Website owner: James Miller
Soon after I graduated from college with a degree in mathematics I started asking myself some questions about the many rules one learns in arithmetic in grade school (rules for operating on fractions, for example) and realized I couldn't answer them. The more I thought about the questions I had posed for myself, the more I tried to answer them, the more confused I became. When I asked myself to prove this well known rule or that one I couldn't. I was confused by the relationship between fractions and division and by the use of the slash symbol for both. I realized I had a degree in mathematics and didn't even really understand arithmetic. What were the basic definitions and axioms of arithmetic? What were the derived theorems and results? I didn't know. The reason for this is the way arithmetic is taught. The textbooks and teachers present the rules and show you how to apply them. They give you a working knowledge of arithmetic. But they don't go into the reasons for the rules, the whys. I got good marks in arithmetic, was good in it, liked it, never had trouble with it. But I only knew how to use it, how to apply the rules. I didn't really understand it at a basic level. So realizing my lack of understanding of the basic fundamentals of arithmitic I set myself a project of trying to set down in a systematic and logical way the basic definitions and axioms of arithmetic and then trying to derive the theorems and results from them, in the manner of Euclidean Geometry. The first problem -- naming. The first problem that ancient man met in his encounter with the number concept was a problem of naming. For example suppose an ancient farmer had 16 bushels of wheat that he wanted to sell to a merchant but had no names for the integers. How could he communicate the information to the merchant? So the first thing needed was names for the integers. Now suppose the integers 1, 2, 3, ... have been named. And suppose a farmer has 16 2/3 bushels of wheat that he wants to sell to a merchant. How does he communicate this information? He is able to communicate the fact that he has between 16 an 17 bushels to sell but he does not have any word or symbol or means to convey the fact that he has 2/3 of a bushel. What type of naming system could be devised that would name all the possible fractional portions between zero and one? Ancient man had met a sizable problem. It was probably one that went unsolved for centuries. However, eventually someone came up with the idea of using the already named integers to name the fractional portions of unity. The idea was to divide unity up into a number of equal parts, say n parts, and then specify a fractional portion through multiples of the n-th part of unity. One can then specify any number as the sum of a whole number plus a fraction. Thus he can now specify that he has, for example, 16 2/3 bushels of wheat. The first two steps: Step 1. Create a logical and easily used naming system for the integers. Step 2. Create a system for designating the fractional parts of a whole (i.e. of unity). __________________________________________________________________ NOTE. In our discussion when we speak of an integer we mean a positive integer 1, 2, 3, 4, ... . Negative integers are introduced in the study of algebra and are important there but are not a part of arithmetic and have no meaning there. __________________________________________________________________ Different systems have been used for naming the integers and the one we currently use traces back to the Hindus, employing a zero and the decimal system. ___________________________________________________________________ FRACTIONS AND MIXED NUMBERS ___________________________________________________________________ To define the concept of a fraction one must first define the concept of a "unit fraction". Definition - Unit Fraction. Conceptually a unit fraction is a number such as 1/5, 1/6, 1/7, etc. (i.e. it is a fraction whose numerator is 1). We define 1/5 to be the fifth part of 1, 1/6 to be the sixth part of 1, and, in general, 1/n to be the n-th part of 1. Logically this concept comes first and the usual concept of a fraction is built up from it. For clarity of thought and development we introduce the special notation n* for 1/n. Thus 1/5 is represented by 5*, 1/6 by 6*, etc. Definition - Fraction. A fraction, designated by the symbol "m/n" is by definition the quantity m(n*) -- which means it is m n*'s (we can speak of m n*'s in the same way we can speak of m cows, m pencils, or m dogs). Thus the fraction 4/5 is, by definition, 4(5*) and represents four "fifth parts of 1", the fraction 5/6 is 5(6*) and represents five "sixth parts of 1", etc.. Definition - Mixed Number. A mixed number is, by definition, a whole number (i.e. a integer) plus a fraction. Thus the mixed number 2 5/8 is defined as 2 + 5/8, the mixed number 1 2/3 is defined as 1 + 2/3, etc.. We now have a way of designating any positive rational number p/q where p and q are integers. Any rational number can be expressed as a mixed number and vice versa. Let us note that any mixed number can be expressed as a fraction. For example the mixed number 3 5/6 can be expressed as 3x6(6*) + 5(6*) = 3x6/6 + 5/6 = 18/6 + 5/6 = 23/6. ___________________________________________________________________ ARITHMETIC DEVELOPED AS A SET OF PROCEDURES FOR SOLVING PROBLEMS ___________________________________________________________________ Ancient man faced a need to solve certain mathematical type problems that he encountered in life. He needed to measure distances, areas, quantities of fluid or grain, weights; to compute sums, differences, etc.. Let us list some of the mathematical type problems he needed to cope with: 1. To measure continuous quantities such as bushels of grain, distances, areas, weights, etc.. 2. To compute sums of whole numbers or sums of continuous quantities. 3. To compute differences of whole numbers or quantities. 4. To multiply an integer by another integer. 5. To multiply some quantity by an integer e.g. How much is 5 times 8 7/8? 6. To find a fractional part of some quantity e.g. How much is 3/8 of 8 7/8? 7. To scale a quantity e.g. scale the quantity 8 7/8 by a factor of 5 3/8 (this problem arises in computing the area of a rectangle). 8. To divide some quantity by an integer e.g. What is the result of dividing 8 7/8 by 7? 9. To determine the number of times one quantity is contained in another i.e. solve the equation ax = b where the unknown x is the number of times a is contained in b. In the beginning early man probably didn't use symbols to indicate the arithmetic operations of addition, subtraction, multiplication or division as we do today. And his procedures for multiplication and division were possibly different from those of our own. He didn't yet have algebra with its use of symbols and procedures and didn't think in terms of algebraic ideas or techniques. But he did have mathematical problems and he needed methods and procedures for dealing with them. He developed procedures for solving his different mathematical problems and from these procedures evolved arithmetic. We now consider the arithmetic operations of addition, subtraction, multiplication and division in a logical order (presumably the order in which they naturally evolved). ___________________________________________________________________ ADDITION ___________________________________________________________________ ADDITION. SUM OF TWO OR MORE INTEGERS. The ancients didn't use the decimal system that we use for naming the integers and because of that their procedure for adding numbers would have been different than ours (the procedure needed is dependent on the naming system). They devised procedures based on their naming system. ADDITION. SUM OF TWO OR MORE MIXED NUMBERS. The sum of two or more mixed numbers can be determined by adding the integral parts and fractions separately, or by reducing each mixed number to a fraction and then adding. One can only add two fractions if they have the same denominator. If he wishes to add two fractions with different denominators he must change the fractions to equivalent fractions, both of which have the same denominator. To do this he needs the following theorem: Theorem. p/q = np/nq [ or p(q*) = np(nq)* ] where p, q and n are integers. In words: We can multipy both the numerator and denominator of a fraction by the same integer without changing the value of the fraction (i.e. multiplying both numerator and denominator of a fraction by the same integer gives an equivalent fraction). Example: 3/5 = 2x3/2x5 = 6/10 Using the above theorem, if one wishes to add the fractions a/b and c/d the sum will be a/b + c/d = ad/bd + cb/db = (ad + cb)/bd ___________________________________________________________________ SUBTRACTION ___________________________________________________________________ In arithmetic you can only subtract a smaller number (integer or mixed number) from a larger one. Subtracting a larger number from a smaller one doesn't make any sense and is not allowed. SUBTRACTION. SUBTRACTING A SMALLER INTEGER FROM A LARGER INTEGER. As with addition the procedure used in subtracting integers is dependent on the naming system used in naming the integers. SUBTRACTION. SUBTRACTING A SMALLER MIXED NUMBER FROM A LARGER MIXED NUMBER. In general, to subtract a smaller mixed number from a larger mixed number you reduce both mixed numbers to fractions and then subtract. To subtract them the denominators of the fractions must be the same. If they are not you must change the fractions to equivalent fractions whose denominators are the same. Example. 5 3/7 - 2 2/3 = 5x7/7 + 3/7 - 2x3/3 + 2/3 = (5x7 + 3)/7 - (2x3 + 2)/3 = 38/7 - 8/3 = 3x38/3x7 - 7x8/7x3 = (3x38 - 7x8)/7x3 = (114 - 56)/21 = 58/21 ________________________________________________________________________ MULTIPLICATION ________________________________________________________________________ MULTIPLICATION. PRODUCT OF TWO INTEGERS. The product m times n, of two integers m and n, expressed as mn, is given by repeated addition. It is defined as "n + n + n + ... " where n is repeated m times. Examples. 3x5 = 5 + 5 + 5 4x6 = 6 + 6 + 6 + 6 MULTIPLICATION. The product of two numbers (i.e. two mixed numbers) can be defined in terms of the products of two fractions since mixed numbers can be converted to fractions (any mixed number is equivalent to some fraction) and that is the course we will take. First we need to define what we mean by the product of two unit fractions m* and n*. We know intuitively what the sum of two unit fractions is but do we know what the product should be? No. It is something that has to be defined. DEF. PRODUCT OF TWO UNIT FRACTIONS. The product (m*)(n*) of two unit fractions m* and n* is defined as (m*)(n*) = (mn)* In our usual notation this means that we define the product of 1/m times 1/n as the quantity given by 1/mn ( e.g. 1/5 X 1/3 = 1/15). PRODUCT OF TWO FRACTIONS. We are now in a position to define the product of any two fractions. By definition, the product of two fractions m(n*) and p(q*), expressed as m(n*)p(q*), is given as: m(n*)p(q*) = mp(nq)* In our usual notation this says (m/n)(p/q) = mp/nq. Products of mixed numbers. Mixed numbers can be multiplied by converting them to fractions and multiplying the fractions. They can also be multiplied by multiplying each term of one number by each term of the other number and adding i.e. mixed numbers e + f/g and h + i/j can be multiplied as: (e + f/g)(h + i/j) = eh + ei/j + fh/g + fi/gj Either method of evaluating a product of two mixed numbers gives the same result. Examples. (3/5)(2/7) = 3x2/5x7 = 6/35 (2 3/8)(3 2/3) = 2x3 + 2(2/3) + (3/8)3 + (3/8)(2/3) = 6 + 4/3 + 9/8 + 6/24 = 6 + (4x8)/(3x8) + (9x3)/8x3 + 6/24 = 6 + 32/24 + 27/24 + 6/24 = 6 + 65/24 = 8 17/24 NOTES ON THE PRODUCT OF TWO FRACTIONS. Intuitively the product (m/n)(p/q) of a fraction m/n times a fraction p/q represents a scaling of the second fraction by the first. For example when the multiplier is 1/3, the product is 1/3 the size of the multiplicand; when the multiplier is 5/6, the product is 5/6 the size of the multiplicand; when the multiplier is 5 1/4, the product is 5 1/4 times the size of the multiplicand. To see why this is true note that (m/n)(p/q) = mp/nq is equivalent to (m/n)(p/q) = m(1/n)(p/q) where (1/n)(p/q) represents the n-th part of p/q and "m" plays the role of a scaling factor that scales this n-th part of p/q. ________________________________________________________________________ DIVISION ________________________________________________________________________ DIVISION. QUOTIENT OF TWO MIXED NUMBERS. To divide two mixed numbers it is necessary to first reduce them to fractions. The quotient a/b of two numbers a and b is defined as the solution to the equation bx = a. Thus the quotient is x = (1/b)a. When a and b are fractions the rule for computing the quotient is "invert the divisor and multiply". Example 1. The quotient of 5/8 divided by 2/3 is the solution to the equation 2/3 x = 5/8 so x = (5/8)(3/2) which we obtained by multiplying the equation by 3/2. Example 2. Find the quotient of 7/8 divided by 2/5 (we invert the divisor and multiply): (7/8)/(2/5) = (7/8)(5/2) = 5x7/8x2 = 35/16 NOTES ON DIVISION. When the divisor is an integer, n, it should be noted that this definition coincides with the concept of division as an operation that divides the dividend into n equal parts i.e. an operation that yields a quantity that is the n-th part of the dividend. For example, 23/5 represents the 5-th part of 23 (the part obtained by dividing 23 into 5 equal parts). Division also gives the number of times the denominator is contained in the numerator. Or equivalently, the quotient represents the number of divisors in the dividend. For example, 9/2 represents the number of 2's contained in 9. Or said another way, the quotient is the scaling factor that must be applied to the denominator to yield the numerator. The well known rule for dividing a fraction a/b by the fraction c/d, invert the divisor and multiply, follows directly from the equation (c/d)x = a/b. We solve for x by multiplying both sides of the equation by d/c to get x = (d/c)(a/b). ___________________________________________________________________ INTUITIVE INTERPRETATIONS ___________________________________________________________________ Operation Interpretation ___________________________________________________________________ (integer) x (integer) repeated addition (integer) x (any number) repeated addition (any number) x (any number) scaling of second number by first (any number) / (integer) divide numerator into equal parts (any number) / (any number) gives number of times the denominator is contained in the numerator
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