Gaussian empirical formula

The Gaussian sum formula (not to be confused with a Gaussian sum ), also called the small Gaussian , is a formula for the sum of the first consecutive natural numbers : ${\displaystyle n}$

${\displaystyle 1+2+3+4+\dotsb +n=\sum _{k=1}^{n}k={\frac {n(n+1)}{2}}={\frac {n^{2}+n}{2}}}$

This series is a special case of the arithmetic series and its sums are called triangular numbers. ${\displaystyle 1,3,6,10,\dotsc }$

Illustrations

Numerical illustration

The formula can be illustrated as follows: Write the numbers from 1 to ascending in a line. Below you write the numbers in reverse order: ${\displaystyle n}$

${\displaystyle {\begin{array}{ccccc}1&2&\ldots &n-1&n\\n&n-1&\ldots &2&1\\\hline n+1&n+1&\ldots &n+1&n+1\end{array}}}$

The sum of each column is Since there are columns, the sum of the numbers in both rows is the same To find the sum of the numbers in one row, the result is halved and the formula above is given: ${\displaystyle n+1.}$ ${\displaystyle n}$ ${\displaystyle n\cdot (n+1).}$

${\displaystyle 1+2+3+4+\dotsb +n={\tfrac {1}{2}}\cdot n\cdot (n+1)}$

Geometric illustration

In the picture below, the individual summands are arranged as rows of green boxes to form a triangle , which is expanded by the white boxes to form a square with side length . Simply halving the square along one of its diagonals would also divide the boxes exactly on the diagonal, which is undesirable. Therefore, the square on the right is supplemented by a column with blue boxes to form a rectangle whose halving along the red line splits off the green boxes exactly as desired . ${\displaystyle n}$ ${\displaystyle n}$

You only need to halve the number of all boxes, which immediately leads to the number of green boxes you are looking for. ${\displaystyle n\cdot (n+1)}$ ${\displaystyle {\frac {n\cdot (n+1)}{2}}}$

Origin of the designation

This molecular formula as well as the molecular formula for the first square numbers was already known in pre-Greek mathematics. ${\displaystyle n}$

Carl Friedrich Gauß rediscovered this formula when he was nine years old. The story is passed down by Wolfgang Sartorius von Waltershausen :

“The young Gauss had barely entered the arithmetic class when Büttner gave up the summation of an arithmetic series. The task, however, was hardly pronounced when Gauss threw the blackboard on the table with the words spoken in the lower Brunswick dialect: "Ligget se '." (There she lies.) "

- Wolfgang Sartorius von Waltershausen

The exact task has not been passed down. It is often reported that Büttner had the students add up the numbers from 1 to 100 (according to other sources, from 1 to 60). While his classmates began to diligently add, Gauss found that the 100 numbers to be added can be grouped into 50 pairs, each of which has the sum 101: up to So the searched result had to be equal to the product . ${\displaystyle 1+100,2+99,3+98}$ ${\displaystyle 50+51.}$ ${\displaystyle 50\cdot 101}$

Sartorius also reports:

“At the end of the lesson the tables were turned upside down; that of Gauss with a single number was on top and when Büttner checked the example, to the amazement of all those present, his was found to be correct ... "

- Wolfgang Sartorius von Waltershausen

Büttner soon realized that Gauss could no longer learn anything in his class.

proof

There is ample evidence for this molecular formula. In addition to the proof of the forward and backward summation presented above, the following general principle is also of interest:

To prove that for all natural ${\displaystyle n}$

${\displaystyle \sum _{k=1}^{n}f(k)=g(n)}$

applies, it is sufficient

${\displaystyle g(n)-g(n-1)=f(n)}$

for all positive and ${\displaystyle n}$

${\displaystyle g(0)=0}$

to show. Indeed, this is true here:

${\displaystyle g(n)-g(n-1)={\frac {n(n+1)}{2}}-{\frac {(n-1)n}{2}}={\frac {n(n+1-n+1)}{2}}={\frac {n\cdot 2}{2}}=n=f(n)}$

for everyone and ${\displaystyle n}$

${\displaystyle g(0)={\frac {0\cdot 1}{2}}=0}$

A proof of the Gaussian empirical formula with complete induction is also possible.

Related sums

From the Gaussian sum formula, by applying the distributive law and other similar elementary calculation rules, formulas for the sum of the even or the odd numbers can easily be obtained.

${\displaystyle \sum _{k=1}^{n}2k=2\cdot \sum _{k=1}^{n}k=2\cdot {\frac {n(n+1)}{2}}=n(n+1)}$

returns the sum of the first consecutive even numbers: ${\displaystyle n}$

${\displaystyle \sum _{k=1}^{n}2k=n(n+1)}$

The formula for the sum of the first consecutive odd numbers ${\displaystyle n}$

${\displaystyle \sum _{k=1}^{n}(2k-1)=n^{2}}$

results like this:

${\displaystyle \sum _{k=1}^{n}(2k-1)=2\cdot \sum _{k=1}^{n}k-\sum _{k=1}^{n}1=2\cdot {\frac {n(n+1)}{2}}-n=(n^{2}+n)-n=n^{2}}$

The sum of the first consecutive square numbers ${\displaystyle n}$

${\displaystyle \sum _{k=1}^{n}k^{2}={\frac {1}{6}}n(n+1)(2n+1)}$

is called the quadratic pyramidal number . A generalization to any positive integer as an exponent is Faulhaber's formula .

literature

• Wolfgang Sartorius von Waltershausen : Gauss to the memory. S. Hirzel, Leipzig 1856, pp. 12-13 (anecdote about Gauss, Google book ).
• Otto Neugebauer : Lectures on the history of the ancient mathematical sciences. First volume. Pre-Greek mathematics. Springer, 1969, pp. 172-173.
• Brian Hayes: Gauss's Day of Reckoning. In: American Scientist. 94, 2006, p. 200, doi: 10.1511 / 2006.3.200 .

Web links

Wikibooks: Math for Non-Freaks: Gaussian Sum Formula  - Learning and Teaching Materials

• Derivation of the Gaussian empirical formula simply explained in two ways ( YouTube video)
• Geometric proof of the Gaussian empirical formula on Vimeo
• Versions of the Gauss Schoolroom Anecdote. ( Memento from March 22, 2014 in the Internet Archive ).
• Video: The Gaussian Sum Formula (Part 1) . University of Education Heidelberg (PHHD) 2012, made available by the Technical Information Library (TIB), doi : 10.5446 / 19756 .
• Video: The Gaussian Sum Formula (Part 2) . University of Education Heidelberg (PHHD) 2012, made available by the Technical Information Library (TIB), doi : 10.5446 / 19757 .
• Video: The Gaussian Sum Formula (Part 3) . University of Education Heidelberg (PHHD) 2012, made available by the Technical Information Library (TIB), doi : 10.5446 / 19758 .

Individual evidence

1. ^ Sartorius von Waltershausen: Gauss for memory. 1856, p. 12 ( excerpt (Google) )
2. ^ Sartorius von Waltershausen: Gauss for memory. 1856, p. 13 ( excerpt (Google) )
3. ↑ Marko Petkovsek, Herbert Wilf, Doron Zeilberger: A = B . 1997, p. 10 ( math.upenn.edu ).