# Natural Number Representation

(整数の表現)

## Discrete Mathematics I

### 2nd lecture, September 23, 2017

http://www.sw.it.aoyama.ac.jp/2017/Math1/lecture2.html

### Martin J. Dürst # Today's Schedule

• Last week's homework, Moodle registration
• How to watch videos of this lecture
• History of numbers and numerals
• The historic origin of numbers: 1
• Creating natural numbers starting from 1
• The discovery of 0
• Positional notation (decimal, binary, ...)

# Summary of Last Lecture

• 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

# Problems with Moodle Registration

• Wrong name (Latin letters),...: Click on your name in the upper left corner, then choose Settings → My profile settings → Edit Profile (exception: username)
• Account not created? Just try to log in, anyway
• Enrollment key:
• Not yet enrolled: Enrollment open again; deadline: Sept. 22 (today!), 22:00
(if you miss that, come to my office on Monday)
• First quiz not completed: Do your best for the rest of the course!
• Special problems: Talk to me after this lecture

# How to Watch Videos

The video of the first lecture is available via a link from Moodle.

The video plugin/application is widely supported. Please contact me if you have problems with your OS or browser.

Userid and password are available on the Moodle course page.
They are the same for all students, but only for students of this lecture!

Please use the video soon to review the lecture.
Please be careful when watching the video on a mobile device (may be expensive!).

The video can be watched at different speeds, and you can jump easily to the next slide.

# 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
• 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. 1 is a natural number
(1∈ℕ)
2. 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)
3. There is no natural number x so that s(x) = 1
4. If two natural numbers are different, then their successors are different
(a∈ℕ, b∈ℕ, abs(a) ≠ s(b))
5. 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 (aB: a is an element of set B)

=: Equality (a = b: a is equal to b)

≠: Inequality (ab: a is not equal to b)

⇒: Implication (ab: If a, then b)

# Number Representation using Peano Axioms

 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)))))) ... ...

# Addition using Peano Axioms

Axioms of addition:

1. a + 1 = s(a)
2. 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:

(ab) △ c = a △ (bc)

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
((a-b) - ca - (b-c))
• Exponentiation ((ab)ca(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 (e.g. 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
• The less axioms and the more interesting theorems, the better (from a mathematical viewpoint)

# The Discovery of 0

• The latest (natural/integer) number discovered by humans
• Discovered around 800 A.D. in India
• Discovery spread West to Arabia and Europe, Easts 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.

Five to the power of four is six hundred twenty-five.

Three raised to the power of four is eight-one.

### Modulo operation (remainder)

Twenty modulo six is two.

Twenty-five modulo seven is four.

# Positional Notation: Decimal Notation

Number representations before positional notation:

Chinese (Han) numerals: 二百五十六、二千十七

Roman numerals: CCLVI, MMXVII

Example of decimal notation:
256 = 2·102 + 5·101 + 6·100

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

Generalization: dn...d1d0 = dn·10n+...+d1·101+d0·100

Example with decimal point:
34.56 = 3·101 + 4·100 + 5·10-1 + 6·10-2

# Binary Numeral System

(the base of a number is often given as a subscript)

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 =

1·64 + 1·16 + 1·2 + 1·1 =

64 + 16 + 2 + 1 =

83

# Base Conversion: Base b to Base 10

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.

dn...d1d0 (in base b) = dn·bn+...+d1·b1+d0·b0

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

Take the number to convert as the first quotient.

Repeatedly:

• Take the quotient of the previous division
• Divide that quotient by the base b
• Add the remainder of the division as a digit to the left of the result
dividend divisor quotient remainder digits of the result
23↙
23 2 11↙ 1 1
11 2 5↙ 1 11
5 2 2↙ 1 111
2 2 1↙ 0 0111
1 0 1 10111

23 divided by 2 is 11 remainder 1

11 divided by 2 is 5 remainder 1

5 divided by 2 is 2 remainder 1

2 divided by 2 is 1 remainder 0

1 divided by 2 is 0 remainder 1

23 = 11·21 + 1·20
= 5·22 + 1·21 + 1·20
= 2·23 + 1·22 + 1·21 + 1·20
= 1·24 + 0·23 + 1·22 + 1·21 + 1·20 = 10111

Using Horner's rule: 23 = (((1×2 + 0)×2 + 1)×2 + 1)×2 + 1

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

• It is possible to start from the most significant digit
• When starting with a, first find n so that bn+1 > abn
• Divide by bn, then by bn-1, and so on
dividend divisor quotient remainder digits of the result
23↙
23 16 1 7↙ 1
7 8 0 7↙ 10
7 4 1 3↙ 101
3 2 1 1↙ 1011
1 1 1 0↙ 10111

# 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 (or the other way round), 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, 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 47623 (base 8) 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 1 0 3 3 2 1 0 3 convert two base-2 digits to one base-4 digit

→ 103321034

# Numbers Represented with Hexadecimal Digits

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

 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

# Number of Digits in Base b

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

# Bases Frequently Used in IT

base name (adjective) and abbreviation (reason for) use constants in programming languages
2 binary, bin used widely in logic and circuits (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 (8bits) can be represented with two digits `0xA3b5` (C and many others)

10 2 16
0 0000 0
1 0001 1
2 0010 2
3 0011 3
4 0100 4
5 0101 5
6 0110 6
7 0111 7
8 1000 8
9 1001 9
10 1010 A/a
11 1011 B/b
12 1100 C/c
13 1101 D/d
14 1110 E/e
15 1111 F/f
16 10000 10

# Powers of 2

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

# Homework: Jokes

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

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

# This Week's Homework

• Solve Arithmetic and Base Conversion (repeat until you get it 100% correct; deadline September 28, 22:00, 15 minutes time limit for each attempt)
• Learn binary and hexadecimal numbers up to 16, and powers of 2 up to 212
• Try to find an answer to the joke questions (no need to submit)
• Use highschool texts or the Web to refresh your knowledge about propositions, logic, and functions

# 今週の宿題

• Moodle で Arithmetic and Base Conversion のクイズを解く (満点まで繰り返す、締切: 9月28日 (木) 22:00, 一回ごとに15分の時間制限)
• 冗談の問題の解答を考える (提出不要)
• 数学の「命題」、「論理演算」、「関数」について高校で学んだことを再確認し、ウェブで調べる

# Glossary

number
numeral

natural number

discovery

origin

positional notation

perfect score

confusion

representation

exponentiation
べき乗演算
Modulo operation
モジュロ演算
remainder

decimal notation (decimal numeral system)

Chinese numerals

Roman numerals
ローマ数字
discovery

axiom

Peano axioms
ペアノの公理
successor

formula (plural: formulæ)
symbol

set membership

equality (sign)

inequality (sign)

implication

property

arithmetic

associative law (property)

counterexample

operation

operator

operand

binary operation

proof

prove

Q.E.D. (quod erat demonstrandum)

parenthesis
(丸・小) 括弧、複数 parentheses
mathematical induction

originality

automated proof checking

imagination

self-evident

remainder

subscript

generalization

decimal point

base

base conversion

positional weight
(その桁の) 重み
dividend (or numerator)

divisor (or denominator, modulus)

quotient

Horner's rule
ホーナー法
digit

shortcut

upper case

lower case

binary

octal

decimal

hexadecimal

circuit

constant

joke

submit

proposition

function