The document discusses numbers, including whole numbers like even and odd numbers. It explains that: - The sum of two even numbers or two odd numbers is even, because even numbers can be written as 2m and odd numbers can be written as 2m+1, so their sums are even. - The product of two even numbers is even, because when multiplying two even numbers written as 2m and 2n, the result is 2(mn) which is even. - It also investigates other number properties and operations like the primes related to a game of lockers.

2. 5 ? What does the number 5 mean?
How could you explain the meaning
of the word „five“ to someone who
doesn‘t know it? (e.g. a child or an alien)

3. 5 ?

4. What is a number? • Representation of
Numbers
Paring of sets
10011000111000011

5. DIGITS are to _ _ _ _ _ _ _
LETTERS are to _ _ _ _ _.
The same AS

6. DIGITS are to NUMBERS
LETTERS are to WORDS.
The same AS

7. DIGITS are the symbols,
with which we write NUMBERS.
LETTERS are the symbols,
with which we write WORDS.
JUST AS

9. What kind of whole numbers do you know?
e.g. Even and Odd Numbers
Denote two even numbers by g1 and g2 and two odd numbers by
u1 and u2. What can we say about
„ g1 + g2 “, „ g1 * g2 “, „ u1 + u2 “, „ u1 * u2 “ etc. ?
Whole Numbers

10. Even+Even = Even – why?
• Every even number can be written as 2m, i.e. it is a double of some whole
number m (e.g. 8=2∙4, here m = 4)…. and vice versa - every number that can
written as 2m is even
• So every two even numbers g1 und g2, can be represented by 2m und 2n,
respectively (z.B. 8, 14 sind 2∙4, 2∙7)
• Now let‘s add these two even numbers: g1 + g2 = 2m+2n
• g1 + g2 =2m+2n=2(m+n) - > we obtain 2∙(whole number), i.e. an even number

11. Even ∙ Even = Even – why?
• This time we multiply two arbitrary even numbers:
g1∙ g2 =2m ∙ 2n e.g. 8∙14 = 2∙4 ∙ 2∙7 = 2∙(4∙2∙7)
• In general, g1∙ g2 = 2m ∙ 2n = 2∙(m∙2∙n) What have we obtained?
And what about dividing two even numbers?

12. • odd + odd = even – why??
• How can we represent an arbitrary odd number?
• u = 2m+1 (e.g. 9=2∙4+1)
• Any two odd numbers u1 und u2 can be written as
u1 =2m+1 and u2 =2n +1 (e.g. 9, 15 can be written as 2∙4+1, 2∙7+1)
• Now we add these two arbitrary odd numbers:
u1 + u2 = 2m+1+2n+1 = 2m+2n+2 = even + even + even = even !
e.g. 2∙4+1+ 2∙7+1 = 2∙4+2∙7+2

13. The set of even numbers is closed under the operations of addition and
multiplication.
The set of odd numbers is closed under multiplication but not under
addition.
What about other operations? − , ÷ ?? Investigate to see if the set of even
numbers and the set of odd numbers are closed under them.

14. The Game of Lockers: Intro to Prime Factorisation
One hundred students are assigned 100 lockers.
In the beginning all the lockers are closed.
The Student #1 runs through and opens all the lockers. Then the Student
#2 runs through, closes her locker and every second locker after that. The
Student #3 similarly goes and changes every third locker. That means if he
finds Locker #3 open, he closes it, and if he finds it closed, he opens it. And
the same with Locker #6, #9, etc.
Then comes Student #4 and she does the same for Locker #4 and every
4th locker after that.
All the way up until Student #100.
Which lockers are open and which are closed at the end?