Class 3

Went of homework problems and had homework quiz.

Next homework:

2.3 - 2, 3, 6, 11, 17

2.4 - 2, 8

On the half open interval (a,b], the number of integers between b and a is N= floor(b)-floor(a).

On the half open interval [a,b), N=ceiling(b)-ceiling(a)

On the open interval (a,b), consider 4 cases:

1. Both a and b are not integers N=floor(b)-floor(a).

2. a is an integer and b is not, N=floor(b)-floor(a), just like in 1. So we can generalize: if b is not an integer, N=floor(b)-floor(a).

3. a not an integer, b is an integer N=floor(b)-floor(a)-1

4. a is an integer, b is an integer, N=floor(b)-floor(a)-1, just like in 3. So we can generalize: if b is an integer, N=floor(b)-floor(a)-1

So we have two general formulas:

N=floor(b)-floor(a)

and

N=floor(b)-floor(a)-1

We want something that is 1 when b is an integer and 0 when it's not.

Notice that ceiling(b)-floor(b) = {1 if b is not an integer; 0 if b is an integer}

N=ceiling(b)-floor(a)-1

Unique Factorization

The fundamental theorem of arithmetic: Every positive integer greater than 1 can be factored uniquely into a product of primes.

n = p₁^a₁p₂^a₂p₃^a₃...

This is called the canonical form.

We can find the total number of divisors: first find the prime factorization.

252 = 2²3²7¹

# of divisors = (e1+1)(e2+1)(e3+1) = 3 x 3 x 2

i.e. multiply each of the exponents plus 1

see proposition 2.3.5 - square root of 2 is irrational

proved using unique factorization

see proposition 2.3.8 - find the exponent of p in the factorization of n!