Ben Seifert
Calculus 120
Professor Webster
December 4th, 2015
Newton’s Method
Sir Isaac Newton developed many famous theories and methods in his
lifetime that advanced the world at an incredible rate. It is Newton that we can
attribute the laws of gravity, the foundations of calculus, and even the idea behind
the brain of a calculator. I am talking about the computer inside of these devices
that use Newton’s method every time a root is looking to be found. In this day and
age, we often use calculators mindlessly without ever wondering how something so
small could have just solved an equation that our brain could not even comprehend.
It is important to realize how this method works because it shows up every single
day of our lives. An analysis of this method and its various components will provide
a better understanding of how calculators work, as well as how advanced Newton
was for his time period.
Newton discovered his method for finding roots algorithmically in 1669 and
it became known to the general public at about 1711 (Newton’s Method). He came
about this discovery when he could not find the root to a quadratic equation. His
exact equation was f(x)=x^3-2x+5. With a little math it is easy to see that one
cannot solve this problem with simply with algebra. In fact, before Newton’s time no
one could solve this problem (probably because no one had calculators).
A Norwegian mathematician named Niels Abel proved in 1824 that there was
no general formula for solving for the roots of a 5th degree equation. Later on, a
French mathematician, Evariste Galois, proved the impossibility of finding a general
formula for the roots of an “nth” degree equation where n is larger than 4 (Stewart).
Newton came along used a method of approximation to solve for an answer very
close to the root trying to be solved.
Newton’s method was very simple, but it worked. He found that by looking
for the point that crosses the x-axis (positive) is nearly impossible unless one can
zoom in extremely close to that point. Instead, he used a method of approximation
that generated a lot of answers for x values, each getting either farther or closer to
