Natural Number Representation

(整数の表現)

Discrete Mathematics I

2nd lecture, September 23, 2016

http://www.sw.it.aoyama.ac.jp/2016/Math1/lecture2.html

Martin J. Dürst

© 2005-15 Martin J. Dürst Aoyama Gakuin University

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

Last Week's Homework

• Create your account on http://moo.sw.it.aoyama.ac.jp
• Enroll for Discrete Mathematics I (deadline: September 22)
• Solve the Quiz Simple Arithmetic in English (290 attempts, 116 complete (all problems solved))
• Review decimal number representation, binary number representation, and n-ary number representation based on high-school notes and Web resources
• Check out/buy/lend a textbook or reference book

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. 23 (today!)
• 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, a ≠ b ⇒ s(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 with 0 rather than with 1.)

(We will learn how to express axioms 3 and 5 as formulæ in the lesson about Predicate Logic)

Number Representation using Peano Axioms

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:

(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 ((a-b) - c ≠ a - (b-c))
• Exponentiation ((a^b)^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'
• 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 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、二千十六

Roman numerals: CCLVI, MMXVI

Example of decimal notation: 256 = 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

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 b, the zeroth 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

Base Conversion: Base 10 to Base b

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

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 > a ≧ bⁿ
• Divide by bⁿ, then by b^n-1, and so on

Base Conversion: Base b to Base c

• General method: Convert via base 10
  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 →

100111110010011₂

→ 10332103₄

Hexadecimal Numbers

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

Bases Frequently Used in IT

Correspondence between binary and hexadecimal numbers

Powers of 2

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 29, 22:00)
• 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29日 (木) 22:00)
• 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

数

numeral

数字

natural number

自然数

discovery

発見

origin

原点

positional notation

位取り表現

perfect score

満点

confusion

混乱

representation

表現

exponentiation

べき乗演算

Modulo operation

モジュロ演算

remainder

剰余 (余り)

decimal notation (decimal numeral system)

十進法

Chinese numerals

漢数字

Roman numerals

ローマ数字

discovery

発見

axiom

公理

Peano axioms

ペアノの公理

successor

後者

formula (plural: formulæ)

式

property

性質

arithmetic

算術

associative law (property)

結合律

counterexample

反例

operation

演算

operator

演算子

operand

被演算子

binary operation

二項演算

proof

証明

prove

証明する

Q.E.D. (quod erat demonstrandum)

証明終了

parenthesis

(丸・小) 括弧、複数 parentheses

mathematical induction

数学的帰納法

self-evident

自明

remainder

余り、剰余

subscript

下付き文字 (添え字)

generalization

一般化

decimal point

小数点

base

基数

base conversion

奇数変換

positional weight

(その桁の) 重み

dividend (or numerator)

被除数、実 (株の配当という意味も)

divisor (or denominator, modulus)

除数、法

quotient

商 (割り算の結果)

Horner's rule

ホーナー法

digit

数字

shortcut

近道

upper case

大文字

lower case

小文字

binary

二進数 (形容詞)

octal

八進数 (形容詞)

decimal

十進数 (形容詞)

hexadecimal

十六進数 (形容詞)

circuit

回路

constant

定数

joke

冗談

submit

提出する

proposition

命題

function

関数