problem 1 I cannot live without...
Include your favorite passage from a book.
problem 2 The Finest Gambit
Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest
weapons. It is a far finer gambit than any chess play: a chess player may offer the
sacrifice of a pawn or even a piece, but a mathematician offers the game. [Excerpt
from A Mathematician’s Apology, G.H. Hardy, 1940, p. 94]
Learn how to write math and construct proofs by reproducing the proof below. You will need to
use the “align” environment, as well as the “align*” environment.
Definition 1 A rational number is a fraction a
b
where a and b are integers.
Show √
2 is irrational.
Proof.
For a rational number a
b
, without loss of generality we may suppose that a and b are integers
which share no common factors, as otherwise we could remove any common factors (i.e. suppose
a
b
is in simplest terms). To say √
2 is irrational is equivalent to stating that 2 cannot be expressed
in the form (
a
b
)
2
. Equivalently, this says that there are no integer values for a and b satisfying
a
2 = 2b
2
(1)
We argue by reductio ad absurdum (proof by contradiction). Assume toward reaching a contradiction that Equation 1 holds for a and b being integers without any common factor between
them. It must be that a
2
is even, since 2b
2
is divisible by 2, therefore a is even. If a is even, then
for some integer c
a = 2c
a
2 = (2c)
2
2b
2 = 4c
2
b
2 = 2c
2
therefore, b is even. This implies that a and b are both even, and thus share a common factor of 2.
This contradicts our hypothesis, therefore our hypothesis is false.
✷
problem 3 Vanity
Learn how to include drawings in your documents with the \includegraphics{file} command
by submitting a caricature of your professor.
