# Cousin Calculator

We understand that we are more closely related to our siblings than our first cousins, and we are closer to our first cousins than our second cousins. But what if I was third cousins with a person on my mother's side and fifth cousins with them on my father's side. Intuitively we have the sense that we are more closely related than either a fifth or a third cousin. But let's see how close.

(The following calculator is revised from the original given to me by Theron Smith and Bill Massey of Lewisburg, Tennessee.)

## Finding Ancestry

We define the following:

C = Cousinship.
We will let C=1, for a person's first cousin; C=2 for a second cousin, etc. For the purposes of this exercise, we let C=0 for a person's siblings.
A = Ancestry.
A person shares 100% of their ancestry with their full siblings; so for siblings, A=1.00. A person shares 1/2 of their grandparents with their full first cousins; so, for first cousins, A=0.50. For second cousins, A=.25, etc.

If a person knows their cousinship number, we can calculate the proportion of their shared ancestry. Because parents come in pairs, the number of ancestors double every generation. In mathematical terms, something that doubles every time period, is said to be growing at an exponential rate. The exponential base is 2 (for doubling). This also means the familial closeness falls away (decreases) at an exponential rate. It decreases by half with each generation. To turn exponential growth into exponential decreases, we divide into 1, thusly:

A = 1 / (2^C)
The proportion of shared ancestors is equal to one divided by two raised to the power of cousinship

## Ancestry Table

Here is a table showing the results

A = 1 / (2^C)
C A
0 1.000
1 0.500
2 0.250
3 0.125
4 0.063
5 0.031
6 0.016
7 0.008
8 0.004
9 0.002
10 0.001

This table tells us that a person shares 25 percent of his/her ancestry with their second cousins; 3.1 percent shared ancestry with fifth cousins, and 1/10 of 1 percent with 10th cousins.

## Double Cousin Example

Suppose a person is double cousins with another person. They are second cousins on their mother's side and third cousins on their father's side. Their total ancestry value would be equal to the mother's side cousinship plus their father's side cousinship.

A(total) = A(motherside) + A(fatherside)

In this case A(total) would be equal to the sum of A(secondcousin) and A(thirdcousin)

A(total) = 0.250 + 0.125
A(total) = 0.375

So in this case, the two people share 37.5% of their ancestry. Looking at the table above, we see they are somewhere between first and second cousins. They are something like first and a half cousins.

## Finding Cousinship

What we would like to do is find a formula that will take our ancestry number (0.375) and turn it into a cousinship number. We need to take the first formula

A = 1 / (2^C)

and fiddle with it so that it is

C = (some function of) A

We need to do two major operations: 1) get C out from under the 1; and 2) convert C from an exponent with a base of 2 to a stand-alone variable. To get C out from under the 1, we use the cross-multiply operation, which yields:

2^C = 1/A

To convert C from an exponent with a base of 2, we need to take the base2 logarithm of 2^C. (The base2 logarithm is also known as the binary logarithm). Applying Log2 to both sides we get:

C = Log2(1/A)

Because most calculators and spreadsheets don't have a Log2 function, we need to convert our log function to the most common or "natural" logarithm. This logarithm uses Euler's constant (e) as the base. The value of e is 2.718281... To convert the value 2 to e (to withn 6 decimal places) we need to multiply by 1.442695. Thus by adding this term to the equation, we can change from log2 to loge, which is known in most calculators and spreadsheets as "log" or "LN."

C = Log(1/A) * 1.442695

For our double cousin example, A = 0.375. The equation is solved as follows:

C = Log(1/0.375) * 1.443
C = Log(2.667) * 1.443
C = 0.981 * 1.443
C = 1.416

A cousinship number of 1.4 means that the a person who is a double cousin (2nd and 3rd) is as close as a 1.4th cousin (about a first and a half cousin, or in other words, they're about halfway between first cousins and second cousins).

## An Exercise

We would like some help thinking through the impact (in mathematical terms) of a cousin being once or twice removed. Leave us your ideas on the discussion page.