Representation of Natural Numbers

Seating Order

Today's Schedule

Introduction of TA

Summary of Last Lecture

Registrations and Homework

Last Lecture's Homework

Languages Necessary for Information Technology

History of Numbers and Numerals

The Shape of Numerals

Creating the Natural Numbers Starting with 1

Symbols Used

Number Representation using Peano Axioms

Addition using Peano Axioms

Associative Law

Proof of Associativity of Addition using Peano Axioms

Proof of Associativity (continued)

Comments on Proof

Comments on Axioms

The Discovery of 0

More Arithmetic Operations

Positional Notation: Decimal Notation

Binary Numeral System

Base Conversion: Base `b` to Base 10

Base Conversion: Base 10 to Base `b` (first method)

Base 10 to Base `b` (first method), Example

Reference: Terms for Operands and Results

Base Conversion: Base 10 to Base `b` (second method)

Base 10 to Base `b` (second method), Example

Base Conversion: Base `b` to Base `c`

Base Conversion Shortcut Example

Addition in Different Numeral Systems

Numbers Represented with Hexadecimal Digits

Number of Digits in Base `b`

Bases Frequently Used in IT

Small Numbers in Various Bases

Powers of 2

Homework: Jokes

Discrete Mathematics I

2nd lecture, September 30, 2022

Martin J. Dürst

Exponentiation (e.g. 2³)

Modulo operation (remainder)

First Joke

Second Joke

(自然数の表現)

https://www.sw.it.aoyama.ac.jp/2022/Math1/lecture2.html

Only use two outer seats of each table

各テーブルの外側ノイズのみ使用！

Moodle registration, last lecture's homework
History of numbers and numerals
The historic origin of numbers: 1
Creating natural numbers starting from 1 (Peano arithmetic)
The discovery of 0
Positional notation (decimal, binary, hexadecimal, ...)
Base conversion

Teaching Assistant (TA): Tensho Miyashita (宮下天祥, M1)

(apologies for delay on September 16 (Friday); reason: software troubles)

Mathematics is an important base for Information Technology
Discrete Mathematics I mainly covers number representation, logic, sets, relations, proofs
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

Students listed in Course Power: 180
Students enrolled in Moodle Course: 154
Homework 1 (5' limit)
- Solution submitted: 132
- Solution non-empty: 112
Homework 2 (repeat until 100% correct)
- Solutions submitted: 128
- Solutions 100% correct: 118

Make sure you are officially registered for the course and enrolled in Discrete Mathematics I (https://moo.sw.it.aoyama.ac.jp/course/view.php?id=58)
Review decimal number representation, binary number representation, and n-ary number representation based on high-school notes and Web resources
Solve the Quiz Languages Necessary for Information Technology (one attempt, maximum 5 minutes; deadline: September 27, 22:00 (no extensions))
Solve the Quiz Simple Arithmetic in English (repeat until you get it 100% correct; deadline: September 27, 22:00 (no extensions))
Check out/buy/borrow a textbook or reference book

(4 in total, order irrelevant)

Mathematics
English
Mother tongue (e.g. Japanese)
Programming language(s)

Besides a mathematical inclination, an exceptionally good mastery of one's native tongue is the most vital asset of a competent programmer.

Edsger W. Dijkstra, EWD498, Selected Writings on Computing: A Personal Perspective, Springer-Verlag, 1982, pp. 129-131.

(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

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,...

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)

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)

1	1
2	`s`(1)
3	`s`(`s`(1))
4	`s`(`s`(`s`(1)))
5	`s`(`s`(`s`(`s`(1))))
6	`s`(`s`(`s`(`s`(`s`(1)))))
7	`s`(`s`(`s`(`s`(`s`(`s`(1))))))
...	...

Axioms of addition:

If a is a natural number, then a + 1 = s(a)
(a∈ℕ ⇒ a + 1 = s(a))
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 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:
(a^b)^{^c}≠ a^{(b^c)}

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)]

We have to be careful that we only use 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.

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)

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,...

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.

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)

Number representations before positional notation:

Chinese (Han) numerals: 二百五十六、二千二十二

Roman numerals: CCLVI, MMXXII

Example of decimal notation (the base of a number is often given as a subscript):
256₁₀ = 2·10² + 5·10¹ + 6·10⁰

Example containing 0: 206 = 2·10² + 0·10¹ + 6·10⁰

Generalization: d_n...d₁d₀ = d_n·10ⁿ+...+d₁·10¹+d₀·10⁰
(here the subscript indicates the position, not the base)

Example with decimal point:
34.56₁₀ = 3·10¹ + 4·10⁰ + 5·10^-1 + 6·10^-2

1010011₂ = 1·2⁶ + 0·2⁵ + 1·2⁴ + 0·2³ + 0·2² + 1·2¹ + 1·2⁰ =

1·64 + 0·32 + 1·16 + 0·8 + 0·4 + 1·2 + 1·1 =

64 + 16 + 2 + 1 = 83

Calculate the sum of each of the digits multiplied by its positional weight.

The positional weight is a power of the base b, the 0th power for the rightmost digit.

The power increases by one when moving one position to the left.

d_n...d₁d₀ (in base b) = d_n·bⁿ+...+d₁·b¹+d₀·b⁰ = ∑ⁿ_i=0 d_i·bⁱ

Take the number to convert as the first quotient. Start with an empty list of result digits.

Repeatedly, as long as the quotient is positive:

Take the quotient of the previous division as the dividend
Divide that dividend by the base b
Add the remainder of the division as a digit to the left of the previous result digits

Example: Convert 65 to base 3
dividend	divisor	quotient	remainder	result digits
		65↙		(empty)
65	3	21↙	2↑	2
21	3	7↙	0↑	02
7	3	2↙	1↑	102
2	3	0→done!	2↑	2102

65 divided by 3 is 21 remainder 2

21 divided by 3 is 7 remainder 0

7 divided by 3 is 2 remainder 1

2 divided by 3 is 0 remainder 2

65 = 65·3⁰
= 21·3¹ + 2·3⁰
= 7·3² + 0·3¹ + 2·3⁰
= 2·3³ + 1·3² + 0·3¹ + 2·3⁰ = 2102₃

Using Horner's rule: 65 = (((2×3 + 1)×3 + 0)×3 + 2)

	left operand	operator	right operand	result
Addition	augend (also: summand or addend)	+	addend (also: summand)	sum
Subtraction	minuend	-	subtrahend	difference
Multiplication	multiplicand (also: factor)	×, ·, `*`, nothing	multiplier (also: factor)	product
Division	dividend	÷, /	divisor	quotient
Modulo operation	dividend	mod, `%`	divisor	remainder
Exponentiation	base	(superscript), `^`, `**`	exponent	power

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 ≦ a < bⁿ⁺¹

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 b^n-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: 3³ ≦ 65 < 3⁴ ⇒ `n`=3
dividend	exponent	divisor	quotient	remainder	digits of the result
				65↙
65	3 (=`n`)	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

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 (3² = 9, therefore make groups of 2 digits and convert to a single digit)
Example 2: base 8 → base 2 (8 = 2³, therefore convert each digit to a group of 3 digits)
Shortcut 2: If both b and c are powers of d (b = dⁿ, c=d^m), 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)

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 3 base-2 digits

100111110010011₂

1	00	11	11	10	01	00	11	split base 2 into groups of 2 digits (start at the right)
1	0	3	3	2	1	0	3	convert two base-2 digits to one base-4 digit

→ 10332103₄

Works the same as in the decimal system:

Progress from least signifinant digit to more significant digits
Carry over when base (e.g. 10) is reached

Example (base 7):

1AF₁₆ = 1×16² + A×16¹ + F×16⁰ = 1×256 + 10×16 + 15×1 = 256 + 160 + 15 = 431

Values of hexadecimal (base 16) digits
digit (upper case)	digit (lower case)	value (decimal)
A	a	10
B	b	11
C	c	12
D	d	13
E	e	14
F	f	15

The number of different digits in base b is b.

The lowest digit is 0, the highest digit is b-1.

Base	Number of Digits	Lowest Digit	Highest Digit	Digits
2	2	0	1	0, 1
3	3	0	2	0, 1, 2
4	4	0	3	0, 1, 2, 3
5	5	0	4	0, 1, 2, 3, 4
8	8	0	7	0, 1, 2, 3, 4, 5, 6, 7
10	10	0	9	0, 1, 2, 3, 4, 5, 6, 7, 8, 9
12	12	0	B (11)	0123456789 A B
16	16	0	F (15)	0123456789 A B C D E F
22	22	0	L (21)	0123456789 ABCDEFGHIJ KL

base	name (adjective) and abbreviation	(reason for) use	constants in programming languages
2	binary, bin	used widely in logic and circuits (computer hardware)	`0b101100` (Ruby,...)
8	octal, oct	shortened form of binary (rare these days)	`024570` (C and many others)
10	decimal, dec	for humans	`1234567` (all languages)
16	hexadecimal, hex	shortened form of binary, 1 byte (8 bits) can be represented with two digits	`0xA3b5` (C and many others)

10	2	3	4	5	6	7	8	9	16
0	0000	0	0	0	0	0	0	0	0
1	0001	1	1	1	1	1	1	1	1
2	0010	2	2	2	2	2	2	2	2
3	0011	10	3	3	3	3	3	3	3
4	0100	11	10	4	4	4	4	4	4
5	0101	12	11	10	5	5	5	5	5
6	0110	20	12	11	10	6	6	6	6
7	0111	21	13	12	11	10	7	7	7
8	1000	22	20	13	12	11	10	8	8
9	1001	100	21	14	13	12	11	10	9
10	1010	101	22	20	14	13	12	11	A/a
11	1011	102	23	21	15	14	13	12	B/b
12	1100	110	20	22	20	15	14	13	C/c
13	1101	111	21	23	21	16	15	14	D/d
14	1110	112	22	24	22	20	16	15	E/e
15	1111	120	23	30	23	21	17	16	F/f
16	10000	121	100	31	24	22	20	17	10

`n`	2ⁿ	in base 16
0	1	1
1	2	2
2	4	4
3	8	8
4	16	10
5	32	20
6	64	40
7	128	80
8	256	100
9	512	200
10	1'024 ≈10³ (kilo)	400
11	2'048 (the game)	800
12	4'096	1000
16	65'536	1'0000
20	1'048'576 ≈ 10⁶ (mega)	10'0000
30	1'073'741'824 ≈ 10⁹ (giga)	4000'0000
40	1'099'511'627'776 ≈ 10¹² (tera)	100'0000'0000

Question: Why do computer scientist always think Christmas and Halloween are the same?
(Hint: In the USA, Halloween is October 31st only)

Your answer:

sum in base 10

sum in base 7

Question: At what age do Information Technologists celebrate "Kanreki" (還暦)