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## Solution of the Equations
Let r be a large enough radius to make p(x) ≈ x Now we apply our two-dimensional counterpart to the IVT:
Since y = 0 is contained yay! The area of mathematics which seeks to understand
continuity and connectedness • Far-reaching generalizations of theorems like our
“Two-dimensional Intermediate • Precise notions of what it means for a curve to “wrap n
times around • Generalizations of winding numbers for wrapping
higher-dimensional shapes • Knot theory. • Bottles whose insides and outsides are indistinguishable. . . . to name a few. It has been said that topologists
can’t tell their donuts from You might think, then, that algebraists would have a tough
time coping with Well, yes. People can be cruel. But this fact really just serves to illustrate that no
field of mathematics is entirely Consider the polynomial p(x) = x
Theorem. Suppose p(x) is any polynomial of degree n > 0.
Then, if we Explanation: If p(x) has degree n > 0, the
regular-strength theorem shows Now, if n were 1, then p Otherwise, p Repeating this process exactly n times, we finally get to
the stage where p Double-yay! Here is another way we can interpret the Fundamental
Theorem of Algebra: The God created the natural numbers, and all the rest is the
work of In fact, we can think of the logical development of number
systems as a gradual • In the natural numbers, we can always add any pair of
numbers. But Z = {. . . − 3,−2,−1, 0, 1, 2, . . .}. • In the integers, we can always multiply, but only
sometimes divide. To
• But the rational numbers don’t include numbers like π
and
and also • And in R, equations like x
The Fundamental Theorem of Algebra then says:
We now have as many solutions as we need to any sort of
polynomial equation So, as you can see, the algebraists really have nothing to
be ashamed of. But Hug an Algebraist Today! |