Gauss: The Prince of Mathematics

As you progress further into college math and physics, no matter where you turn, you will repeatedly run into the name Gauss. Johann Carl Friedrich Gauss is one of the most influential mathematicians in history. Gauss was born on April 30, 1777 in a small German city north of the Harz mountains named Braunschweig. The son of peasant parents (both were illiterate), he developed a staggering number of important ideas and had many more named after him. Many have referred to him as the princeps mathematicorum, or the “prince of mathematics.”

As part of his doctoral dissertation (at the age of 21), Gauss was one of the first to prove the fundamental theorem of algebra. He went on to publish seminal works in many fields of mathematics including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy, optics, etc. Number theory was Gauss’s favorite and he referred to number theory as the “queen of mathematics.”

One of the reasons why Gauss was able to contribute so much math over his lifetime was that he got a very early start. There are many tales of his childhood precociousness. The most famous anecdote of young Gauss is the time he found the shortcut for calculating the sum of an arithmetic progression at the tender age of 10.

The anecdote involves his schoolteacher who wanted to take a rest and asked the students to sum the integers from 1 to 100 as busy work. After a few seconds, the teacher saw Gauss sitting idle. When asked why he was not frantically doing addition, Gauss quickly replied that the sum was 5050. His classmates and teacher were astonished, and Gauss ended up being the only pupil to calculate the correct answer.

The story may be apocryphal, and is told different ways in different sources. Nobody is sure which method of summing an arithmetic sequence Gauss figured out as a child. Though there are several ways young Gauss might have solved it, one of them has a concise, intuitive, and elegant visual representation.

\[ 1+2+3+ \cdots +(n-1)+n = ?\]

Fig 1: Arithmetic Progression

Consider two sets of marbles as shown in the Figure 1. The left pile has \( n\) rows of blue marbles, where the \( j^\text{th}\) row contains \( j\) marbles. The right pile has \( n\) rows of red marbles, where the \( j^\text{th}\) row contains \( n+1-j\) marbles.

The total number of blue marbles is given by

\[ 1+2 +3+ \cdots + (n-1)+n,\]

while the total number of red marbles is given by

\[ n+(n-1)+(n-2) + \cdots + 2 + 1,\]

and clearly both contain the same number of marbles. Now if we were to add these piles together as shown in Figure 2, we would then get a stack with \( n\) rows, where each row contains \( n + 1\) marbles:

Fig 2: Arithmetic Progression

The total number of marbles in the added pile would be \( n(n + 1)\). Since both the red pile and the blue pile have an equal number of marbles, each pile must have contributed \( \frac{n(n + 1)}{2}\) marbles. Hence, we obtain

\[ 1+2+3+...+(n-1)+n= \frac{n(n+1)}{2}.\]

To sum all the numbers from 1 to 100, Gauss simply calculated \( \frac{100\times (100+1)}{2}=5050\), which is immensely easier than adding all the numbers from 1 to 100. Note that \( 1+2+3 + \cdots +(n-1)+n\) must always be a positive integer. Even though the above formula divides by 2, the result will always be a positive integer. This is because the numerator will always be conveniently even due to the multiplication properties of parity. For example, \( n\) could either be even or odd. If \( n\) is even, then \( n+1\) is odd and hence

\[ n \times (n + 1) = even \times odd = even.\]

Similarly, if \( n\) is odd, then \( n + 1\) is even and hence

\[ n \times (n + 1) = odd \times even = even.\]

Therefore, the numerator is always even and \( \frac{n(n + 1)}2\) is always a positive integer.

Numbers of the form \( \frac{n(n + 1)}{2}\) are called triangular numbers, for reasons well illustrated in the above figures. The first few triangular numbers are

\[ 1,3,6,10,15,21,28,36, \ldots. \]

It is commonplace to encounter an application of summing an arithmetic sequence, both in classroom problems, and in describing the broader world. It is less common to meet 10 year olds who figure out the tricks of arithmetic progression for themselves. It is even less common for a precocious 10 year old to grow up to be nearly as prolific as Gauss.

• Proved the law of quadratic reciprocity. The law of quadratic reciprocity is a theorem about quadratic residues modulo an odd prime. It states:

\[\large \left(\dfrac{p}{q}\right)\left(\dfrac{q}{p}\right)=(-1)^{\frac{p-1}{2} \frac{q-1}{2} },\]

where \(p\) and \(q\) are odd prime numbers, and \(\left(\dfrac{p}{q}\right)\) denotes the Legendre symbol.

• Formulated Gauss' lemma, which gives a condition for an integer to be a quadratic residue. Gauss' lemma states:

\(\frac{a}{p}\) = 1 iff an an even number of least positive residues of \({a, 2a, 3a, ..., ((p-1)/2)a}\) exceed \(\frac{p}{2}\)

• Proved that every positive integer is the sum of at most 3 triangular numbers. The proof follows from the result that every positive integer \(\equiv\) 3 (mod 8) can be written as a sum of three squares.

• Proved the Theorema Egregium, a major theorem in the differential geometry of curved surfaces. This theorem states that the Gaussian curvature is unchanged when the surface is bent without stretching.

• Made important contributions to statistics and probability theory. The Gaussian probability distribution is named after Gauss.