Here are the notes for last semester.

We'll start lecture 13 by fixing the broken proof that every number has a prime decomposition that we started in lecture 12.

Be sure you understand why we are stuck at the end of that lecture.

We'll also prove the existence and uniqueness of the quotient and remainder. Come to lecture knowing the following definitions and how they relate to your existing notion of quotient and remainder:

We say

[math]q \href{/cs2800/wiki/index.php/%5Cin}{\in} \href{/cs2800/wiki/index.php/%E2%84%A4}{ℤ}
[/math] is a

quotient of

[math]a
[/math] over

[math]b
[/math] if

[math]a = qb + r
[/math] for some

[math]r \href{/cs2800/wiki/index.php/%5Cin}{\in} \href{/cs2800/wiki/index.php/%E2%84%A4}{ℤ}
[/math] with

[math]0 \leq r \lt b
[/math]. We write

[math]q = quot(a,b)
[/math] (note that

quot is a

well defined function).

We say

[math]r \href{/cs2800/wiki/index.php/%5Cin}{\in} \href{/cs2800/wiki/index.php/%E2%84%A4}{ℤ}
[/math] is a

remainder of

[math]a
[/math] over

[math]b
[/math] if

[math]a = qb + r
[/math] for some

[math]q \href{/cs2800/wiki/index.php/%5Cin}{\in} \href{/cs2800/wiki/index.php/%E2%84%A4}{ℤ}
[/math] and

[math]0 \leq r \lt b
[/math]. We write

[math]r = rem(a,b)
[/math] (note that

rem is a

well defined function).

Time permitting, we may start discussion the base b representation of a number; for this we will need the following definitions:

numbers [math]d
[/math] satisfying

[math]0 \leq d \lt b
[/math] are called

base b

digits.

If

[math]d_k, d_{k-1}, \dots, d_1, d_0
[/math] are all

natural numbers satisfying

[math]0 \leq d_i \lt b
[/math] for all [math]i
[/math], then the

base b interpretation of

[math]\href{/cs2800/wiki/index.php/Sequence_notation}{(d_i)}
[/math], written

[math](d_kd_{k-1}\cdots{}d_1d_0)_b
[/math] is given by

[math]\href{/cs2800/wiki/index.php/Base}{(d_i)_b} := \sum_{i} d_ib^i
[/math]