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  • Hello. I'm Professor Von Schmohawk and welcome to Why U.

  • In the last lecture, we explored the dawn of number systems.

  • These early number systems were concerned only with numbers used to count objects.

  • In mathematics, we call these counting numbers the "natural numbers".

  • The smallest natural number is 1

  • and there is no limit to the largest natural number.

  • As we also saw in the previous lecture

  • there are many number systems which could be invented to represent natural numbers.

  • For instance, the Romans used a natural number system

  • which by today's standards seems quite complicated.

  • In the Roman system, the symbols I, V, X, L, C, D, and M

  • represent the quantities 1, 5, 10, 50, 100, 500, and 1000.

  • The quantities 2 and 3 are represented by two or three I's.

  • The quantities 6, 7, and 8 are represented by the symbol for five, V

  • followed by one, two, or three I's.

  • And the quantities 4 and 9

  • are represented by the symbols for 5 or 10, V and X, preceded by an I.

  • The numbers 10 through 100 follow the same pattern

  • except that the symbols X, L, and C are used to represent 10, 50, and 100.

  • The same pattern is used for the numbers 100 through 1000.

  • using the symbols C, D, and M to represent 100, 500, and 1000.

  • In addition, the symbol M may be repeated up to three times

  • to represent 1000, 2000, or 3000.

  • These groups of numerals can be combined

  • to form any number up to 3999.

  • For example, this number is written as

  • three-thousand

  • plus nine-hundred

  • plus ninety

  • plus nine.

  • The Romans rarely needed numbers larger than this.

  • When they did, they used the standard symbols with a bar over them

  • to indicate a value 1000 times greater.

  • At first look, it seems like it would be very difficult

  • to do calculations using Roman numerals.

  • For instance, take the following simple addition problem.

  • Using Roman numerals, this same problem looks quite complicated.

  • However, the Roman number system is actually not all that different from ours

  • if you think of groups of Roman symbols

  • being the equivalent to our single numeric symbols.

  • If we arrange the symbols into columns of ones, tens, and hundreds

  • the two number systems look a little more similar.

  • The first difference that is apparent

  • is that the Roman number system had no symbol for zero.

  • An even more important difference

  • is that our modern number system uses the same symbol

  • to represent different values depending on its position in the number.

  • For instance, in this problem, the number "2" represents 2, 20, and 200

  • depending upon which column the "2" is in.

  • On the other hand, in the Roman system, 2, 20, and 200

  • are represented by different symbols.

  • The important difference between the Roman number system and our modern system

  • is that in the Roman system

  • the position of a symbol within a number doesn't determine the value.

  • Since symbols do not have to fall into particular columns

  • zeros are not needed as a column placeholder.

  • Our modern number system is an example of "positional notation".

  • In positional notation, the same symbol represents different quantities

  • depending on its position in the number.

  • For example, the symbol "1" can represent 1, 10, 100, 1000, and so on.

  • Consequently, the numbers 10, 100, and 1000

  • require zeros as column place holders following the one.

  • The Roman number system is an example of "sign-value notation".

  • Sign-value notations do not require a symbol for zero

  • since different quantities such as 1, 10, 100, and 1000

  • each have unique symbols whose value does not depend on their position in the number.

  • The natural number system used today, with which most everyone is familiar

  • is called the "decimal" or "base-10" number system.

  • In the next lecture we will explore these numbers

  • as well as other natural number systems using other bases such as binary, octal, and hexadecimal

  • which are often used when working with digital computers.

Hello. I'm Professor Von Schmohawk and welcome to Why U.

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A2 初級

代数2 - ローマ数字.符号-値と位置表記 (Pre-Algebra 2 - Roman Numerals: Sign-Value vs Positional Notation)

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    Chun Sang Suen に公開 2021 年 01 月 14 日
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