Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Abuse of notation wikipedia, lookup

Mathematics of radio engineering wikipedia, lookup

Georg Cantor's first set theory article wikipedia, lookup

Four color theorem wikipedia, lookup

Fermat's Last Theorem wikipedia, lookup

Fundamental theorem of calculus wikipedia, lookup

Large numbers wikipedia, lookup

Wiles's proof of Fermat's Last Theorem wikipedia, lookup

Theorem wikipedia, lookup

Mathematical proof wikipedia, lookup

Collatz conjecture wikipedia, lookup

Elementary mathematics wikipedia, lookup

Number theory wikipedia, lookup

Fundamental theorem of algebra wikipedia, lookup

Proofs of Fermat's little theorem wikipedia, lookup

Transcript
```HOMEWORK 1: DUE THURSDAY SEPTEMBER 29
INSTRUCTOR: PAUL APISA
For each problem, communicating how you reasoned your way to the answer you’re
submitting is far more important than the answer itself. “Show” is shorthand for “Write a
proof or produce a counterexample”.
1. Practice with the Rational Numbers
Remember that a number r is said to be rational if r = ab where a and b are integers
and b is nonzero. Recall that the integers are the counting numbers along with 0 and their
negatives, i.e. {. . . , −2, −1, 0, 1, 2, . . .}.
Exercise 1: If a and b are irrational numbers is a + b irrational? What about ab?
√
√
Exercise 2: Imitate the proof of the irrationality of 2 to show that 3 is irrational.
2. Practice with Induction and Combinatorics
Recall that
is the number
Pof ways to choose k objects from a collection of n objects.
Recall too that, by definition, nk=0 ak = a0 + a1 + . . . + an
P
Exercise 3: Show that nk=0 nk = 2n . (Hint: As it’s written, this is an intimidating
be equivalent), you’re at a family reunion with n people and your family wants to
photograph every possible combination of people (including a photo with no one in
it). Show that they will have to take 2n photos.)
Note that a reformulation of the previous result is that the sum of the nth row of Pascal’s
triangle is 2n .
Exercise 4: Let Sn = 1 + 3 + 5 + . . . + (2n − 1) be the sum of the first n odd numbers.
Guess a closed formula for Sn and prove that it holds using induction.
n
Exercise 5: Show that n+1
= k−1
+ nk . (Hint: There is a symbol pushing way
k
of showing
this. However, the approach I hope you take goes back to the definition
n
of k . In fact, I challenge you to write a proof of this statement using only words
(no equations)).
Note that a reformulation of the previous result is the rule that the sum of two horizontally adjacent entries in Pascal’s triangle gives the entry underneath them.
Exercise
Binomial Theorem: Use induction to show that (x + y)n =
6 -k The
Pn
n
n−k
k=0 k x y
The binomial theorem will be very important in the coming weeks. For instance, we’ll
come back to it to find the derivative of f (x) = xn for n a positive integer.
n
k
1
```
Related documents