http://www.sw.it.aoyama.ac.jp/2012/Math1/lecture2.html
© 2005-12 Martin J. Dürst 青山学院大学
(資料: Georges Ifrah: The Universal History of Numbers, John Wiley & Sons, 1998)
(ペアノの公理; Peano Axioms; Guiseppe Peano, 1858-1932)
(現在は 1 のではなく 0 からのスタートが普通)
| 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)))))) |
| ... | ... |
公理 (Axiom):
2 + 3 をペアノ風で計算:
足算の結合律: (a + b) + c = a + (b + c)
c = 1 の場合: (a + b) + 1 = s(a + b) = a + s(b) = a + (b + 1)
c = k+1 の場合 (c = k の場合を前提に):
(a + b) + k = a + (b + k) ⇒? (a + b) + (k+1) = a + (b + (k+1))
(a + b) + (k+1) = (a +
b) + s(k) = s((a + b) +
k) = s(a + (b + k)) = a
+ s(b + k) = a + ((b +
k) + 1) = a + (b + (k + 1))
Q.E.D.
Exponentiation:
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 three is eight-one.
Modulo operation (remainder):
Twenty modulo six is two.
Twenty-five modulo seven is four.
Number representations before positional notation:
Chinese (Han) numerals: 二百五十六、二千十二
Roman numerals: CCLVI, MMXII
Example of decimal notation: 256 = 2×102 + 5×101 + 6×100
Example containing 0: 206 = 2×102 + 0×101 + 6×100
Generalization: dn...d1d0 =
Example with decimal point: 34.56 = 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 b, the zeroth power for the rightmost digit.
The power increases by one when moving one position to the left.
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.
| dividend | quotient | remainder | digits of the result |
|---|---|---|---|
| 23 | |||
| 23 | 11 | 1 | 1 |
| 11 | 5 | 1 | 11 |
| 5 | 2 | 1 | 111 |
| 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
1AF = 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 |
| base | name (adjective) and abbreviation | (reason for) use | constants in programming languages |
|---|---|---|---|
| 2 | binary, bin | used widely in logic and circuits | 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 | 8 | 16 |
|---|---|---|---|
| 0 | 0000 | 0 | 0 |
| 1 | 0001 | 1 | 1 |
| 2 | 0010 | 2 | 2 |
| 3 | 0011 | 3 | 3 |
| 4 | 0100 | 4 | 4 |
| 5 | 0101 | 5 | 5 |
| 6 | 0110 | 6 | 6 |
| 7 | 0111 | 7 | 7 |
| 8 | 1000 | 10 | 8 |
| 9 | 1001 | 11 | 9 |
| 10 | 1010 | 12 | A |
| 11 | 1011 | 13 | B |
| 12 | 1100 | 14 | C |
| 13 | 1101 | 15 | D |
| 14 | 1110 | 16 | E |
| 15 | 1111 | 17 | F |
| 16 | 10000 | 20 | 10 |
| n | 2n | 16 進法 |
|---|---|---|
| 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 | 1024 ≈103 (kilo) | 400 |
| 11 | 2048 | 800 |
| 12 | 4096 | 1000 |
| 16 | 65536 | 10000 |
| 20 | 1048576 ≈ 106 (mega) | 100000 |
| 30 | 1073741824 ≈ 109 (giga) | 40000000 |
| 40 | 1099511627776 ≈ 1012 (tera) | 10000000000 |
exponentiation: べき乗演算
Modulo operation: モジュロ演算
remainder: 剰余 (余り)
positional notation: 位取り表現
decimal notation (decimal numeral system): 十進法
Chinese numerals: 漢数字
Roman numerals: ローマ数字
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)
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
Q: Why do computer scientist always think Christmas and Halloween are the same ?
質問: なぜ情報テクノロジーの専門家はクリスマスとハロウィーンをいつも誤解するか。
質問: 情報テクノロジーで還暦は何歳か