Place Value and Integer Arithmetic: A Study from First Principles

Transcription

1 Place Value and Integer Arithmetic: A Study from First Principles 0. Introduction This article attempts to point out ways to make the study of integer arithmetic more unified and more conceptual through a systematic emphasis on structure of place- value numbers. In the process, we illustrate how the goal of mathematical theory is to make problems easier to understand and solve. This paper is a reworking of the first two sections of a larger paper by Roger Howe and Susanna Epp called Taking Place Value Seriously. The general approach and the key ideas are due to Roger Howe. In many places, the text comes straight from the Howe- Epp paper with their permission. Section one lays out the structure of place- value notation. This structure makes possible arithmetic operations that in almost all cases reduce multi- digit calculations to a set of single- digit calculations. Sections two through five discuss in detail how place- value notation leads to efficient algorithms for addition, subtraction, multiplication and division of integers, respectively. The crucial point is that the all these procedures follow naturally when we apply the Rules of Arithmetic 1 to the expanded form of place- value numbers. While the subtraction and multiplication algorithms are natural modifications of the addition algorithm, the division algorithm requires a different approach that makes division the most complex operation. Moreover, long division cannot be reduced to a set of single- digit division problems. In section two, we explain the huge advance that place- value notation represents over previous number systems. The great mathematician Carl Friedrich Gauss wrote, The greatest calamity in the history of science was the failure of Archimedes to invent positional notation. 1. Place-Value Notation Following the approach of Euler, a non- negative number is a magnitude that exists independently of the notation used to write it. The place- value system is an efficient notation for writing numbers in terms of powers of 10. It uses only 10 symbols- - the base 10 digits: 0,1,2,3,4,5,6,7,8,9- - to express any number, no matter how large the number. In this paper, we assume numbers are expressed in place- value notation. For emphasis, we sometimes use the term place-value number. While humans use numbers in base 10 (based on counting with our 10 fingers), computers use numbers in base 2, since the base- 2 digits, 0 and 1, correspond naturally to the two states of an electronic system, OFF and ON. While place value is a great invention simply for the ease of writing all numbers, its real power comes from the way it simplifies arithmetic calculations. This efficiency of arithmetic with place- value numbers is possible because of the mathematical structure 1 By the Rules of Arithmetic, we mean what mathematicians refer to as the Field Axioms, and what are often called number properties, or just properties. They are nine in number: four (Commutative, Associative, Identity and Inverse Rules) for addition, four parallel ones for multiplication, and the Distributive Rule to connect addition and multiplication.

2 underlying place- value notation. We hope that making this structure more explicit will increase conceptual understanding and improve computational flexibility. Finally, the structure of place- value notation can illuminate the parallels between arithmetic and algebra, thus making arithmetic a preparation for algebra, rather than the impediment that it sometimes appears to be. Initially, all numbers will be nonnegative integers. A place- value number has a natural representation as a sum of terms that are each the product of a non- zero digit and a power of 10. We call this sum the expanded form of the number, and terms in the sum the singleplace components of the number; for short, its components. For example 7,402 has the expanded form, 7, x x or 7, We equivalently write single- place components as 7, The leading (leftmost) digit must be non- zero in a place- value number and in each of its single- place components. The order of a component is the power of 10 in the product; for example, the component of order 3 in 7,402 is 7 x 1000 or 7,000. In a place- value number, each place with a non- zero digit has an associated component in the expanded form. For example in 7,402, the third place from the right, with digit 4, has the associated component 400. A place with a 0 digit has no associated component. Each place- value number has a unique expanded form. Conversely, each sum of single- place components uniquely specifies a place- value number, provided there is at most one component of any order. While one normally writes numbers in place- value notation, arithmetical calculations are best understood with numbers written in expanded form. 2. Addition The development of arithmetic is a model for the way mathematical subjects should develop: a simple concept, addition, leads directly to other concepts, subtraction and multiplication, that in turn collectively build higher other concepts in this case, division. The key algorithm in arithmetic is addition. The algorithms for the three other operations are based on it, with the algorithms for subtraction undoing addition and multiplication repeated addition being straightforward extensions of the addition algorithm, while division is more complicated. The standard multiplication algorithm is remembered so easily, although it is not used often, because it is so similar to the often- used addition algorithm. The basic version of addition algorithm using the expanded form is stated as follows. We assume that all integers as positive; negative numbers are introduced in section 4. Step 1. Write each of the addends, the numbers to be added, in expanded form. Step 2. Group together the single- place components of the same order. Step 3. (Simple version) For each group of components of the same order, factor out their common power of 10 and then add together the digits. Step 4. Rewrite the terms of the expanded form in place- value notation. 2

3 Here is a simple example: (200704)(600203) step 1 - write each addend into its expanded form (200600)(7020)(43) step 2 - group the associated components of each order (26) x 100 (72) x 10 (43) step 3 - factor out the common unit component and 8 x x 10 7 add together the digits for each power of step 4 rewrite in place- value notation. This addition example avoided the problem of regrouping, or carrying. We shall address this complication shortly, but for now our focus is on gaining familiarity with expanded forms. In general, it is always best to introduce a new idea in the simplest setting possible to allow the reader to focus on gaining familiarity with the idea in this case, expanded forms without other things to worry about. However, even in this simple example of addition with expanded forms, there is some subtle mathematics at work that is easily taken for granted. Two Basic Rules for Addition: The steps in the addition algorithm are justified by means of the two basic rules of addition, the Commutative Rule and the Associative Rule. The Commutative Rule for Addition: The value of a sum does not depend on the order of the addends; that is, for any two numbers, a and b, a b b a. The Associative Rule for Addition: When adding three numbers, the value of the sum does not depend on the way the numbers are combined into pairwise sums. More precisely, for any three nonnegative integers a, b, and c, (a b) c a (b c). A variety of methods are commonly used to justify these rules. For instance, if we think of a number as corresponding to a length, say the length of a bar, then the process of adding two numbers is modeled by placing the corresponding bars end to end and measuring the total length. In this model, the commutative rule simply says that swapping the order of the two bars does not change their total length. The associative rule says, take a bar composed of three sub- bars of lengths a, b, and c, and observe that it can be thought of as being made of two sub- bars of length a b and c, or of two sub- bars of lengths a and b c. The associative rule justifies the common way we write sums, e.g., a b c d, omitting the parentheses that would indicate the order in which we add pairs of numbers or partial sums. In other words, addition can be treated an n- ary operation on a list of numbers, instead of a binary operation. Repeated application of the commutative rule means that we can re- order a list of addends any way we want. The commutative and associative rules together give us great flexibility in how we add a set of numbers. We summarize this flexibility with the following rule. 3

4 The Any-Which-Way Rule for Addition: When a list of numbers is to be added, it does not matter in what sequence we do the pairwise sums or how we order the addends in any of the intermediate sums: the final result will always be the same. The following sum illustrates the way the Any- Which- Way Rule is used Standard binary addition of this sum would be performed as follows: 5 plus 3 equals 8; 8 plus 8 equals 16; 16 plus 7 equals 23; 23 plus 5 equals 28. The Any- Which- Way Rule allows for mental shortcuts such as noting that there are two 5 s which sum to 10; similarly for the 7 and 3. So we reduce the sum to: two 10 s plus 8, equal 28. We also need the following common- sense principle. The Units Principle: The resulting value of an addition calculation does not depend on the units we use. For example, whether we add income in dimes, or one- dollar bills, or ten- dollar bills does not affect the monetary value of the answer. The Units Principle is a special case of the Distributive Law of arithmetic. We postpone full discussion of Distributive Law, which says that a x ( b c) a x b a x c, until we deal with multiplication. The Units Principle justifies the way we reduce the addition of components to adding their non- zero digits in step 3 of the addition algorithm. Here is an example: 2,000 5,000 1,000 (2 5 1) x 1,000 switching to units measured in 1,000 s 8 x switching back to units measured in 1 s Now we go through the first three steps of the addition algorithm in detail to show how the Any- Which- Way Rule and the Units Principle justify the steps in the addition algorithm. Consider the problem: Step 1. Write the addends in expanded form: ( ) ( ) (100 3) ( ) Step 2: Use the Any-Which-Way Rule to re-order the addends by grouping together components of the same order: We can write this grouping linearly- - ( ) ( ) ( ) 4

5 or we can do this grouping vertically (the standard way) Step 3: For each group of components of the same order, use the Units Principle to factor the common power of 10 and then add together the digits: x 100 x x x Step 4: Write terms of the expanded form in place-value notation I M P O S S I B L E Terms like 22 x 10 are not single- place components of an expanded form, because, as their name implies, single- place components can have only one non- zero digit. Technical Note: if the first term was 20 x 100, we would still have a problem because it forms a single- place component of a higher order. INTERLUDE: Before correcting Step 3, we pause to reflect on the achievement of placevalue notation that reduced the addition of large numbers to a set of single-digit additions. Place-value notation with its associated expanded form is arguably the greatest mathematical accomplishment in human history up until a few hundred years ago. Before the development of place-value notation, first developed by Indian and Persian mathematicians around 1500 years ago, arithmetic was a highly technical skill. The complex arithmetic calculations required for engineering projects in ancient Egypt, Greece and Rome were performed by specialists using a device like an abacus. Modern commerce and science are heavily dependent on the ease of calculating with place-value numbers. Place-value notation seems natural and obvious to us today. But the great Greek mathematicians did not discover this obvious notation. The 19th century mathematician Carl Friedrich Gauss wrote, The greatest calamity in the history of science was the failure of Archimedes to invent positional notation. Greeks used Roman numerals. Consider the number 849 in Roman numerals, DCCCXLIX. The 100 s place digit, 8, is expressed by DCCC, where D 500 and C 100. The 10 s place digit, 4, is expressed by XL, where L 50 and X 10 and putting the X before for L instead of after it means we subtract 10 from 50 instead of adding it. Finally the 1 s place digit, 9, is represented by IX, where again X 10 and I 1 and putting the I before instead of after X means we subtract 1 from 10. There are different symbols for each different power of 10, along with additional symbols for 5 times each different power of 10. Conversely, each symbol in a Roman numeral has a unique value. There is no way to reduce the 5

6 addition of large numbers into easier sums of small numbers because large numbers cannot be decomposed into small numbers the way the expanded form works for placevalue numbers. The key to making place value notation work is the digit 0. Without 0, there is no way to express single-place components. While the digit 0 is first learned by children as representing none of something, this interpretation of 0 is almost incidental to its real role in associating a unique power of 10 with each digit in a place-value number. For thousands of years, it was taken for granted that the symbols used in numbers had to have unique values. With place value, now the places in a number have the unique values, and a digit can represent different values depending on its place. It truly is an incredibly innovative change. And the key to it all is the digit 0. The goal of mathematical theory is to come up with insights and structures to make complex problems easy, ideally to make their solution obvious, and to extend the solution of the simplest problems in a reasonable step-by-step process to solve the most difficult problems. By this measure, no mathematical result comes close to the creation of place-value notation for performing arithmetic calculations. We return now to our addition example. Applying the first three steps of the addition algorithm to , we obtained the preliminary answer 18 x x (*) We shall refer to the terms in (*) as single-place sums. The order of a single- place sum is the order of the single- place components being summed. In (*), the single- place sum of order 2 is Our basic version of the addition algorithm does not work here because the single- place sums in (*) are not single- place components. Addition with Regrouping: We need to regroup the single- place sums by rewriting them in expanded form. If we were adding 50 numbers, the single- place sum for, say, the 100 s place might be 243 x 100, which would be rewritten in expanded form as 20,000 4, The terms 4,000 and 20,000 need to be regrouped with the other components of order 3 and order 4, respectively, in the addition calculation. Because of such regrouping, we need to compute single- place sums and regroup them in sequence from lowest order to highest order. General Version of Addition Algorithm using expanded forms: Step 1. Write each of the addends, the numbers to be added, in expanded form. Step 2. Group together the single- place components of the same order. Step 3: For the group of single- place components of order k 0, then k 1, etc. If there are no single- place components and no carry component for some order k, go to the next larger k: Step 3a. (Single-Place Sum) Factor out 10 k (the common power of 10) and then add the digits together to obtain a single- place sum of the form s x 10 k. Step 3b. (i) If s is a single- digit number, then s x 10 k is a component in the expanded form of the answer. 6

7 (ii) (Regrouping) If s is a two- digit number, the tens digit, call it c, becomes a carry component of order k1 and we group c x 10 k1 with the other components of order k1. The ones digit, call it a, becomes the component a x 10 k in the expanded form of the answer, provided a 0. Step 4. Rewrite the components of the expanded form in place- value notation. The process in Step 3c(ii) is commonly called carrying. In Step 3b(ii), if s has more than two digits, then there could be additional carry components of increasing orders. Let us apply the general version of the addition algorithm to our example Step 1. Write Addends in Expanded Form: ( ) ( ) (100 3) ( ) Step 2: Use the Any-Which-Way Rule to re-order the addends by grouping together singleplace numbers of the same order: Step 3 General version step 3a for order step 3b for order 0 (10 was added to the 10s column) x x 10 5 step 3a for order 1 7

8 step 3b for order 1 (200 was added to the 100s column) x x step 3a for order ( 2000 ) 30 5 step 3b for order step 4 collapse expanded form into place- value notation When this algorithm is performed in practice, one does not write the answer initially in terms of single- place components. Instead the single- place components are directly translated into place- value notation. This means that a 0 digit is written in a place that has no corresponding single- place component Despite our use of single- place components, the reader should be able to recognize the traditional column- by- column addition algorithm. The framework using expanded forms we have developed for addition will extend in a natural way to multiplication, where single- place sums will again be used along with single- place products. When the reader understands the addition algorithm with expanded forms, the more complicated multiplication algorithm with expanded forms will involve only one new step. Pedagogical Comments. 1. The Everyday Math textbook series presents four versions of the addition algorithm, and one version uses partial sums that are the same as our single- place sums. The Everyday Math s partial- sum version of our addition example would look like the following: partial sums The way in which addition skills are developed over grades K- 2 is a nice case study in how mathematical skills evolve. The sequence of skills develops from single- digit addends with single- digit sums, to single- digit addends with two- digit sums, 8

9 to addends that are single- place components (e.g., 20 60) to multi- digit addition with carrying, to skip counting (starting with counting by 2 s and later by 5 s). Further, a growing number of re- enforcing applications of addition appear involving measurement, money, and time. Building on this development of addition, subtraction is introduced and linked to addition. The skip counting leads in grade 3 to single- digit time single- digit multiplication. Each piece of this skills development reinforces previous knowledge and anticipates future learning. For virtually all students, the many pieces of this world of addition and its extensions to subtraction and multiplication interconnect in a way that makes sense. 3. Multiplication Multiplication of whole numbers is easily understood as repeated addition. As a subcase of addition, albeit a messy subcase, multiplication is conceptually simpler than subtraction, and thus we discuss it before subtraction. The standard multi- digit multiplication algorithm for expanded forms of place- value numbers is a natural extension of the addition algorithm for expanded forms. We will be start, as is done in school, with the algorithm for a single digit number times a multi- digit number. However, we first discuss the rules for multiplication. And just as the commutative and associative rules for addition are critical for addition procedures for place- value numbers, so the commutative and associative rules for multiplication play a crucial role in multiplication procedures for place- value numbers. We assume again that all numbers are positive; multiplication with negative numbers is discussed at the end of section 2C. The Commutative Rule for Multiplication: The value of a product does not depend on the order of the factors; that is, for any two numbers n and m: m n n n. The Associative Rule for Multiplication: When multiplying three numbers, the value of the product does not depend on the way the number are combined into pairwise products; that is, for any nonnegative integers l, m, and n, l (m n) (l_ m) n. The best way to develop an intuitive sense for the commutative rule is through the Array Model. This model is also important because it helps prepare students develop intuition for the concept of area. For example, consider the arrays used to represent 4 3 and 3 4: O O O O O O O O O O and O O O O O O O O O O O O O O Reflecting the diagram on the left across its diagonal produces the diagram on the right: 4 9

10 rows of 3 objects give the same total number of objects as 3 rows of 4 objects: The same reasoning can be applied to any array with n rows and m columns to help visualize the general commutative rule for multiplication. Unlike the associative law for addition, the associative law for multiplication is far from obvious. Accordingly, there is no simple geometric model to illustrate this rule. However, we can extend the Array Model to imagine l (m n) as a rectangular 3- dimensional array consisting of l layers of flat arrays of size m n. This 3- dimensional array can also be viewed as n perpendicular slices of flat arrays of size l m. The associative rule generalizes to the product of any sequence of numbers, in which the product of two adjacent numbers is computed and then the result is repeatedly multiplied by a third number just to the right or left. No matter how this is done, the associative rule guarantees that the final result is always the same Any-Which-Way Rule for Multiplication: As was the case with the commutative and associative rules for addition, the commutative and associative rules for multiplication can be stated an Any- Which- Way rule for multiplication: The Any-Which-Way Rule for Multiplication: When a list of numbers is to be multiplied together, it does not matter in what sequence we compute the pairwise products or how we order the factors in any of the intermediate products: the final result is always the same. As an example, the Any- Which- Way rule says that (3 8) (7 8) (7 3) (8 8). We verify this equality in a step- by- step fashion with the commutative and associative rules. (3 8) (7 8) 3 (8 (7 8)) by the Associative Rule 3 ((8 7) 8) by the Associative Rule 3 ((7 8) 8) by the Commutative Rule 3 (7 (8 8)) by the Associative Rule (3 7) (8 8) by the Associative Rule (7 3) (8 8) by the Commutative Rule The Any- Which- Way Rule is useful for mental arithmetic. For instance, a person who finds it difficult to compute (7 5) (6 2) might see that it is much easier to find the product if 5 and 2 are grouped together to give a factor of 10: (7 5) (6 2) (5 2) (7 6) The General Distributive Rule: The Distributive Rule tells us how to multiply a number times a sum of two numbers: The Basic Distributive Rule: For any numbers a, b and c, we have the equation a (b c) a b a c. Just as repeated uses of the commutative and associative rules can be combined to produce an Any- Which- Way rule, so repeated uses of the basic distributive rule, combined with the commutative and associative rules, produce the General Distributive Rule. 10

11 The General Distributive Rule: If A and B are each sums of several numbers, then the product AB may be computed by multiplying each addend of B by each addend of A, and adding all the resulting products. As an example of the general distributive, if A a b and B c d e, then AB (a b)(c d e) equals ac ad ae bc bd be. The following diagram illustrates the general distributive rule for: (3 2) (1 4) The left side counts the number of o s as the total number of rows times the total number of columns. The right side counts the number of o s in terms of 4 subarrays. o o o o o o o o o o o o o o o o o o o o o o o o o Computing Products. The Any- Which- Way rules for addition and multiplication plus the distributive rule, which links multiplication and addition, are all necessary tools for multiplying place- value numbers. Since each number in expanded form is a sum of single- place components, the distributive rule tells us that the product of two numbers may be found by multiplying each single- place component of the first number times each single- place component of the second number, and then adding all the products together. Moreover, the Any- Which- Way Rule for addition says that we have great latitude how we sum these numbers up. As just noted, the product of two place- value numbers in expanded form involves products of a single- place components. Thus, multi- digit multiplication is reduced to a set of single- digit products. Product of Two Single-Place Components: The product of two single- place components, M d 1 10 m and N d 2 10 n, equals the product of their leading digits, d 1 d 2, times 10 raised to the sum of their orders, 10 mn ; that is, MN d 1 d 2 10 mn. Reminder: By the Law of Exponents, when we multiply (integer) powers of 10, we add the exponents; that is, 10 m 10 n 10 mn. Extending the concept of a single- place sum in addition, we call the product of two single- place components a single-place product. The order of a single- place product is the sum of the orders of the two components being multiplied together. Product of a Single Digit Times A Multi-Digit Number: Consider the product In keeping with the interpretation of whole- number multiplication as repeated addition, is equivalent to The goal here is to reduce a new problem in multiplication to a familiar problem in addition. Now we can use the addition algorithm from 2A to solve this multiplication problem. We re- state the addition algorithm. Then we simplify it for the case where we can use 11

12 multiplication to perform the addition of each group of single- place components. Addition Algorithm Step 1. Write each of the addends in expanded form. Step 2. Group together the single- place components of the same order. Step 3: For the group of single- place components of order k 0, then k 1, etc. If there are no single- place components (or carry component) for some order k, go to the next larger k: Step 3a. (Single-Place Sum) Factor out 10 k (the common power of 10) and then add the digits together to obtain a single- place sum of the form s x 10 k. Step 3b. (i) If s is a single- digit number, then s x 10 k is a component of the expanded form of the answer. (ii) (Regrouping) If s is a two- digit number, the tens digit, call it c, becomes a carry component of order k1 and we group c x 10 k1 with the other components of order k1. The ones digit, call it a, becomes the component a x 10 k in the expanded form of the answer, provided a 0. Step 4. Rewrite the components of the expanded form in place- value notation. Now consider how to reformulate this algorithm for the case of repeated addition of a common addend. In Step 1 there is only one multi- digit number to write in expanded form. In Step 2, we use the distributive law to form products of the single- digit factor times each single- value component. In step 3a we use digit- times- digit multiplication to perform the repeated addition. Carrying will be handled the same way. In step 3, we replace the term single- place sum by single- place product. Here is the modified algorithm. Multiplication Algorithm for a single digit d times multi-digit number n: Step 1. Write n in expanded form. Step 2. By the distributive law, form products of d times each single- place component of n. Step 3: For k 0, then k 1, etc. Step 3a. (Augmented Product) Compute the augmented product p x 10 k of order k by forming the single- place product of order k d times the number s component of order k, call it b x 10 k and then adding the carry component of order k, call it c x 10 k. Possibly one of these two components is missing; if both are missing, go to the next larger value for k. Factoring out the common unit of 10 k, we have p d x b c. Step 3b. (i) If p is a single- digit number, then p x 10 k is the component of order k in the answer, and there is no carry component. (ii) (Regrouping) If p is a two- digit number, then its tens digit, call it c, becomes the carry component c x 10 k1 of order k 1. The ones digit, call it a, forms the component a x 10 k of order k in the answer, provided a 0. Step 4. Rewrite the components in the expanded form in place- value notation. While in addition, the carry component can be added with other components of the next higher order, this is not possible in multiplication algorithm because we have single- place products not single- place sums. Thus the need for augmented products. On the other hand, regrouping is exactly the same for multiplication as for addition. 12

13 We illustrate this algorithm with the product : ( ) Step Step (42) Step 3a, order ( ) 2 Step 3b(ii), order 0- Regrouping ( 28 ) 10 2 Step 3a, order Step 3b(ii), order 1-- Regrouping Step 3a,b(i), order 2 (30) Step3a order Step 3b, order 3-- Regrouping Step 3ab(i), order Step 4 Observe that this algorithm is easily extended to the product of a single- place component times a multi- digit number, such as , since by the any- which- way rule, ( ) 100. All that is required is that in step 3a we modify the power of 10 in the single- place products. General Multiplication It is a natural step now to go from the preceding calculation to the product of two multi- digit numbers, such as the product By using the distributive law, such a product can be broken into a set of products of a single- place component times a multi- digit number (30 6) Then we apply the preceding algorithm to each product on the right side and add the result of these multiplications together for the final answer. We shall compute shortly. As the reader surely recognizes, we are describing here the algorithm that most children are taught for multiplying two multi- digit numbers. Observe the incremental evolution algorithms, from simple addition (without carrying), to general addition (with carrying), to multiplication by a single- digit, to multiplication by a single- place component to to multiplication of two multi- digit numbers. It is important to realize that the mathematical structure for place- value arithmetic, such as expanded forms and the Any- Which- Way rule, is all introduced with addition. Addition is a good, simple setting for developing this important mathematical structure. With the general addition algorithm, we introduced the key step of carrying. There was little mathematically new when we move to multiplication, only the shortcut of using multiplication facts, along with the distributive law, to speed up repeated addition. Of course, multiplication represents a major increase in the amount of computation compared 13

14 to addition, and single- digit multiplication facts are harder than the corresponding addition facts. Still, the multiplication algorithm is easy for students to learn because it follows in a natural incremental way from addition. Let us now present a very general algorithm for multiplying two multi- digit numbers. It is known in schools as the Every Product method. General Multiplication Algorithm using expanded forms Step 1. Write each factor in extended form. Step 2. Using the general distributive rule, form all the single-place products of a single-place component in the first factor times a single-place component in the second factor. Step 3. Use the addition algorithm to sum all the single-place products. As an example, consider (30 6) ( ) Step 1. write in expanded form Step 2. form and compute all 12, single- place products 16,452 Step 3. sum the single- place products 4: Subtraction, Negation, and Negative Numbers Conceptually, increasing some quantity or decreasing the quantity seem like equally likely activities and so one might expect some sort of symmetry between addition and subtraction. However, when it comes to calculating how to add and how to subtract, there is a huge difference. Subtraction is a very different operation from addition in terms of its mathematical structure. The commutative law does not apply for subtraction, e.g Similarly, the associative law does not apply, e.g., (7 4) 2) 7 (4 2). So when we write numbers in expanded form, we do not have an Any- Which- Way Rule for subtraction to group single- place components of the same order together, as we did with addition. On the other hand, the reader knows that there is a column- by- column subtraction algorithm, similar to the column- by- column addition algorithm, that reduces subtraction of two multi- digit numbers to a set of single- digit subtraction problems. So how do we can have a subtraction algorithm that is so similar to addition when subtraction is so different mathematically from addition? The key to resolving this paradox is to treat subtraction as the addition of a positive number and a negative number. So we now discuss negative numbers. To understand negative numbers, we need the operation of negation, for a negative number is created by the negation of a positive number. (Place- value notation only produces positive numbers, plus 0.) Note that negation acts on a single number it is a unary operation- - while subtraction involves two numbers it is a binary operation. Unfortunately, the minus sign ( ) is commonly used to denote both subtraction and negation, i.e., a negative number. We shall focus on negation and not refer to subtraction for a while. We work with negative numbers so much in mathematics that it is easy to forget that negative numbers are not real quantities; one cannot have 5 apples. Rather, negative 14

15 numbers are a convenient mathematical invention, with practical uses in situations where we have increases and decreases in a quantity, as for example deposits and withdrawals from a bank account. They allow us create a single scale to register values greater and less than a pre- set value of 0, as for example temperatures above and below 0 Fahrenheit. On the number line the negative number 8 is represented by a point that 8 units to the left while the positive number 8 is a point 8 units to the right of 0. The magnitude of a quantity is always a non - negative number. Whether a number represents an increase or decrease or an amount higher than 0 or lower than 0 is determined by its sign. What type of change is considered an increase depends on the context. With credit cards, purchases, which are equivalent to withdrawals (decreases) in a bank account, are treated as positive changes on a credit card statement, while payments, which are equivalent to deposits (increases), are treated as negative changes in the credit card statement. This interchange of the roles of positive and negative numbers means that adding a collection of numbers that are all negative say, withdrawals from a bank account is equivalent to adding a collection of numbers that are all positive treating the withdrawals like purchases on a credit card statement. No matter what the context, a negative number ( a) always has the reverse effect from a positive number a in addition, in the sense that the sum of a positive number and its negation is 0. In formal mathematical language, ( a) is the additive inverse of a. This assertion is the first of the two laws of negation: Laws of Negation for any positive numbers a, b: (i) a ( a) 0 (ii) ( a) ( b) (a b) For emphasis, e will always put parentheses around the negation of a positive number, e.g., ( 741), as we do in the Laws of Negation above. Note that law (i) implies a is the negation of ( a). The second law of negation says that addition to the left of 0 on the number line is similar to addition to the right; namely, moving a units to the left of 0 and then b more units to the left is the same as moving a b units to the left. Law (ii) can be generalized to a sum of any size. Note that it is a distributive law for negation. We turn now to subtraction. In light of the preceding discussion, we see that subtraction of positive numbers is the operation of adding a positive number and the negation of a positive number. To avoid any confusion, we will use a long dash,, for subtraction and a short dash,, for negation. Definition of Subtraction For any two positive numbers a, b: a b a ( b) To understand subtraction with place- value numbers, we first need to define the expanded form for a negative number. The single- place components will now be negative numbers. ( 741) (741) ( ) by the expanded form of a positive number ( 700) ( 40) ( 1) by Law of Negation (ii) Note that the Unit Principle in section 2A is valid for negative as well as positive 15

16 numbers. Thus we can rewrite the preceding expanded form as ( 7) x 100 ( 4) x 10 ( 1) Likewise, the Any-Which-Way Rule for adding positive numbers extends to adding a mixture of positive and negative numbers. That is, the order in which we enter a set of deposits and withdrawals does not change the final balance; similarly for the order of the moves to the right and left along the number line. Now we can present the standard subtraction algorithm. We assume that one knows single- digit subtraction facts, based on one s knowledge of single- digit addition facts, i.e., because In light of our definition of subtraction, we state it as an algorithm for adding a positive number a and a negative number (- b). We assume that a b. In its basic version, it is exactly the same as the simple version for the addition algorithm for positive numbers. Algorithm for adding a positive number and a negative number (basic version) Step 1. Write the two numbers in expanded form, putting the positive number first. Step 2. Group together the single- place components of the same order. Step 3. (Basic version) For each pair of components of the same order, factor out their common unit component and then add together the positive and negative digit. Step 4. Rewrite the terms of the expanded form in place- value notation. Consider the following example 894 ( 742) ( ) ( 700) ( 40) ( 2) Step 1 ( 800 ( 700) ) ( 90 ( 40) ) ( 4 ( 2) ) Step 2 (8 ( 7)) x 100 (9 ( 4)) x 10 (4 ( 2)) Step 3 1 x x Step 4 As in the addition of two positive numbers in section 2, we assume that we know how to add a single- digit positive number and a single- digit negative number, that is, we know how to subtract one digit from another digit. Subtraction with Regrouping Next we present the general case where regrouping is needed. As with the re- grouping in addition for positive numbers, we need to perform Step 3 in sequence for the single- place components of increasing orders. In addition of positive numbers, regrouping was needed when a single- place sum was not a single- place component; that is, when the sum of the leading digits exceeded 9. When adding a positive and a negative number, regrouping will be needed when a single- place sum is negative; that is, when the sum of the leading digits is negative. Algorithm for adding a positive number and a negative number (general version) Step 1. Write the two numbers in expanded form, putting the positive number first. 16

17 Step 2. Group together the single- place components of the same order. Step 3 (General Version): For the group of single- place components of order k 0, then k 1, etc. If there are no single- place components and no borrow component for some order k, go to the next larger k: Step 3a. (Single-Place Sum) Factor out 10 k (the common power of 10) and then add the signed digits together to obtain a single- place (signed) sum of the form s x 10 k. Step 3b: (i) If s is positive, d x 10 k is a component in the expanded form of the answer. (ii) (Regrouping) If s is negative, we borrow a unit component of order k1 and add this unit component (written as 10 x 10 k ) to the single- place difference of order k; the result is a component in the answer. To compensate for the borrowing, we add the negation of the unit component of order k1 called the borrow component to the other components of order k1. Step 4. Write the terms from step 3 in place- value notation. We illustrate this general version with the following subtraction problem, which we first re- state as the addition of a positive and a negative number. To make the connection with the regular format for subtraction, we group the components of the same magnitude in a column > ( 742) ( ) Step 1 ( ( 700) ( 40) ( 2) ) 300 ( 700) ( 40) 8 ( 2) Step ( 700) ( 40) 6 k 0, Step 3a, b(i), 300 ( 700) ( 4) x 10 6 k 1, Step 3a ( 100) 300 (10 ( 4) ) x 10 6 ( 700) ( 100) x 10 6 ( 700) k1, Step 3b (ii), 17

18 ( 1) 3 ( 7) x x 10 6 ( 5) x x 10 6 k2, Step 3a ( 1000) 1000 (10 ( 5) ) x100 6 x10 6 ( 1000) x100 6 x10 6 k2, Step 3b (ii) ( 1) x x x x x 10 6 k 3, Step 3a (no 3b) Step 4 Adding a Collection of Positive and Negative Numbers. Suppose we have to add a large set of numbers, some positive and some negative, as in calculating a bank account balance. There are two strategies that can be used. First, we can compute a series of partial sums, by repeatedly adding the next number in the set to the current partial sum, using the appropriate version of the addition algorithm. Second, we can use the Any- Which- Way Rule to group the positive numbers together and the negative numbers together. Then we add together all the positive numbers and add together all the negative numbers (as mentioned earlier in this section, we add a group of negative numbers that same way we add a group of positive numbers). Finally, we add the sum of the positive numbers to the sum of negative numbers with the preceding algorithm. There is a third possible strategy that would combine our two algorithms so that at step 3 there are four cases for the value of the single- place sum: a) a single positive digit; b) a number greater than 9, c) a negative number, and d) 0. However, in case c), if the negative single- place sum is less than - 9, a more complicated version of regrouping is needed. The reader may want to try to work out the details. Connecting Subtraction with Addition: Our formal treatment of subtraction as the addition of a positive number and the negation of a positive number highlights the similarities between addition and subtraction in how they reduce multi- digit problems to a set of single- digit problems, i.e., the single- place sums. We illustrate how regrouping in subtraction reverses the effect of regrouping in addition, with the following addition problem and reversing subtraction problem , or equivalently 1000 (- 1). 18

19 Addition Subtraction ( 1) Steps 1 & ( 1) Step 3, k ( 10) (10 ( 1) ) (A) x 10 ( 1) x 10 9 Step 3, k ( 100) (10 ( 1)) x 10 9 (B) 10 x 100 ( 1) x Step 3, k ( 1000) (10 ( 1)) x (C) Step 4 Notice in line A of these computations how in the subtraction problem the process of borrowing 10 from the set of components of order 1 (the 10s place) to add to the single- place sum of order 0 (the 1s place) reverses the process in the addition problem of carrying of 10 from the single- place sum of order 0 to the set of components of order 1. The same situation occurs in line B with 100 moving in reverse directions between the set of components of order 2 and the single- place sum of order 1; similarly in line C with 1000 moving in reverse directions between the set of components of order 3 and the single- place sum of order 2. Shortcuts for Borrowing. When borrowing is needed during subtraction of one place- value number from another, it is often performed with one of the following two shortcuts. 19

20 Consider for k 1, Step 3b(ii) in the problem We wrote the borrowing in expanded form as: ( 100) 300 (10 ( 4) ) x 10 ( 700) Instead of borrowing by creating the new order- 2 component ( 100), one can decrease the positive component from 300 to 200 that is, cross out the 3 in the 100s column and replace by a 2. The other shortcut is, one can decrease the negative component from ( 700) to ( 800) that is, cross out the 7 and replace it by an - 8. We believe that writing a ( 1) at the top of the column is best because this step most closely matches how place- value notation is used in subtraction. We note that in subtraction algorithms that students are taught, students are supposed to determine whether a single- place sum will be negative and, if so, to borrow first before performing the sum. In the example of problem , since 0 4 is negative, students would borrow from the 100s place and turn the 0 into a 10. To parallel the addition algorithm, the subtraction algorithm presented here summed first and then borrowed. Multiplication with Negative Numbers: Multiplication with one or two negative number is easily accommodated. The sign of the multiplicands only influences the sign of the product. The digit- by- digit calculations are not affected. The product of a negative and a position number is a negative number repeated addition of a negative number will be a negative number. The product of a negative times a negative number is a positive number. This result follows from the following additional law of negation: ( a) ( 1) x a, that is, multiplying a number by ( 1) yields its negation; in particular, ( 1) ( 1) 1. Then ( a) x ( b) ( 1) x a x ( 1) x b ( 1) x ( 1) x a x b 1 x a x b a x b. 5. Division Division reverses the effect of multiplication, as subtraction does for addition. That is, if a x b c, then c a b and c b a. Like subtraction, division is neither commutative nor associative. For example, and (16 4) 2) 16 (4 2). However, unlike subtraction, there is no backdoor way to divide in the way that subtraction can be restated as adding the negation of a number. The multiplicative inverse of a whole number a is the reciprocal 1/a. Reciprocals, for a 0, are fractions that are not whole numbers, and we are restricted here to working only with integers. So we need a new strategy not based on a variation of addition to perform division. We shall derive the standard single- digit division algorithm by reversing the single- digit multiplication algorithm. As with other operations, this division algorithm reduces multi- digit division to a set of single- digit division problems. However, in a major exception to other operations, we will see that there is no way to reduce long division (with a multi- digit divisor) into a collection of single- digit division problems. This is why long division is so difficult and often minimized in current school mathematics curricula. 20

21 Recall that in the division problem a b, a is called the dividend, b the divisor, and the answer is called the quotient. We assume that one knows single- digit division facts (that is, problems where the divisor and the quotient are single digits) based on multiplication facts, e.g., because 7 x Extending this division knowledge, we assume one also knows single- digit division with remainder, e.g., r 3. We restate the single- digit multiplication algorithm, which we will be reversing. Multiplication Algorithm for a single digit d times a multi-digit number: Step 1. Write the multi- digit number in expanded form. Step 2. By the distributive law, form products of d times each single- place component of the number. Step 3: For k 0, then k 1, etc. Step 3a. (Augmented Product) Compute the augmented product p x 10 k of order k by forming the single- place product of order k d times the number s component of order k, call it b x 10 k and then adding the carry component of order k, call it c x 10 k. Possibly one of these two components is missing; if both are missing, go to the next larger value for k. Factoring out the common unit of 10 k, we have p d x b c. Step 3b. (i) If p is a single- digit number, then p x 10 k is the component of order k in the answer, and there is no carry component. (ii) (Regrouping) If p is a two- digit number, then its tens digit, call it c, becomes the carry component c x 10 k1 of order k 1. The ones digit, call it a, forms the component a x 10 k of order k in the answer, provided a 0. Step 4. Rewrite the components in the expanded form in place- value notation. The heart of the multiplication algorithm is step 3 that involves the augmented product, p x 10 k, where p d x b c. In multiplication, d was the single- digit factor; in division, d is the divisor. In multiplication, b was leading digit of the component of order k in the multi- digit factor; in division, it is the leading digit in the component of order k in the quotient (our desired answer). In multiplication, c was the leading digit of the carry component of order k. We shall explain its role in division shortly. To reverse the calculation of the augmented product, we rewrite the augmented product formula p d x b c as a division problem: p d b r c (*) We use the following division problem, , to illustrate how we reverse the multiplication algorithm. Let q to denote the quotient for Our goal is to determine the single- place components of q. Recall that in section 3 we applied the multiplication algorithm to the product 6 x Thus q Undoing the last step in the multiplication algorithm, we write the dividend in expanded form: , The multiplication algorithm processed components in increasing order. To reverse procedure we process components in decreasing order. We start with the largest component of the dividend, 30,000 or 3 x 10 4, which is of order 4. 21