### What is Math - Houston Independent School District

```What is
Math?
Niels Østergård, Birkerød Gymnasium, May 2006
Lesson 1
1
Math = Language
Syntax / grammar
Vocabulary
Communication between
mathematicians, student/teacher, ...
Math > Language
Math < Language
You can make statements
that are difficult or
impossible to make in
natural languages.
Truth-values of statements
can be decided within the
system.
You can’t say “I love you”
Niels Østergård, Birkerød Gymnasium, May 2006
Mathematical statements do
not refer to the real World.
...or do they?
Lesson 1
2
Scale of abstraction
Crafts
Engineering

Exact
Sciences

Concrete
Specific
Synthesis
Matter
Reality
Niels Østergård, Birkerød Gymnasium, May 2006
Applied
Math

Lesson 1

Pure
Math
Meta
Math

Abstract
General
Analysis
Ideas
Model
3
Mathematical modelling
Reality
Model
Ferryboat
Plastic boat
Paper on
desktop
Classical
geometry
Classical
geometry
Coordinate
geometry
All sets with
three elements
Number three
Union of
disjoint sets
3
3+2=5
 Abstraction

Lesson 1
Niels Østergård, Birkerød Gymnasium, May 2006
4
Math is a formal system
Whenever you ask “why” and get an answer, you can ask “but why that”?
This leads to infinite regress.
You have to stop somewhere, take something on faith, as rules of the game.
In math, besides the laws of logic, we have axioms or postulates,
i.e. simple (primitive) properties of simple undefined concepts.
The axioms are postulated, not proved.
New theorems (more complicated properties of the concepts) are proved by
applying deductive logic to axioms and to already proven theorems.
Mathematical theorems are true in an absolute sense different from scientific truth
(i.e. empirical or inductive truthsLesson
like 1“The sun will rise tomorrow”).
Niels Østergård, Birkerød Gymnasium, May 2006
5
Euclidean geometry
Fundamental concepts
point
distance
circle
line
angle
parallel
Axioms / postulates
1. Unique line segment between any two points
2. Unique indefinite extension of line segment
3. Unique circle with any centre and radius
4. All right angles are equal
5. Parallel postulate, e.g.:
If two lines are drawn which intersect a third in such a way that the sum
of the inner angles on one side is less than two right angles, then the two
lines inevitably must intersect each other on that side if extended enough.
Or:
Given any line and any point not on the line, one and only one
line exists through the point that doesn't intersect the first line.
Niels Østergård, Birkerød Gymnasium, May 2006
Lesson 1
6
Euclidean plane geometry vs. spherical geometry
Concept
Plane
Sphere
"point"
point
pair of opposite points
"line"
"circle"
Parallel
postulate
straight line
circle
great circle
minor circle
Valid
Invalid
Niels Østergård, Birkerød Gymnasium, May 2006
Lesson 1
7
Sum of angles
Proof of angular sum A+B+C = 180° for arbitrary triangle:
This proof uses the parallel postulate, so it is not valid for spherical triangles!
Niels Østergård, Birkerød Gymnasium, May 2006
Lesson 1
8
Division of circles
Problem: A number of points are
marked on the perimeter of a circle,
and all the lines connecting pairs of
these points are drawn.
Into how many regions can the lines cut the circle
for a certain number of points?
Notation: Let us call the number of points n, and let
us call the largest possible number of regions for a
given number of points R(n).
Niels Østergård, Birkerød Gymnasium, May 2006
Lesson 1
9
Solution:
Let us consider the first few cases:
n Figures R(n)
n Figures R(n)
1
1
4
8
2
2
5
16
3
4
6
?
Niels Østergård, Birkerød Gymnasium, May 2006
Lesson 1
10
The case n = 6
2
9
3
11 12 13
10
26
14
27
15
16
1
8
25
28
31
18
29
7
24
23
17
4
19
30
22 21
20
5
6
R(6) = 31
Niels Østergård, Birkerød Gymnasium, May 2006
Lesson 1
11
Number of subsets
Problem: How many different subsets does a set with n elements have?
Note: The empty set (Ø) and the set itself are always counted as subsets.
Notation: The number of subsets is called S(n).
N
0
1
2
3
4
Set
Subsets
Ø
Ø
{a}
Ø, {a}
{a,b}
Ø, {a}, {b}, {a,b}
{a,b,c}
Ø, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}
abcd
Ø, a, b, c, d, ab, ac, ad, bc, bd, cd, abc, abd, acd, bcd, abcd
5
abcde
Ø, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abc, abd,
abe, acd, ace, ade, bcd, bce, bde, cde, abcd, abce, abde,
acde, bcde, abcde
6
abcdef
Niels Østergård, Birkerød Gymnasium, May 2006
S(n)
1
2
4
8
16
32

Lesson 1
12
Better approach
Listing the subsets in a different order, a pattern emerges:
n
Set
Subsets
S(n)
0
Ø
1
a
2
ab
3
abc
4
abcd
5
abcde
1
Ø
Ø
2
Ø  {a} = {a}
Ø
a
Ø  {b} = {b}
{a}  {b} = {a,b}
4
Ø
{a}
{b}
{a,b}
Ø  {c} = {c}
{a}  {c} = {a,c}
{b}  {c} = {b,c}
{a,b}  {c} = {a,b,c}
Ø
a
b
ab
c
ac
bc
abc
Ød = d
bd = bd
... = abd
cd
acd
bcd
abcd
8
16
32
Mathematical induction
For n = 0
For any
n>0
Obviously S(0)=1.
Given the S(n-1) subsets of the set with n-1 elements, the S(n) subsets of the
set with n elements are: The same S(n-1) sets, plus the same S(n-1) sets with
the new element added to each of them. Thus S(n) = 2·S(n-1).
n
As
this
argumentBirkerød
works for
any valueMay
of n,2006
the doubling sequence
Niels
Østergård,
Gymnasium,
Lesson 1 (given by the formula S(n) = 2 ) will go on for ever. 13
What is
Good
Math?
Niels Østergård, Birkerød Gymnasium, May 2006
Lesson 2
14
Tennis tournament
A tennis club arranges a knockout tournament in which
the winner takes it all. There are 843 participants.
Round
Matches
No.
1st
In the first round of the tournament, the 843 players are matched; one odd
player is left out in this round. This means that 842/2=421 matches are played.
421
2nd
In the second round, the 421 winners plus the odd player are matched. Thus,
422/2=211 matches are played.
211
3rd
In the third round, the 211 winners from the second round are matched; again
one odd player must be left out.
105

 and so on 
Last

In the last round, only two winners are left so only one match is played – the
final!
Problem: How many matches were played before the final winner
was found? (A relevant question if you have to pay the ballboys!)
Solution: 421+211+105+ . . . +1 =
Niels Østergård, Birkerød Gymnasium, May 2006
Lesson 2
1
Sum
?
15
Better solution
At the end of the tournament, all players except the final
winner will have lost exactly one match.
In each match there is exactly one looser.
Therefore, the number of matches equals the number of
players excluding the final winner.
Suppose the tournament has n participants.
Then, n-1 matches will be played before the winner is
found!
What qualities make this a better solution?
Niels Østergård, Birkerød Gymnasium, May 2006
Lesson 2
16
Qualities
Beautiful
Elegant
Eye opener
Aha!
General
Productive
Practical
Useful
Niels Østergård, Birkerød Gymnasium, May 2006
Lesson 2
17
You need math
Most occupations today involve math
Numbers and calculations, e.g. related to
economy
Formal systems, e.g. computer programs and
instruction manuals
Many news stories involve math (and very often get it wrong)
Economy,
large numbers
Science and
technology
Graphs, diagrams
and tables
Statistics and
percentages
In a democracy, as many citizens as possible should understand as much as possible about society.
Your everyday life involves math
Interests, compound interests
and mortgage
Percentages, e.g. moms
(VAT) and discounts
Niels Østergård, Birkerød Gymnasium, May 2006
Lesson 2
Measurements for interior
decoration...
18
You need math
Most occupations today involve math
Numbers and calculations, e.g. related to
economy
Formal systems, e.g. computer programs and
instruction manuals
Many news stories involve math (and very often get it wrong)
Economy,
large numbers
Science and
technology
Graphs, diagrams
and tables
Statistics and
percentages
In a democracy, as many citizens as possible should understand as much as possible about society.
Your everyday life involves math
Interests, compound interests
and mortgage
Percentages, e.g. moms
(VAT) and discounts
Niels Østergård, Birkerød Gymnasium, May 2006
Lesson 2
Measurements for interior
decoration...
19
```