C. and A.D. date system, there is no zero year between B.C. and A.D.  You go from 1 B.C. to 1 A.D. without going past zero.  Zero simply is not there on the time line.  Therefore, here is a graphic to illustrate the calculation.  Assume you are adding 10 years to 5 BC.  What year does it take you to on the A.D. scale?  The answer is 6 AD.  Here is a time line of the numbers with zero on it,

Standard Number Line

                    negative                                    positive numbers

        |-5  -4  -3  -2   -1  0  1  2  3  4   5  6  7  8  9--|

      0  1  2  3   4  5  6  7  8  9  10      

This is how the math appears to work on the standard number line.  As you can see, with the starting number at -5, you count forward 10 and it takes you to 5 (note that the zero count is under the -5 on the number line and you count forward from that to 10).  But this is a standard number line which has a zero.  The actual B.C. - A.D. year scale does not have a zero.

Here is how it should work out on the B.C. to A.D. scale     

                    B.C. dates                             A.D. dates

       |-5  -4  -3   -2  -1  1  2  3  4  5   6  7  8  9--|  no zero year

        0  1  2   3  4  5  6  7  8  9  10

As you can plainly see, counting forward 10 years from 5 B.C. on the B.C. - A.D. time scale takes you to 6 A.D.  Therefore, the formula for computing dates from B.C. forward past the B.C. - A.D. border is to do the following:

    A.D. Date = B.C. date + number of years + 1 = -5 + 10 + 1 = 6 A.D.

However, in the case of our example above:

     -5 + 10 + 1 = 5 + 1 = 6 A.D.  Remember that B.C. Dates have a negative value in this formula.

This converts the numbers computed using the number line, which has a zero, to a B.C. - A.D. time scale, which does not have a zero.  But remember this is for adding years to a BC date that will result in an AD year.

Now, lets do the reverse.  Subtract years from an A.D. year to derive a B.C. date.

Use a similar example to what was used above.  6 A.D. - 10 years = 4 B.C.

First, here is the calculation using a standard number line:

Standard Number Line

                               negative                              positive numbers

     |-7  -6  -5  -4  -3   -2  -1  0  1  2  3   4  5  6  7  8-|

                10  9  8  7  6  5  4   3  2  1  0

The result would appear to be -4 or 4 B.C.  The number line above has zero in it, but the actual B.C. - A.D. time scale has no zero year, so here is how it should actually look:

                                  B.C. Dates                         A.D. Dates

        |-7  -6  -5  -4   -3  -2  -1  1--2--3--4--5--6--7--8-|

                 10  9   8  7  6  5  4  3  2   1  0

As you can plainly see, counting backwards on the B.C. - A.D. time scale leads you to 5 B.C. as the correct answer.  So the general formula for calculating such dates would be:

B.C. Date = A.D. date - number of years - 1 

Drop any minus sign that is on the result and call it a BC date.  This only works for dates where the AD-BC date line is crossed.

So, in the case of the example given above, subtract 10 years from 6 A.D.  6 A.D. - 10 years - 1 = - 4 - 1 = -5 = 5 B.C.

Now suppose you want to know the number of years between an A.D. Date and a B.C. Date.  Here again is the formula:

Number of Years = A.D. Date - B.C. Date - 1

Using the same example, 6 A.D. and 5 B.C, here is what the result would be:

Number of Years = 6 - (-5) - 1= 6 + 5 - 1 = 10, which is consistent with the calculations above.

Hopefully this helps you understand the calculations for B.C. and A.D. dates.