Code 7 우리가 사용하는 10개의 숫자들

CodeCh7. 우리가 사용하는 열 개의 숫자들

chois79

13년 1월 27일 일요일

개요

• 지금까지는...

• 1 ~ 3: 부호에 대한 고찰

• 부호는 의사 전달을 위한 수단

• 4 ~ 6: 신호를 전달하는 방법

• 전하, 전신기, 릴레이

• 지금부터는...

• 7 ~ 9: 수 체계에 대한 고찰 ~ Bit

• 10 ~ : 논리 회로 ...


언어는 단지 부호에 불과

문화에 따라 표현이 다름.

ex) 고양이, Cat


그에 반에 숫자는?물론, 다르게 말하고, 발음 됨하나, one, 둘, two

그러나, 동일한 표기법이 존재1, 2, 3, 4, 5, 6, 7, 8, 9, 0


숫자는 부호들 중 가장 추상화된 부호

47

Chapter Seven

Our Ten Digits

he idea that language is merely a code seems readily acceptable.Many of us at least attempted to learn a foreign language in highschool, so we’re willing to acknowledge that the animal we call a

cat in English can also be a gato, chat, Katze, !"#!$, or !"##".Numbers, however, seem less culturally malleable. Regardless of the lan-

guage we speak and the way we pronounce the numbers, just about everybodywe’re likely to come in contact with on this planet writes them the same way:

1 2 3 4 5 6 7 8 9 10Isn’t mathematics called “the universal language” for a reason?

Numbers are certainly the most abstract codes we deal with on a regu-lar basis. When we see the number

3we don’t immediately need to relate it to anything. We might visualize 3apples or 3 of something else, but we’d be just as comfortable learning fromcontext that the number refers to a child’s birthday, a television channel, ahockey score, or the number of cups of flour in a cake recipe. Because ournumbers are so abstract to begin with, it’s more difficult for us to understandthat this number of apples

$T

3


숫자의 유래는?

왜 10이 특별한 의미를 가질까?

숫자를 모른다고 가정해보자


어떤 것을 세기 위하여 위한 방법이 필요 했을 것

손가락을 먼저 사용하지 않았을까?

“digit” 어원 손가락“five”, “fist”는 같은 어원을 가짐


현재까지 사용되고 있는 로마 숫자는?

48 Chapter Seven

doesn’t necessarily have to be denoted by the symbol

3Much of this chapter and the next will be devoted to persuading ourselvesthat this many apples

can also be indicated by writing

11Let’s first dispense with the idea that there’s something inherently spe-

cial about the number ten. That most civilizations have based their numbersystems around ten (or sometimes five) isn’t surprising. From the very be-ginning, people have used their fingers to count. Had our species developedpossessing eight or twelve fingers, our ways of counting would be a littledifferent. It’s no coincidence that the word digit can refer to fingers or toesas well as numbers or that the words five and fist have similar roots.

So in that sense, using a base-ten, or decimal (from the Latin for ten),number system is completely arbitrary. Yet we endow numbers based on tenwith an almost magical significance and give them special names. Ten yearsis a decade; ten decades is a century; ten centuries is a millennium. A thou-sand thousands is a million; a thousand millions is a billion. These numbersare all powers of ten:

101 =10102 =100103 =1000 (thousand)104 =10,000105 =100,000106 =1,000,000 (million)107 =10,000,000108 =100,000,000109 =1,000,000,000 (billion)

Most historians believe that numbers were originally invented to countthings, such as people, possessions, and transactions in commerce. For ex-ample, if someone owned four ducks, that might be recorded with drawingsof four ducks:

오리를 다 그려야해?

Our Ten Digits 49

Eventually the person whose job it was to draw the ducks thought, “Whydo I have to draw four ducks? Why can’t I draw one duck and indicate thatthere are four of them with, I don’t know, a scratch mark or something?”

And then there came the day when someone had 27 ducks, and the scratchmarks got ridiculous:

Someone said, “There’s got to be a better way,” and a number system wasborn.

Of all the early number systems, only Roman numerals are still in com-mon use. You find them on the faces of clocks and watches, used for dateson monuments and statues, for some page numbering in books, for some itemsin an outline, and—most annoyingly—for the copyright notice in movies.(The question “What year was this picture made?” can often be answeredonly if one is quick enough to decipher MCMLIII as the tail end of the creditsgoes by.)

Twenty-seven ducks in Roman numerals is

The concept here is easy enough: The X stands for 10 scratch marks and theV stands for 5 scratch marks.

The symbols of Roman numerals that survive today are

I V X L C D MThe I is a one. This could be derived from a scratch mark or a single raisedfinger. The V, which is probably a symbol for a hand, stands for five. TwoV’s make an X, which stands for ten. The L is a fifty. The letter C comes fromthe word centum, which is Latin for a hundred. D is five hundred. Finally,M comes from the Latin word mille, or a thousand.

Although we might not agree, for a long time Roman numerals wereconsidered easy to add and subtract, and that’s why they survived so longin Europe for bookkeeping. Indeed, when adding two Roman numerals, yousimply combine all the symbols from both numbers and then simplify theresult using just a few rules: Five I’s make a V, two V’s make an X, five X’smake an L, and so forth.

오리와 손가락 모양으로표현하니 편함

Our Ten Digits 49

Eventually the person whose job it was to draw the ducks thought, “Whydo I have to draw four ducks? Why can’t I draw one duck and indicate thatthere are four of them with, I don’t know, a scratch mark or something?”

And then there came the day when someone had 27 ducks, and the scratchmarks got ridiculous:

Someone said, “There’s got to be a better way,” and a number system wasborn.

Of all the early number systems, only Roman numerals are still in com-mon use. You find them on the faces of clocks and watches, used for dateson monuments and statues, for some page numbering in books, for some itemsin an outline, and—most annoyingly—for the copyright notice in movies.(The question “What year was this picture made?” can often be answeredonly if one is quick enough to decipher MCMLIII as the tail end of the creditsgoes by.)

Twenty-seven ducks in Roman numerals is

The concept here is easy enough: The X stands for 10 scratch marks and theV stands for 5 scratch marks.

The symbols of Roman numerals that survive today are

I V X L C D MThe I is a one. This could be derived from a scratch mark or a single raisedfinger. The V, which is probably a symbol for a hand, stands for five. TwoV’s make an X, which stands for ten. The L is a fifty. The letter C comes fromthe word centum, which is Latin for a hundred. D is five hundred. Finally,M comes from the Latin word mille, or a thousand.

Although we might not agree, for a long time Roman numerals wereconsidered easy to add and subtract, and that’s why they survived so longin Europe for bookkeeping. Indeed, when adding two Roman numerals, yousimply combine all the symbols from both numbers and then simplify theresult using just a few rules: Five I’s make a V, two V’s make an X, five X’smake an L, and so forth.

오리가 많으면...


손 하나를 표현하는 부호 등장!

VX = V V

L = XXXXXC = LL

D = CCCCCM = DD

5, 10 단위로의미를 가짐


로마 숫자의 장점(?)

덧셈이 쉬웠을 것.

XIII + LXIIII

= LXXIIIIIII (차례로 표시)

= LXXVII (정리)

아마도.. 쉬운듯...


로마 숫자로

곱셈과 나눗셈을 하려면?

...


힌두-아라비아 숫자(인도-아라비아 숫자)

서기 828년 수학자 무하마드 이반 알 카와리즈미에 의해 널리 사용되기 시작


다른 숫자 체계와의 차이점

숫자의 위치에 따라 의미를 부여

십을 나타내기 위한 특별한 부호가 없음

초기 모든 숫자 체계에 없는 ‘0’이 존재(0을 사용하여 위치에 따른 의미를 부여)

‘0‘은 숫자 체계에서 가장 중요한 발명중 하나십 = 1 + 0 (위치부여)


‘0’ 사용의 장점

위치에 따른 표기법 지원25, 205, 2050

수 많은 연산을 쉽게 수행( 곱셈, 나눗셈 ... )


아라비아 숫자의 구조

4,825발음을 통해 전체 구조가 파악됨

사천 팔백 이십 오

4,825 = 4,000 + 800 + 20 + 5

4,825 = 4 X 1,000 + 8 X 100 + 2 X 10 +

5 X 1

4,825 = 4 X 103 + 8 X 102 + 2 X 101 +

5 X 100


숫자의 위치는 특별한 의미를 가짐

Our Ten Digits 51

Or we can write the components like this:

4825 = 4000 + 800 + 20 + 5

Or breaking it down even further, we can write the number this way:

4825 = 4 !1000 +8 !100 +2 !10 +5 !1

Or, using powers of ten, the number can be rewritten like this:

4825 = 4 !10 3 +8 !10 2 +2 !10 1 +5 !10 0

Remember that any number to the 0 power equals 1.Each position in a multidigit number has a particular meaning, as shown

in the following diagram. The seven boxes shown here let us represent anynumber from 0 through 9,999,999:

,,Number of onesNumber of tensNumber of hundredsNumber of thousandsNumber of ten thousandsNumber of hundred thousandsNumber of millions

Each position corresponds to a power of ten. We don’t need a special sym-bol for ten because we set the 1 in a different position and we use the 0 asa placeholder.

What’s also really nice is that fractional quantities shown as digitsto the right of a decimal point follow this same pattern. The number42,705.684 is

4 !10,000 +2 !1000 +7 !100 +0 !10 +5 !1 +6 ÷10 +8 ÷100 +4 ÷1000

9,999,999까지 어떤 숫자라도 표현 가능각 위치는 10의 거듭제곱수에 해당


소수점 부분의 숫자들도 같은 규칙을 따름

42,705.684

4 X 10,000 + 2 X 1000 + 7 X 100 + 0 X 10 + 5 X 1 + 6 / 1 + 8 / 10 + 4 / 1000

42,705.684

4 X 104 + 2 X 103 + 7 X 102 + 0 X 101 + 5 X 100 + 6 X 10-1 + 8 X 10-2 + 4 X 10-3

10의 거듭 제곱으로 표현


52 Chapter Seven

This number can also be written without any division, like this:

4 ! 10,000 +2 ! 1000 +7 ! 100 +0 ! 10 +5 ! 1 +6 ! 0.1 +8 ! 0.01 +4 ! 0.001

Or, using powers of ten, the number is

4 ! 104 +2 ! 103 +7 ! 102 +0 ! 101 +5 ! 100 +6 ! 10-1 +8 ! 10-2 +4 ! 10-3

Notice how the exponents go down to zero and then become negativenumbers.

We know that 3 plus 4 equals 7. Similarly, 30 plus 40 equals 70, 300 plus400 equals 700, and 3000 plus 4000 equals 7000. This is the beauty of theHindu-Arabic system. When you add decimal numbers of any length, youfollow a procedure that breaks down the problem into steps. Each step involvesnothing more complicated than adding pairs of single-digit numbers. That’swhy someone a long time ago forced you to memorize an addition table:

+ 0 1 2 3 4 5 6 7 8 9

0 0 1 2 3 4 5 6 7 8 9

1 1 2 3 4 5 6 7 8 9 10

2 2 3 4 5 6 7 8 9 10 11

3 3 4 5 6 7 8 9 10 11 12

4 4 5 6 7 8 9 10 11 12 13

5 5 6 7 8 9 10 11 12 13 14

6 6 7 8 9 10 11 12 13 14 15

7 7 8 9 10 11 12 13 14 15 16

8 8 9 10 11 12 13 14 15 16 17

9 9 10 11 12 13 14 15 16 17 18

Our Ten Digits 53

Find the two numbers you wish to add in the top row and the left column.Follow down and across to get the sum. For example, 4 plus 6 equals 10.

Similarly, when you need to multiply two decimal numbers, you followa somewhat more complicated procedure but still one that breaks down theproblem so that you need do nothing more complex than adding or multi-plying single-digit decimal numbers. Your early schooling probably alsoentailed memorizing a multiplication table:

x 0 1 2 3 4 5 6 7 8 9

0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9

2 0 2 4 6 8 10 12 14 16 18

3 0 3 6 9 12 15 18 21 24 27

4 0 4 8 12 16 20 24 28 32 36

5 0 5 10 15 20 25 30 35 40 45

6 0 6 12 18 24 30 36 42 48 54

7 0 7 14 21 28 35 42 49 56 63

8 0 8 16 24 32 40 48 56 64 72

9 0 9 18 27 36 45 54 63 72 81

What’s best about the positional system of notation isn’t how well itworks, but how well it works for counting systems not based on ten. Ournumber system isn’t necessarily appropriate for everyone. One big problemwith our base-ten system of numbers is that it doesn’t have any relevancefor cartoon characters. Most cartoon characters have only four fingers oneach hand (or paw), so they prefer a number system that’s based on eight.Interestingly enough, much of what we know about decimal numberingcan be applied to a numbering system more appropriate for our friends incartoons.

덧셈표 곱셈표

아라비아 숫자의 연산

긴 십진수를 더하든 각 단계로 나누어 풀어가는 과정을 반복


위치에 기반한 10진수 숫자 체계는 매우 잘 작동 한다

그런데, 만약 손가락이 4개였다면?아마 8진수 체계를 사용했을 것

10진수 기반 방법을 8진수에도 적용할 수 있을까?

Next Chapter ...13년 1월 27일 일요일

Code 7 우리가 사용하는 10개의 숫자들

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