The Number System And Its Function (1961)

we could not get along very well without a concept of number and the symbols that make up a number system we are going to investigate the structure and characteristics of our number system the idea of number probably began when man first became aware of collections or groups of objects the fingers on his hand represented a group when man saw a relationship between the five fingers on one hand and a matching group of objects such as five fish he was becoming aware of number his first scheme of counting was probably to match up fingers with objects and his first method of recording numbers was probably to make one mark for each object counted later the marks developed into symbols symbols we call numeral today with only 10 symbols or digits we can represent any number each symbol is the representation of a number or set of things so for example the numeral five can represent a set of five single apples or a set of five bags of single apples or a set of five baskets made up of bags of apples in our number system the position of a digit in a numeral tells how large a set it describes this characteristic is called place value so for example this numeral because of place value equals five units five groups of 10 no hundreds and 5 thousands the zero is a placeholder in our system because the value of each place from right to left increases 10-fold our number system is called a decimal system but a number system with place value doesn't have to be based on 10 the ancient Babylonians used a number system based on 60 place value increased 60 fold for each place to the left modern electronic computers on the other hand operate on a number system based on two this is the binary system in which there are only two symbols one and zero corresponding to the on and off positions of the computer's electronic circuits in a binary system place value doubles each place to the left so the first place shows units the second place shows sets of twos then fours in the third place then eights in the fourth place and so on each placed to the left doubling in value here's how we would represent some decimal numerals in the binary system one two red one old three which is one unit and one group of two units and red one one 41 511 6 1 1 7 which is 1 + 2 + 4 11 1 and 8 1 we can perform the same operations with the binary system as with the decimal system in our daily lives we use different kinds of numbers for different purposes for instance when we count we use whole numbers or integers sometimes they are called natural numbers we can represent whole numbers as points on a number line to illustrate one important fact about whole numbers the sum of any two or more whole numbers is always a whole number the product of any two or more whole numbers is always a whole number our system of whole numbers is complete or closed for addition and for multiplication these are called closure principles the system of natural whole numbers is not closed for division or subtraction for example there is no whole number on the line that equals 3 / 4 to provide for this and similar divisions fraction are added to our number line there is also no number on the natural number line that equals 2 - 3 to give meaning to this and similar subtractions to the number line are added negative whole numbers and fractions 2 - 3 = -1 positive and negative whole numbers and fractions are only part of what is called the real number system there are also fundamental laws that govern our number system to illustrate one let's put six eggs into a carton in this order first two then four would the total have been different if we had changed the order and put in the two groups like this four then two of course not this has been a simple illustration of a law called called the commutative law for addition which can be stated using either numerals or letters according to this law the order in which you add numbers does not affect the sum addition is commutative and so is multiplication in ordinary arithmetic and algebra 4 * 2 = 2 * 4 in our number system Division and subtraction are not commutative 4 / 2 which equals 2 is not the same as 2 / 4 which equal 24s or 12 5 - 3 which equal 2 is not the same as 3 - 5 which equal -2 let's illustrate another fundamental law by by putting a dozen eggs into the carton in groups of two eggs four eggs and six eggs we'll fill the carton in two different ways first we'll put in two then four and six together make a dozen now we will put two and four then six and again we have a dozen now we Illustrated this law called the associative law for addition you can associate or group quantities in a continued sum in any way without changing the sum addition is associative in ordinary arithmetic and algebra and so is multiplication the way in which you group multipli can doesn't affect the product now we'll illustrate another and perhaps most fundamental law here are two dark cartons and three light ones let's put 12 eggs in each the total number of eggs in the five cartons could be represented by this statement 12 * the quantity 2 + 3 to find that number we could add the number of cartons five and multiply by the number of eggs in each carton 12 but we would get the same answer if we did it this way multiply 12 by the number of dark cartons and multiply 12 by the number of light cartons and then add the products here is the arithmetic now we've Illustrated this law called the distributive law it makes no difference whether the letter A appears on the left or right the law still holds multiplication is distributive with respect to addition we have seen how the idea of numbers and their symbols probably began today with only nine symbols and the placeholder zero we can represent any number in our system of numbers our number system contains different kinds of numbers that can be handled according to certain principles and laws a clear understanding of them will give you a better understanding of algebra and a better background for further progress in mathematics [Music]