Carl Friedrich Gauss
Carl Friedrich Gauss
is sometimes referred to as the "Prince of Mathematicians" and the
"greatest mathematician since antiquity". He has had a remarkable
influence in many fields of mathematics and science and is ranked as one
of history's most influential mathematicians.
Gauss was a child prodigy. There are many anecdotes concerning his
precocity as a child, and he made his first ground-breaking mathematical
discoveries while still a teenager.
At just three years old, he corrected an error in his father payroll
calculations, and he was looking after his father’s accounts on a
regular basis by the age of 5. At the age of 7, he is reported to have
amazed his teachers by summing the integers from 1 to 100 almost
instantly (having quickly spotted that the sum was actually 50 pairs of
numbers, with each pair summing to 101, total 5,050). By the age of 12,
he was already attending gymnasium and criticizing Euclid’s geometry.
Although his family was poor and working class, Gauss' intellectual
abilities attracted the attention of the Duke of Brunswick, who sent him
to the Collegium Carolinum at 15, and then to the prestigious
University of Göttingen (which he attended from 1795 to 1798). It was as
a teenager attending university that Gauss discovered (or independently
rediscovered) several important theorems.
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Graphs of the density of prime numbers
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At 15, Gauss was the first to find any kind of a pattern in the
occurrence of prime numbers, a problem which had exercised the minds of
the best mathematicians since ancient times. Although the occurrence of
prime numbers appeared to be almost competely random, Gauss approached
the problem from a different angle by graphing the incidence of primes
as the numbers increased. He noticed a rough pattern or trend: as the
numbers increased by 10, the probability of prime numbers occurring
reduced by a factor of about 2 (e.g. there is a 1 in 4 chance of getting
a prime in the number from 1 to 100, a 1 in 6 chance of a prime in the
numbers from 1 to 1,000, a 1 in 8 chance from 1 to 10,000, 1 in 10 from
1 to 100,000, etc). However, he was quite aware that his method merely
yielded an approximation and, as he could not definitively prove his
findings, and kept them secret until much later in life.
Gaussian Elimination
It was introduce as a procedure for solving a system of linear equations.
Matrix
- is a rectangular array of numbers a, symbol or expression arranged in rows and columns
Entry
- Is the individual items in the matrix
aij
Row Subscript
- The index
i above is called row subscript because it identifies the row in which the entry lies
Column Subscript
- The index
j is called the Column Subscript because it identifies the column in which the entry lies
Square of Order
n
- A matrix with m rows and n columns (an m x n ) is said to be the size. If m=n then the matrix is called square of order n
Main Diagonal Entries
a11, a12, a33...
Augmented Matrix
-the matrix derived from the coefficient and constant terms of a system of linear equation
Coefficient Matrix
- The matrix containing only the coefficients of the system
Elementary Row Operations
1. Interchange two equations
2. Multiply an equation by a non-zero constant
3.Add a multiple of an equation to another equation
Examples:
references : http://www.storyofmathematics.com/19th_gauss.html
http://www.math.dartmouth.edu/archive/m23s06/public_html/handouts/row_reduction_examples.pdf
