Number Representation

(数の表現)

Discrete Mathematics I

2nd lecture, September 25, 2015

http://www.sw.it.aoyama.ac.jp/2015/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

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

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• First quiz not completed: Do your best for the rest of the course!
• Special problems: Talk to me after this lecture

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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.)

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 all operands a, b, and c:

(a△b) △ c = a △ (b△c)

Examples:

• Addition of natural numbers is associative.
• Multiplication of natural numbers is associative.
• Multiplication of matrices is associative

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

• 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.

The Discovery of 0

• The latest (natural/integer) number discovered by humans
• Discovered around 800 A.D. in India
• Discovery spread westwards to Arabia and Europe, eastwards 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、二千十五

Roman numerals: CCLVI, MMXV

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 =

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, and divide it by the base b,

adding the remainder as a digit to the result from right to left.

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 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?

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

This Week's Homework

• Solve Arithmetic and Base Conversion until October 1, 22:00
• 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: 10月1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

property

性質

arithmetic

算術

associative law (property)

結合律

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

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