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ARITHMETIC: A Textbook for Math

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ARITHMETIC: A Textbook for Math 01 3rd edition (2012) Anthony Weaver Department of Mathematics and Computer Science Bronx Community CollegeChapter 1 Whole Numbers The natural numbers are the counting numbers: 1,2,3,4,5,6,... . The dots indicate that the sequence is infinite – counting can go on forever, since you can always get the next number by simply adding 1 to the previous number. In order to write numbers efficiently, and for other reasons, we also need the number 0. Later on, we will need the sequence of negative numbers−1,−2,−3,−4,−5,−6,.... Taken together, all these numbers are called the integers. It helps to visualize the integers laid out on a number line, with 0 in the middle, and the natural numbers increasing to the right. There are numbers between any two integers on the number line. In fact, every location on the line represents some number. Some locations represent fractions such as one-half (between 0 and 1) or four-thirds (between 1 and 2). Other locations represent numbers which cannot be expressed as fractions, such as π. (π is located between 3 and 4 and expresses the ratio of the circumference to the diameter of any circle.) −5 −4 −3 −2 −1 0 1 2 3 4 5 For now, we concentrate on the non-negative integers (including 0), which we call whole numbers. We need only ten symbols to write any whole number. These symbols are the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. We write larger whole numbers using a place-value system. The digit in the right-most place indicates how many ones the number contains, the digit in the second-from-right place indicates how many tens the number contains, the digit in the third-from-right place indicates how many hundreds the number contains, etc. Example 1. 7 stands for 7 ones 72 stands for 7 tens + 2 ones 349 stands for 3 hundreds + 4 tens + 9 ones 6040 stands for 6 thousands + 0 hundreds + 4 tens + 0 ones Notice that when you move left, the place value increases ten-fold. So if a number has five digits, the fifth-from-right place indicates how many ten-thousands the number contains. (Ten-thousand is ten times a thousand.) 91.0.1 Exercises 1. 35 stands for 2. 209 stands for 3. 9532 stands for 4. 21045 stands for 1.1 Adding Whole Numbers When we add two or more integers, the result is called the sum. We assume you know the sums of single-digit numbers. For practice, do the following example. Example 2. Fill in the missing squares in the digit-addition table below. For example, the number in the row labelled 3 and the column labelled 4 is the sum 3 + 4 = 7. + 0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 5 6 7 8 9 1 2 3 7 4 5 6 7 12 8 9 Table 1.1: The digit addition table 1. Do you notice any patterns or regularities in the digit-addition table? Can you explain them? 2. Why is the second line from the top identical to the top line? 3. What can you say about the left-most column and the second-from-left column? 1.1.1 Commutativity, Associativity, Identity The first question leads us to an important property of addition, namely, that for any two numbers x and y, x + y = y + x. Page 10In other words, the order in which two numbers are added does not effect the sum. This property of addition is called commutativity. The last two questions lead us to an important property of 0, namely, for any number x, x + 0 = x = 0 + x. In other words, when 0 is added to any number, x, you get the identical number, x, again. Because of this property, 0 is called the additive identity. One final property of addition is expressed in the following equation (x + y) + z = x + (y + z), which says that if three numbers are added, it doesn’t matter how you “associate” the additions: you can add the first two numbers first, and then add the third to that, or, you could add the second two numbers first, and then add the first to that. This property of addition is called associativity. Example 3. Find the sum of 2, 3, and 5 by associating in two different ways. Solution. Associating 2 and 3, we calculate (2 + 3) + 5 = 5 + 5 = 10. On the other hand, associating 3 and 5, we calculate 2 + (3 + 5) = 2 + 8 = 10. The sum is the same in both cases. 1.1.2 Multi-digit addition To add numbers with more than one digit, we line up the numbers vertically so that the ones places (right-most) are directly on top of each other, and all other places are similarly arranged. Then we add the digits in each place to obtain the sum. Example 4. Find the sum of 25 and 134. Solution. We line up the numbers vertically so that the 5 in the ones place of 25 is over the 4 in the ones place of 134. If we do this carefully, all the other places line up vertically, too. So there is a “ones” column, a “tens” column to the left of it, and a “hundreds” column to the left of that: 2 5 1 3 4 Then we draw a line and add the digits in each column to get the sum: 2 5 + 1 3 4 1 5 9 Page 11Sometimes we need to “carry” a digit from one place to the next higher place. For example, when adding 38 + 47, we first add the ones places, 8 + 7 = 15. But 15 has two digits: it stands for 1 ten + 5 ones. We “put down” the 5 in the ones place, and “carry” the 1 (standing for a ten) to the tens column. So we have: Example 5. Find the sum 3 8 + 4 7 Solution. Put down the 5 in the ones place: 3 8 + 4 7 5 and carry the 1 to the top of the tens column: 1 3 8 + 4 7 5 and finish the job by adding up the tens column, including the carried one: 1 3 8 + 4 7 8 5 The sum is 85. Since carrying alters the sum in the column immediately to the left of the one we are working on, in multi-digit addition, we always start with the right-most column and proceed leftward. 1.1.3 Exercises Find the sums, carrying where necessary. 1. 2 6 + 5 5 Page 122. 3 8 3 + 4 7 3. 3 2 4 5 + 6 4 4. 1 2 9 3 7 7 + 4 5 0 3 5. 9 0 9 7 7 7 + 6 9 6 4 6. 3 2 0 9 8 4 9 0 2 + 4 5 0 3 Page 137. 9 9 9 9 9 0 2 + 9 5 0 2 8. 5 6 3 2 0 9 8 6 4 9 0 4 + 6 5 0 3 9. 3 2 0 0 0 9 8 4 4 9 0 2 + 4 5 0 3 10. 2 9 9 7 9 8 4 4 2 0 5 + 5 4 9 0 8 1.2 Subtracting Whole Numbers Another way to say that 5 + 2 = 7 is to say that 5 = 7− 2. In words, “5 is the difference of 7 and 2,” or “5 is the result of taking away 2 from 7.” The operation of taking away one number from another, or finding their difference, is called subtraction. For now, we have to be careful that the number we take away is no larger than the number we start with: we cannot have 3 marbles and take 7 of them away Later on, when we introduce negative numbers, we won’t have to worry about this. We assume you remember the differences of single-digit numbers. Just to make sure, do the following example. Example 6. Fill in the missing squares in the digit-subtraction table below. Here’s a start: the number in the row labelled 7 and the column labelled 2 is the difference 7−2 = 5. The digit in the row labelled 3 and the column labelled 3 is the difference 3−3 = 0. (Squares that have an asterisk (∗) will be filled in later with negative numbers.) Page 14− 0 1 2 3 4 5 6 7 8 9 0 1 2 3 0 4 5 6 7 5 8 9 4 Table 1.2: The digit subtraction table 1.2.1 Commutativity, Associativity, Identity When we study negative numbers, we will see that subtraction is not commutative. We can see by a simple example that subtraction is also not associative. Example 7. Verify that (7− 4)− 2 is not equal to 7− (4− 2). Solution. Associating the 7 and 4, we get (7− 4)− 2 = 3− 2 = 1, but associating the 4 and the 2, we get 7− (4− 2) = 7− 2 = 5, a different answer. Until we establish an order of operations, we will avoid examples like this It is true that x− 0 = x for any number x. However, 0 is not an identity for subtraction, since 0− x is not equal to 0 (unless x = 0). To make sense of 0− x, we will need negative numbers. 1.2.2 Multi-digit subtractions To perform subtractions of multi-digit numbers, we need to distinguish the number “being diminished” from the number which is “doing the diminishing” (being taken away). The latter number is called the subtrahend, and the former, the minuend. For now, we take care that the subtrahend is no larger than the minuend. To set up the subtraction, we line the numbers up vertically, with the minuend over the subtrahend, and the ones places lined up on the right. Example 8. Find the difference of 196 and 43. Solution. The subtrahend is 43 (the smaller number), so it goes on the bottom. We line up the numbers vertically so that the 6 in the ones place of 196 is over the 3 in the ones place of 43. 1 9 6 4 3 Then we draw a line and subtract the digits in each column, starting with the ones column and working right to left, to get the difference: Page 151 9 6 − 4 3 1 5 3 1.2.3 Checking Subtractions Subtraction is the “opposite” of addition, so any subtraction problem can be restated in terms of addition. Using the previous example, and adding the difference to the subtrahend, we obtain 1 5 3 + 4 3 1 9 6 which is the original minuend. In general, if subtraction has been performed correctly, adding the difference to the subtrahend returns the minuend. This gives us a good way to check subtractions. Example 9. Check whether the following subtraction is correct: 9 4 − 5 1 3 3 Solution. Adding the (supposed) difference to the subtrahend, we get 3 3 + 5 1 8 4 which is not equal to the minuend (94). Thus the subtraction is incorrect. We leave it to you to fix it 1.2.4 Borrowing Sometimes, when subtracting, we need to “borrow” a digit from a higher place and add its equivalent to a lower place. Example 10. Find the difference: 8 5 − 4 6 Solution. The digit subtraction in the ones column is not possible (we can’t take 6 from 5). Instead we remove or ”borrow” 1 ten from the tens place of the minuend, and convert it into 10 ones, which we add to the ones in the ones place of the minuend. The minuend is now represented as 715, Page 16standing for 7 tens + 15 ones. It looks funny (as if 15 were a digit), but it doesn’t change the value of the minuend, which is still 7× 10 + 15 = 85. We represent the borrowing operation like this: 715 8 5 − 4 6 Ignoring the original minuend, we have 15−6 = 9 for the ones place, and 7−3 = 4 for the tens place, as follows: 715 8 5 − 4 6 3 9 The difference is 39. We can check this by verifying that the difference + the subtrahend = the original minuend: 3 9 + 4 6 8 5 Sometimes you have to go more than one place to the left to borrow successfully. This happens when the next higher place has a 0 digit – there is nothing to borrow from. Example 11. Find the difference: 2 0 7 − 6 9 Solution. In the ones column we can’t take 9 from 7, so we need to borrow from a higher place. We can’t borrow from the tens place, because it has 0 tens. But we can borrow from the hundreds place. We borrow 1 hundred, and convert it into 9 tens and 10 ones. The minuend is now represented as 1917, standing for 1 hundred + 9 tens + 17 ones, (as if 17 were a digit). We represent the borrowing as before: 1917 2 0 7 − 6 9 Ignoring the original minuend, we have 17−9 = 8 for the ones place, 9−6 = 3 for the tens place, and 1− 0 = 1 for the hundreds place: 1917 2 0 7 − 6 9 1 3 8 Page 17The difference is 138. To check, we verify that the difference + the subtrahend = the original minuend: 1 3 8 + 6 9 2 0 7 (You may have noticed that the carrying you do in the addition check simply reverses the borrowing done in the original subtraction) 1.2.5 Exercises Find the differences, borrowing where necessary. Check that the difference + the subtrahend = the minuend. 1. 9 4 − 3 7 2. 2 7 5 − 1 8 1 3. 3 5 0 − 7 6 4. 5 0 0 − 1 9 1 5. 6 0 0 − 1 9 9 Page 186. 1 5 0 0 − 1 1 9 1 7. 5 6 7 8 − 4 5 6 7 8. 5 0 0 0 0 − 4 9 9 9 9. 8 0 1 − 7 9 0 10. 6 3 8 9 − 9 9 9 11. 5 0 0 0 0 0 − 4 3 2 1 0 12. 9 0 0 1 0 1 0 − 1 1 1 1 1 1 1 Page 191.3 Multiplying Whole Numbers Multiplication is really just repeated addition. When we say “4 times 3 equals 12,” we can think of it as starting at 0 and adding 3 four times over: 0 + 3 + 3 + 3 + 3 = 12. We can leave out the 0, since 0 is the additive identity ( 0+3 = 3). Using the symbol× for multiplication, we write 3 + 3 + 3 + 3 = 4× 3 = 12. The result of multiplying two or more numbers is called the product of the numbers. Instead of the× symbol, we often use a central dot (·) to indicate a product. Thus, for example, instead of 2× 4 = 8, we can write 2· 4 = 8. Example 12. We assume you remember the products of small whole numbers, so it should be easy for you to reproduce the partial multiplication table below. For example, the number in the row labelled 7 and the column labelled 5 is the product 7· 5 = 35. × 0 1 2 3 4 5 6 7 8 9 10 11 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 10 11 12 2 0 2 4 6 8 10 12 14 16 18 20 22 24 3 0 3 6 9 12 15 18 21 24 27 30 33 36 4 0 4 8 12 16 20 24 28 32 36 40 44 48 5 0 5 10 15 20 25 30 35 40 45 50 55 60 6 0 6 12 18 24 30 36 42 48 54 60 66 72 7 0 7 14 21 28 35 42 49 56 63 70 77 84 8 0 8 16 24 32 40 48 56 64 72 80 88 96 9 0 9 18 27 36 45 54 63 72 81 90 99 108 10 0 10 20 30 40 50 60 70 80 90 100 110 120 11 0 11 22 33 44 55 66 77 88 99 110 121 132 12 0 12 24 36 48 60 72 84 96 108 120 132 144 Table 1.3: Multiplication table 1. Do you notice any patterns or regularities in the multiplication table? Can you explain them? 2. Why does the second row from the top contain only 0’s? 3. Why is the third row from the top identical to the first row? 4. Is multiplication commutative? How can you tell from the table? Page 201.3.1 Commutativity, Associativity, Identity, the Zero Property An examination of the multiplication table leads us to an important property of 0, namely, when any number, N, is multiplied by 0, the product is 0: 0· N = N · 0 = 0. It also shows us an important property of 1, namely, when any number, N, is multiplied by 1, the product is the identical number, x, again: 1· N = N · 1 = N. For this reason, 1 is called the multiplicative identity. The following example should help you to see that multiplication is commutative. Example 13. The figure shows two ways of piling up twelve small squares. On the left, we have piled up 3 rows of 4 squares (3× 4); on the right, we have piled up 4 rows of 3 squares (4× 3). In both cases, of course, the total number of squares is the product 3× 4 = 4× 3 = 12. Figure 1.1: 3× 4 = 4× 3 = 12 Examples like the following help you to see that multiplication is associative. Example 14. We can find the product 3× 4× 5 in two different ways. We could first associate 3 and 4, getting (3× 4)× 5 = 12× 5 = 60, or we could first associate 4 and 5, getting 3× (4× 5) = 3× 20 = 60. The product is the same in both cases. 1.3.2 Multi-digit multiplications To multiply numbers when one of them has more than one digit, we need to distinguish the number “being multiplied” from the number which is “doing the multiplying.” The latter number is called the multiplier, and the former, the multiplicand. It really makes no difference which number is called the multiplier and which the multiplicand (because multiplication is commutative). But it saves space if we choose the multiplier to be the number with the fewest digits. To set up the multiplication, we line the numbers up vertically, with the multiplicand over the multiplier, and the ones places lined up on the right. Page 21Example 15. To multiply 232 by 3, we write 2 3 2 × 3 We multiply place-by-place, putting the products in the appropriate column. 3× 2 ones is 6 ones, so we put 6 in the ones place 2 3 2 × 3 6 3× 3 tens is 9 tens, so we put 9 in the tens place 2 3 2 × 3 9 6 Finally, 3× 2 hundreds is 6 hundreds, so that we put down 6 in the hundreds place, and we are done: 2 3 2 × 3 6 9 6 Sometimes we need to carry a digit from one place to the next higher place, as in addition. For example, 4× 5 = 20, a number with two digits. So we would “put down” the 0 in the current place, and “carry” the 2 to the column representing the next higher place, as in the next example. Example 16. Multiply 251 by 4. Solution. The steps are 2 5 1 × 4 For the ones place, 2 5 1 × 4 4 For the tens place, 4×5 = 20, so we put down the 0 in the tens place and carry the 2 to the hundreds column: 2 2 5 1 × 4 0 4 Page 22For the hundreds place, 4 × 2 hundreds is 8 hundreds, to which we add the 2 hundreds that were carried. This gives us 10 hundreds, or 1 thousand. We put down 0 in the hundreds place, and 1 in the thousands place. 2 2 5 1 × 4 1 0 0 4 The product of 251 and 4 is 1004. If the multiplier has more than one digit, the procedure is a little more complicated. We get partial products (one for each digit of the multiplier) which are added to yield the total product. Example 17. Consider the product 2 4 × 3 2 Since the multiplier stands for 3 tens + 2 ones, we can split the product into two partial products 2 4 × 2 4 8 and 2 4 × 3 Notice that in the second partial product the multiplier is in the tens column. This is almost exactly like having a 1-digit multiplier. The second partial product is obtained by simply putting down a 0 in the ones place and shifting the digit products one place to the left: 2 4 × 3 7 2 0 (Notice that we put down 2 and carried 1 when we performed the digit product 3×4 = 12.) The total product is the sum of the two partial products: 48 + 720 = 768. We can write the whole procedure compactly by aligning the two partial products vertically 2 4 × 3 2 4 8 7 2 0 Page 23and then performing the addition 2 4 × 3 2 4 8 + 7 2 0 7 6 8 Here’s another example. Example 18. Find the product of 29 and 135. Solution. We choose 29 as the multiplier since it has the fewest digits. 1 3 5 × 2 9 We use the 1-digit multiplier 9 to obtain the first partial product 34 1 3 5 × 2 9 1 2 1 5 Notice that we put down 5 and carried 4 to the tens place, and also put down 1 and carried 3 to the hundreds place. Next we use the 1-digit multiplier 2 (standing for 2 tens) to obtain the second partial product, shifted left by putting a 0 in the ones place 1 1 3 5 × 2 9 1 2 1 5 2 7 0 0 (What carry did we perform?) Finally, we add the partial products to obtain the (total) product 1 3 5 × 2 9 1 2 1 5 + 2 7 0 0 3 9 1 5 Note that the whole procedure is compactly recorded in the last step, which is all that you need to write down. Page 241.3.3 Exercises Find the products. 1. 1 2 2 × 4 2. 8 3 × 5 3. 1 0 4 × 7 4. 3 0 0 8 × 9 5. 2 1 2 × 4 3 6. 8 3 × 5 6 Page 257. 1 3 6 × 2 7 8. 3 0 8 × 1 0 9 9. 2 1 0 3 × 4 4 10. 8 3 7 × 5 4 1.4 Powers of Whole Numbers If we start with 1 and repeatedly multiply by 3, 4 times over, we get a number that is called the 4th power of 3, written 4 3 = 1× 3× 3× 3× 3. The factor of 1 is understood and usually omitted. Instead we simply write 4 3 = 3× 3× 3× 3. 4 In the expression 3 , 3 is called the base, and 4 the exponent (or power). 5 Example 19. The 5th power of 2, or 2 , is the product 2× 2× 2× 2× 2 = 32. 3 The 3rd power of 4, or 4 , is the product 4× 4× 4 = 64. We make the following definition in the cases where the exponent is 0. Page 26For any nonzero number N, 0 N = 1. 0 (0 is undefined.) This makes sense if you remember the harmless factor of 1 that is understood in every exponential expression. (You’ll see another justification when you study algebra.) 0 2 3 Although 0 is undefined, expressions such as 0 , 0 , etc., with 0 base and nonzero exponent, make 3 perfect sense (e.g., 0 = 0× 0× 0 = 0). 0 5 1 0 Example 20. 17 = 1. 0 = 0. 0 = 0. 0 is undefined. 1.4.1 Squares and Cubes Certain powers are so familiar that they have special names. For example, the 2nd power is called the 2 3 square and the 3rd power is called the cube. Thus 5 is read “5 squared,” and 7 is read as “7 cubed.” The source of these special names is geometric (see Section 1.8). The area of a square, x units on a 2 2 side, is x square units. This means that the square contains x small squares, each one unit on a side. 2 For example, the figure below shows a square 6 units on a side, with area 6 = 36 square units. = 3 Similarly, the volume of a cube, y units on a side, is y cubic units. This means that the cube 3 contains y little cubes, each one unit on a side. For example, the volume of an ice cube that measures 3 2 cm (centimeters) on a side is 2 = 8 cubic centimeters. 1.4.2 Exercises 1. Rewrite using an exponent: 8× 8× 8× 8 2. Rewrite using an exponent: 4· 4· 4· 4· 4· 4 5 3. Evaluate 2 0 4. Evaluate 9 7 5. Evaluate 0 4 6. Evaluate 5 2 7. Evaluate 10 Page 27