The Number System and Common and Decimal Fractions
The Number System and Common and Decimal Fractions In our system of numeration we use ten symbols called digitso 1 2 3 4 5 6 7 8 and 9 and combinations of these sym bols Our system of numeration is called the decimal or base ten system There is little doubt that our ten fingers influenced the development of a numeration system based on ten digits Other numeration systems were developed in early cultures and societies Two of the most common were the base five system related to the number of fingers on one hand and the base twenty system related to the number of fingers and toes In some languages the word for five is the same as the word for hand and the word for ten is the same as the word for two hands In our own language the word digit is a syn onym for the word finger that is ten digits ten fingers Still another early system of numera tion was a base sixty system developed by the Mesopotamians and used for centuries These ancient people divided the year into 360 days 6 x 60 To this day we still divide the hour into 60 minutes and the minute into 10 60 seconds Numeration systems of current interest include a binary or base two system 12
NUMERATION SYSTEMS used in computers and a base twelve or duo decimal system It is worthwhile for everyone to become familiar with the principles of the base twelve system of numeration and with those of base two base five or other systems Working with other bases gives deeper insight into the decimal system that most people in the modern world have used since childhood Before considering other sys tems however let us investigate our more familiar decimal system THE DECIMAL SYSTEM The ten digits of our numeration system are used to name the numbers of dots shown in these frames ODO BE E G 0 0 0 0 If we were not familiar with our system of numeration a reasonable method for naming the number of dots in frames showing 10 11 and 12 dots might be somewhat as follows 13
The Number System and common and Decimal Fractions D B 91 A system of numeration based on such a procedure as this is called an additive or sign value system The numeral 995 would rep resent the sum 9 9 5 In an additive system of numeration the value of each numeral is the equivalent of the sum of its individual numeric signs or digits However in our system of numeration we use a positional or place value system invented by ancient Hindu mathemati cians The positional system of numeration has many advantages over a simple additive system In a positional system of numeration the value assigned to a digit depends upon its position in the numeral For example in our decimal positional system the 1 in the numeral 103 refers to one group of 100 the o refers to zero groups of 10 and the 3 refers to three is 10 103 1X 10 X 10 OX 10 3 X 1 11 14
NUMERATION SYSTEMS The symbol 10 is a convenient shorthand notation for the product of 10 X 10 The sym bol is called an exponent and it tells you that 10 is to be multiplied by itself In the same notation IO x 10 x 10 Io3 and 10 x 10 x 10 x 10 104 Using this shorthand notation the numeral 1 572 can be expressed 1 572 1 x 10 5 x 10 7 x 1o 2 x 1 Numbers such as 1 6 230 and 1 572 are called whole numbers The ten digits may be used to represent not only whole num bers but also numbers less than one and numbers which are sums of whole numbers and numbers less than one These numbers can be represented either as fractions or as decimals The value assigned to each digit is again determined by its position in the numeral For example 132 1 and 3x H 2 x 1000 s 23 35 2 x 10 3 x 1 3x x 15
