Sign value

Transcription

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 one 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.