Mathematics is an important base for Information Technology
Discrete Mathematics I mainly covers discrete mathematics (logic,...)
Mathematics is a tool, a language, and a way of thinking
English is very important for Information Technology
English is best learned by diving in and practicing
Last Week's Homework
Make sure you are officially registered for the course and enrolled in Discrete
Mathematics I
Review decimal number representation, binary number representation, and
n-ary number representation based on high-school notes and Web
resources
Write a short report (~1/2 page) about hexadecimal number representation
(16進法) and submit it
to Moodle; deadline: September 23, 22:00
Solve the Quiz Simple
Arithmetic in English (repeat until you get it 100% correct;
deadline: September 24, 22:00 (no extensions))
Check out/buy/lend a textbook or reference book
Comments on Homeworks
Checked scan quality, found the following problems, left comments:
Not white enough background
Does not print as A4
Not enough focus
Missing contrast
Not handwriting
Missing name/student number
Not yet checked: content
In some cases, very thin (e.g. only a table)
How to Watch Videos
The video of the first lecture is available via a link from Moodle.
Please use the video soon to review the lecture. The video will be available
for about a week.
The video can be watched at different speeds:
Slower if you don't understand things well.
Faster to repeat the lecture content.
This video is only intended for students of this lecture. Do not give the
link to anybody outside this lecture.
History of Numbers and Numerals
(Georges Ifrah: The Universal History of Numbers, John Wiley
& Sons, 1998)
Humans have used many different representations of numbers throughout
history
The first number represented was 1
Representations such as |, ||, |||,... are most frequent
For bigger numbers, using groups of 10 is most frequent
(20 (French 80: quatre-vingt=4-20) and 60 (minutes, seconds) also exist
The Shape of Numerals
Chinese numerals: 一、二、三、亖 (or 四)
Roman numerals: I, II, III, IIII (or IV)
(Arabic-)Indic numerals: ١, ٢, ٣ (used in
Arabic)
(European-)Arabic numerals: 1, 2, 3
Chinese numerals: 十、廿、...
Roman numerals: X, XX,...
Creating the Natural Numbers Starting with 1
Peano Axioms (Guiseppe Peano, 1858-1932):
1 is a natural number
(1∈ℕ)
If a is a natural number, then s(a) is a
natural number (s(a) is the successor of
a)
(a∈ℕ ⇒ s(a)∈ℕ)
There is no natural number x so that
s(x) = 1
If two natural numbers are different, then their successors are
different
(a∈ℕ, b∈ℕ, a ≠ b ⇒
s(a) ≠ s(b))
If we can prove a property for 1,
and we can prove, for any natural number a, that if a
has this property then s(a) also has this
property,
then all natural numbers have this property.
(Nowadays, it is usual to start natural numbers with 0 rather than with
1.)
(We will learn how to express axioms 3 and 5 as formulæ in the lesson about
Predicate Logic)
Symbols Used
ℕ: The set of natural numbers
∈: Set membership (a ∈ B: a is an
element of set B)
=: Equality (a = b: a is equal to b)
≠: Inequality (a ≠ b: a is not equal to
b)
⇒: Implication (a ⇒ b: If a, then
b, or: a implies b)
If a and b are natural numbers, a +
s(b) = s(a + b)
(a∈ℕ, b∈ℕ ⇒ a +
s(b) = s(a + b))
Calculate 2 + 3 using Peano arithmetic:
Associative Law
(associative property)
A binary operation (represented by operator △) is associative if
and only if for all operands a, b, and c:
(a△b) △ c = a △
(b△c)
Examples:
Addition of (natural) numbers is associative:
(a+b) + c = a +
(b+c)
Multiplication of (natural) numbers is associative:
(a·b) · c = a ·
(b·c)
Multiplication of matrices is associative.
Counterexamples:
Subtraction of (natural) numbers is not associative:
((a-b) - c ≠ a -
(b-c))
Exponentiation is not associative:
(ab)c≠
a(bc)
Proof of Associativity of Addition using Peano Axioms
What we want to prove:
Associative law for addition: (d + e) + f =
d + (e + f)
Let's prove this for all values of f.
Let's distinguish two cases: f=1 and
f=k+1
If f = 1, then (d + e) + 1 =s(d +
e) [by the 1st axiom of addition, with
(d+e) for a]
= d + s(e) [by the 2nd axiom of addition,
with d for a and e for b]
= d + (e + 1) [by the 1st axiom of addition,
backwards, with e for a]
Assuming that associativity holds for f = k
(i.e. (d + e) + k = d +
(e + k)),
let's prove associativity for f = k+1,
i.e let's prove (d + e) + k = d
+ (e + k) ⇒
⇒(d + e) + (k+1) = d +
(e + (k+1)):
(d + e) + (k+1) = (d +
e) + s(k) [by the 1st axiom of addition,
with k for a]
= s((d + e) + k) [by the 2nd
axiom of addition, with (d+e) for a and k
for b]
= s(d + (e + k)) [using the
assumption]
= d + s(e + k) [by the 2nd
axiom of addition, backwards, with d for a and
(e+k) for b]
= d + ((e + k) + 1) [by the 1st axiom of
addition, with (e+k for a]
= d + (e + (k + 1)) [using the case
f=1, with e for d and k for
e]
Q.E.D. [using the 5th Peano axiom, with f for a and
the property (d + e) + f = d +
(e + f)]
Comments on Proof
We have to be careful that we are only using the axioms, not any 'general
knowledge'.
Proofs include two aspects:
Originality (human imagination)
Mechanics (careful execution or automated proof checking)
Because we have not yet established associativity, we always have to use
parentheses.
Once we have proved associativity, we can eliminate the parentheses.
This proof uses mathematical induction.
Peano Axiom 5 can be seen as the basis for mathematical
induction.
We will look at mathematical induction more closely later.
Comments on Axioms
Mathematics tries to start with very few facts or rules
These are usually called axioms
The axioms should be self-evident
Other facts and rules (theorems) are deduced from the axioms using
proofs
This is called the axiomatic method
The less axioms and the more interesting theorems, the better (from a
mathematical viewpoint)
Quiz
The Discovery of 0
The latest (natural/integer) number (and numeral) discovered by
humans
Discovered around 800 A.D. in India
Discovery spread West to Arabia and Europe, East to China and Japan
0 is very important for positional notations such as decimal,
binary,...
More Arithmetic Operations
Exponentiation (e.g. 23)
Two raised to the power of three is eight.
Two to the power of three is eight.
Two to the three (third) is eight.
The third power of two is eight.
Three raised to the power of three is twenty-seven.
Five to the power of four is six hundred
twenty-five.
Modulo operation (remainder)
Twenty modulo six is two.
Twenty-five modulo seven is four.
(written "25 % 7" in C and many other programming languages,
"25 mod 7" in Mathematics)
Positional Notation: Decimal Notation
Number representations before positional notation:
Chinese (Han) numerals: 二百五十六、二千十九
Roman numerals: CCLVI, MMXIX
Example of decimal notation:
25610 = 2·102 + 5·101 + 6·100
Base Conversion: Base 10 to Base b (second method)
Start with the number to convert (a) as the first remainder.
Start with an empty list of result digits.
Determine the first exponent n so that
bn+1 > a ≧
bn
Repeatedly, as long as the exponent is ≧0:
Take the remainder of the previous division as the dividend
Divide the dividend by bn, then by
bn-1, and so on
Add the quotient of the division as a digit to the right of the previous
result digits
Example:Convert 65 to base 3: 34 > 65 ≧
33 ⇒ n=3
dividend
exponent
divisor
quotient
remainder
digits of the result
65↙
65
3
33=27
2↓
11↙
2
11
2
32=9
1↓
2↙
21
2
1
31=3
0↓
2↙
210
2
0→done!
30=1
2↓
0
2102
Base Conversion: Base b to Base c
General method: Convert via base 10
base b → base 10 → base c
Example: base 3 → base 10 → base 5
Shortcut 1: If b is a power of c (b =
cn) or the other way around (c
= bn), then convert the digits in
groups
Example 1: base 3 → base 9 (9 is 32, therefore make groups of
two digits and convert to a single digit)
Example 2: base 8 → base 2 (8 is 23, therefore convert each
digit to a group of three digits)
Shortcut 2: If both b and c are powers of
d (b = dn,
c=dm), then convert via base
d
Example: base 4 → base 8
because 4 = 22 and 8 = 23, d = 2
therefore, convert base 4 → base 2 → base 8 (use shortcut 1 two
times)
Base Conversion Shortcut Example
Convert 476238 to base 4.
8 = 23, 4 = 22, therefore convert base 8 → base 2
→ base 4
476238 →
4
7
6
2
3
base 8
100
111
110
010
011
convert each base-8 digit to three base-2 digits
1001111100100112
1
00
11
11
10
01
00
11
split base 2 into groups of two digits (start at the right)
