Representation of Natural Numbers

(自然数の表現)

Discrete Mathematics I

2nd lecture, September 30, 2022

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

Martin J. Dürst

AGU

© 2005-22 Martin J. Dürst Aoyama Gakuin University

 

Seating Order

Only use two outer seats of each table

各テーブルの外側ノイズのみ使用!

 

Today's Schedule

 

Introduction of TA

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

Summary of Last Lecture

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

 

Registrations and Homework

 

Last Lecture's Homework

 

Languages Necessary for Information Technology

(4 in total, order irrelevant)

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.

 

History of Numbers and Numerals

(Georges Ifrah: The Universal History of Numbers, John Wiley & Sons, 1998)

 

The Shape of Numerals

 

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, or: a implies 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. If a is a natural number, then a + 1 = s(a)
    (a∈ℕ ⇒ 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:

Counterexamples:

 

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.

 

Proof of Associativity (continued)

 

Comments on Proof

 

Comments on Axioms

 

The Discovery of 0

 

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: dn...d1d0 = dn·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

 

Binary Numeral System

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 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 = ∑ni=0 di·bi

 

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

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:

 

Base 10 to Base b (first method), Example

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)

 

Reference: Terms for Operands and Results

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

 

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 bna < bn+1

Repeatedly, as long as the exponent is ≧0:

 

Base 10 to Base b (second method), Example

Example: Convert 65 to base 3: 33 ≦ 65 < 34n=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

 

Base Conversion: Base b to Base c

 

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 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

 

Addition in Different Numeral Systems

Works the same as in the decimal system:

Example (base 7):

Operand 1 3 6 5 1 2
Operand 2 6 0 3 3 4
carry 1 1 1
sum in base 10 1 10 7 8 4 6
sum in base 7 1 13 10 11 4 6
Result 1 3 0 1 4 6

 

Numbers Represented with Hexadecimal Digits

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

 

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 (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)

 

Small Numbers in Various Bases

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

 

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

First Joke

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

Your answer:

Second Joke

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

Your answer:

 

Summary of This Lecture

 

This Week's Homework

 

今週の宿題

 

Glossary

number
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)
symbol
記号
set membership
集合に属する (こと)
equality (sign)
等式、等号
inequality (sign)
不等式、不等号
implication
含意
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
originality
独創性、独創力
automated proof checking
自動証明検証
imagination
想像 (力)
self-evident
自明
axiomatic method
公理的方法
integer
整数
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