The mathematics you have learned about so

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1 VISUALIZING EXTRAORDINARY WAYS TO MULTIPLY LECTURE 4 The mathematics you have learned about so far has been addition ( adding quantities ); subtraction ( adding negative quantities ); and some ratios and proportions, which are forms of multiplication. And you went extraordinarily far with just those ideas, looking at and solving tricky problems. But in this lecture, you will make sense of multiplication by exploring the area of a rectangle: By chopping a rectangle into pieces, you can make the multiplication much easier by adding the area of the individual pieces. MULTIPLICATION WITH RECTANGLES 4. 7 Multiplication is a problem in geometry. When someone asks you to compute 7, think area. You are being asked to find the area of a rectangle that is 7 units wide and units high. 7 Those numbers, 7 and, are difficult, so let s avoid hard work and split the numbers into more manageable figures. For example, we can think of 7 as + 7 and as +. This means that we re thinking of dividing the rectangles into 4 pieces. And it is easy for us to work out the areas of the individual pieces: gives a piece of area, 7 gives a piece of area 7, gives a piece of area, and 7 gives a piece of area 5. So, the area of the rectangle, 7, must equal = =.

2 THE POWER OF MATHEMATICAL VISUALIZATION People like to believe that multiplication is commutative that is, that the order in which one computes a product does not matter: 7, apparently, gives the same answer as 7. The visual makes it clear why this is so. Turn the 7 rectangle 9 and suddenly we see an 7 rectangle. The area of the rectangle hasn t changed, so the answers to the multiplication problems just have to be the same. 7 7 = 7 Consider another example: 4. Again, it s an area problem. (See figure 4.4.) Let s break the rectangle into manageable pieces: and 4 and, and and. We can work out the areas of the individual pieces of the rectangle quite easily. We can see,, 4, 9,, and. The area of the rectangle must be the sum of these values. 4 4 Before we compute this sum, let s point out that there is some lovely structure in this picture: The pieces of the rectangle line up diagonally by thousands, hundreds, tens, and ones. (See figure 4.5.) 9

3 LECTURE 4 VISUALIZING EXTRAORDINARY WAYS TO MULTIPLY We have plus and 9, making 7; 4 + making, which is tens; and. 9 thousands hundreds tens ones 4 = = 7 Thus, 4 = = 7. Let s compare this to the line method We draw lines, 4 lines, and lines for 4, and lines and lines for, and then count intersection points. We add those counts diagonally and read off the answer in terms of thousands, hundreds, tens, and ones = thousands, 7 hundreds, tens, and = 7 In the rectangle, we compute to get in the top-left corner. In the line method, we compute to get intersection points in the top-left corner. In the rectangle, in the bottom-left corner we compute to get 9. In the line method, we compute to get 9 intersection points in the bottom-left corner. The computations are identical throughout. The rectangle method keeps tracks of the thousands, hundreds, tens, and ones as you go along. The line method ignores the thousands, hundreds, tens, and ones at first, but then it adds the results diagonally, now keeping the thousands, hundreds, tens, and ones in mind, just as one should.

4 THE POWER OF MATHEMATICAL VISUALIZATION The line method is just encoding all the information of the rectangle, so really is doing correct multiplication. In fact, the standard algorithm, long multiplication, is just the rectangle in disguise! 4.7 Sometimes long multiplication is taught in a less compact form For example, for 4, we write =. Then, we look at 4 but note that it is really 4, so we write the answer,. Then, we look at but note that it is really and write 9 and so on. This version of long multiplication is the same as the rectangle, listing all the pieces and adding them. It just doesn t draw the rectangle. The compact version of the long multiplication is designed to save space. It writes all the intermediate computations on top of one another. is, which we write is really 4, which equals. That s tens, which we write, and, which we ll delay writing until we work with the hundreds. We finish off the first row by doing but note that this is really, so the answer is 9, but we still have the we held off on writing down. So, we really have hundreds. Let s write in that same row, for hundreds.

5 LECTURE 4 VISUALIZING EXTRAORDINARY WAYS TO MULTIPLY Next, we do the same thing again: is really. So, let s put in a to capture the idea that we re now working with tens. is 4, that s 4 tens, so we write 4 in the tens place. 4 is, so we write in the hundreds place. is, so we write in the thousands place. Now add all the pieces and get 7, again, as the final answer. It is so much easier to just do the rectangle! USING THE RECTANGLE METHOD The rectangle method is rich with understanding. We can use it to explain an age-old question: Why, in mathematics, is negative times negative computed as positive? To get there, let s check all of the other possibilities first. We re okay with positive times positive. For example, is usually interpreted as groups of ( + ), and that makes positive. We re okay with positive times negative: ( ) is groups of ( + ), which would be. But things are a bit shaky with negative times positive: for example, ( ). The idea of negative groups of doesn t really make sense. But there is a way out of this pickle. We like to believe that multiplication is commutative that is, that you can switch order in a multiplication problem and get the same answer in the end. And most people think that this should be true of all numbers, including negative ones. So, we can switch the order of the and think of it as. And that is something we can interpret: groups of ( + + ) makes.

6 4 THE POWER OF MATHEMATICAL VISUALIZATION So, positive times positive is positive; positive times negative is negative; and negative times positive is negative. Now comes the big question: What is negative times negative? What is, for example? Switching the order doesn t help: is just as confusing as. So, we need something more. Most people like to believe that all of the usual rules of arithmetic should hold for all types of numbers, even negative ones This means that the arithmetic that is encoded in our area model should hold for negative numbers, as well. Even though geometry doesn t allow objects to have negative lengths, the arithmetic that pictures with negative side lengths represent should still be valid. 5 Here s our picture for 7. The answer is. 7 = = But let s be quirky and change how we think of 7. Instead of thinking of it as + 7, think of it as As a piece of geometry, this is strange. But look at the arithmetic. We have =. 4 We have =. We have, which we can do; negative times positive is negative, so the answer is. And we have, giving 4. 7 = + 4 = We get + 4, and that is again, just as it should be. The strange geometry picture is still representing correct arithmetic.

7 LECTURE 4 VISUALIZING EXTRAORDINARY WAYS TO MULTIPLY Instead, we can be quirky with the number and think of it as +. 4 We still get. 4 = ; 7 = 4; = ; 7 = = = These pieces again add to. Let s be really quirky and write 7 as and as simultaneously. What do we get? 4 = 4; = 4; =. 4? Now we see that we have a problem: We need to work out or, negative times negative. 7 = 4 4 +? = ? 7 = 4 4 +? = must be Just prior to this we worked out 7 in different ways and got the answer every time. If we want mathematics to be consistent, then we should get the answer in this fourth example, too. In this case, we see that we have no choice but to declare to be positive. So, if we choose to believe that all numbers, including negative ones, obey the standard rules of arithmetic, then for consistent mathematics, we have no choice but to set negative times negative to be positive.

8 4 THE POWER OF MATHEMATICAL VISUALIZATION FURTHER EXPLORATION WEB Tanton, A Cute Finger ( and Toe! ) Multiplication Trick. Line Multiplication. Multiplication without Multiplying. ( Yet another mysterious multiplication method. ) Why Is Negative Times Negative Positive? READING Tanton, Mathematics Galore!, Thinking Mathematics! Vol.. PROBLEMS. In some curricula, students are taught the lattice method for performing long multiplication = 99 7 To multiply 4 and 7, for example, draw a -by- grid of squares and write the digits of the first number at the head of each column and the digits of the second number at the end of each row. Divide each cell of the grid with a diagonal line and write the product of the column digit and row digit of that cell as a -digit answer that is placed on either side of that diagonal. Add the digits in each diagonal and conduct any necessary carries. a Compute 5 via the lattice method. b Why does this lattice method work?

9 LECTURE 4 VISUALIZING EXTRAORDINARY WAYS TO MULTIPLY 4. Neptunians have hands, which each have 4 fingers. To compute products up to times, they use the following finger trick: A closed fist represents 4. To represent a number between 4 and, raise finger for each count to bring the number 4 up to the desired value. ( Thus, is represented as 4 +, which is fingers raised on a closed fist, and 7 is represented as 4 +, which is fingers raised on a closed fist. ) To compute a product of numbers, represent each number on a hand. Count the total number of raised digits, give each the value, and compute that value of that many s. Multiply the unraised digits on each hand and add this product to the count of s. This value is the answer to the original multiplication problem. For example, in computing 7, there are 5 raised digits in total, making 5 s, or 4. There are and unraised digits in each hand, respectively. Adding = to 4 gives 4. This is indeed the answer to 7. Why does this method work?. Vedic mathematics, established in 9 by Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaja, has students compute the product of two -digit numbers as follows: 4. What do you think this sequence of diagrams means?

10 44 THE POWER OF MATHEMATICAL VISUALIZATION SOLUTIONS. a Use a -by- table this time. b The lattice method is the area method in disguise. Look at the example given in the question computed via both methods side by side as in figure =, = 99 We see that the diagonal lines serve to separate tens from ones, hundreds from tens, thousands from hundreds, and so on, within each cell and therefore organize powers of in a diagonal fashion. This accounting system allows you to let go of drawing all the zeros for the powers of ( whereas the area model keeps track of the powers of by writing all the zeros ).

11 LECTURE 4 VISUALIZING EXTRAORDINARY WAYS TO MULTIPLY 45. Suppose that a digits are raised on hand ( to represent the number 4 + a ) and b on the other ( to represent the number 4 + b ). Then, there are a + b raised digits in all, with 4 a unraised digits on the first hand and 4 b unraised on the second. We claim that the product ( 4 + a ) ( 4 + b ) can be computed as ( a + b ) + ( 4 a ) ( 4 b ). Let s check. ( 4 + a ) ( 4 + b ) = + 4a + 4b + ab ( a + b ) + ( 4 a ) ( 4 b ) = a + b + 4a 4b + ab = 4a + 4b + +ab These are indeed the same! With n fingers on each hand, it is true that ( n + a ) ( n + b ) = n ( a + b ) + ( n a ) ( n b ). Thinking Further: Mercurians have hands, with 5 fingers and with 7 fingers. Is there a ( nonsymmetrical ) multiplication trick they can use?. This is a visual mnemonic for long multiplication. The first diagram says to multiply the units; the second diagram says to multiply units and tens to get answers in tens; the third diagram says to compute all the products that give hundreds; the fourth diagram says to compute all the products that give thousands; and the fifth diagram says to compute all the products that give ten thousands.