## Sunday, July 28, 2013

### Gauss .... and me

Question: what is the sum of the first 100 whole numbers?? how am i supposed to work this out efficiently? --
The question you asked relates back to a famous mathematician, Gauss. In elementary school in the late 1700’s, Gauss was asked to find the sum of the numbers from 1 to 100. The question was assigned as “busy work” by the teacher, but Gauss found the answer rather quickly by discovering a pattern. His observation was as follows:

1 + 2 + 3 + 4 + … + 98 + 99 + 100

Gauss noticed that if he was [sic] to split the numbers into two groups (1 to 50 and 51 to 100), he could add them together vertically to get a sum of 101.

1 +    2 +   3 +    4 +    5 + … + 48 + 49 + 50
100 + 99 + 98 + 97 + 96 + … + 53 + 52 + 51

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
.
.
.
48 + 53 = 101
49 + 52 = 101
50 + 51 = 101

Gauss realized then that his final total would be 50(101) = 5050.

The sequence of numbers (1, 2, 3, … , 100) is arithmetic and when we are looking for the sum of a sequence, we call it a series. Thanks to Gauss, there is a special formula we can use to find the sum of a series:

S=n(n+1)/2
I had essentially worked out this pattern as a very young child, by third or fourth grade, using its principle to more easily sum rows of numbers. One of my sisters also has always used this arithmetic shortcut (we can't recall whether I taught it to her or she independently discovered it), and taught it to her daughters when they were kids.

When I was in eighth grade, our teacher told us all (both eighth and seventh grades shared a class room) to write down the numbers one through one hundred and then sum them. I started applying my habitual method for summing long rows of numbers (finding '5s' and '10s' in each column) ... and then realized that I could group entire rows by '100s'.

That is, whereas Johann Karl Friedrich "Gauss noticed that if he [were] to split the numbers into two groups (1 to 50 and 51 to 100), he could add them together vertically to get a sum of 101", I noticed that I could sum the numbers in this manner: 0+100 1+99 2+98 ... such that a formula to solve the problem was 50(100)+50. I spent more time double- and triple-checking my reasoning ("Is it really that simple?") than on working out the answer.