What is the Gauss formula

Gaussian empirical formula

 

The Gaussian empirical formula, also little gauss called, is a formula for the sum of the first consecutive natural numbers:

This series is a special case of the arithmetic series and its sums are called triangular numbers.

illustration

The formula can be illustrated as follows: Write the numbers from 1 to ascending in one line. Below you write the numbers in reverse order.

The sum of the columns gives the value Because it Columns, the sum of the numbers in both rows is the same To the sum of the numbers one To determine the line, the result is halved and the formula above is given:

Origin of the designation

This molecular formula as well as the empirical formula for the first Square numbers were already known in pre-Greek mathematics.

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 hardly 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 it is.)"

- Wolfgang Sartorius von Waltershausen

The exact task has not been handed 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) and Gauss found that the first and last number (1 + 100), the second and the penultimate number (2 +99) etc. together always result in 101. The value of the sum you are looking for is 101 times 50.

According to the circumstances at the time, Büttner taught about 100 students in one class. At that time, punishments with the so-called Karwatsche (carbatsche, leather whip) were common. Sartorius reports: “At the end of the lesson, the calculation tables were reversed; that of Gauss with a single number was at the top and when Büttner checked the example, to the amazement of all those present, his was found to be correct, while many of the others were wrong and were immediately rectified with the Karwatsche. ”Büttner soon recognized that Gauss was in his Class couldn't learn anything more.

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

applies, it is sufficient

for all positive and

to show. Indeed, this is true here:

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

Related sums

Occasionally, the sum formulas for the sum of the even or the odd numbers are also required:

The first formula is obtained by multiplying the basic formula by 2. The sum of the odd numbers then results as follows:

The similar looking sum of the square numbers

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

Based on an article in: Wikipedia.de
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Date of the last change: Jena, 13.02. 2021