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1.1. INTRODUCTION11In the last example, we were originally frustrated in our attempt toborrow from the ten’s place since there was a zero there, so we had to firstborrow from the hundred’s place.MultiplicationIn multiplication, the two numbers being multiplied are called the factors, and the result is called the product. You may have seen many differentnotations for multiplication: , ·, , (). The “*” is mostly used in computerscience, and we will not use it in this book. The “ ” sign is the one youprobably learned originally, and it is the sign most often used for multiplication in applications (like on your calculator), but there is a problem withit. In an algebra course like this, we will eventually be using the letter x(variable), and the two are easily confused. Therefore, we will mostly usethe “·” notation for multiplication. The one exception will be when we usethe tower method to multiply without a calculator. Here, we will use the“ ” sign, as a sloppy “·” could be mistaken for a decimal point. Parentheses also show multiplication, so that whenever the left parenthesis “(” ispreceded by a number or a right parenthesis “)”, this means multiply.Example 7. Multiply:(4)(3) 122(7) 14.This idea may be extended to any grouping symbol (we will see brackets“[. . .]” and absolute value “ . . . ” later). One curious fact about multiplication is that it is sometimes hidden. You will see several cases in the CoreMath courses where different expressions are right next to each other withno obvious operation indicated, and you will need to understand that youare supposed to multiply. You could treat parentheses like this if it helps.So the last example could have been written:Example 8. Multiply:(4)(3) (4) · (3) 4 · 3 122(7) 2 · (7) 2 · 7 14.Multiplication is defined as repeated addition. Therefore, multiplying

12CHAPTER 1. REAL NUMBERS AND THEIR OPERATIONSfour times three is the same as adding three to itself four times:Example 9. The following are equal:4 · 3 3 3 3 3 12.Multiplication is commutative just like addition, so that when you aremultiplying two numbers, you can multiply in either order:Example 10. The following are equal:4 · 3 3 3 3 3 12 4 4 4 3 · 4.Even though you could do any old-fashioned multiplication problemthrough repeated addition, this would get tedious. Instead, you shouldknow your multiplication table for multiplying any single-digit whole number times any other single-digit whole number. This is probably alreadythe case, but just in case you have forgotten (or are helpless without yourcalculator.), we will include it next.Basic Multiplication 16243240485664729091827364554637281The first test will be much easier if you take the time to memorize thistable as soon as possible. For multiplying larger numbers without a calculator, we will use a tower method. To refresh your memory on how this works,let us look at an example.

1.1. INTRODUCTION13Example 11. Multiply 152 · 8.Solution 11. Scratch work:4 11 5 2 81 2 1 6So 152 · 8 1216.Notice that when we multiplied the 8 times the 2, we got 16, wrote downthe 6 and carried the 1 to the next column. Then, when we multiplied the8 times the 5 and got 40, we added the 1 to get 41, and etc. If the bottomnumber has two-digits, first multiply the one’s digit by the entire top numberas in the last example. Then cancel or erase all of your previous carries,put a zero below the rightmost digit of your previous product (well, thatmight be old-school; some people just leave this spot blank), multiply theten’s digit by the entire top number, and add the two products together.For example:Example 12. Multiply 327 · 47.Solution 12. Scratch work:1 43 2 7 4 72 2 8 9 So 327 · 47 15, 369.3 2 7 4 72 2 8 90 1 23 2 7 4 72 2 8 91 3 0 8 01 5 3 6 9

14CHAPTER 1. REAL NUMBERS AND THEIR OPERATIONSThe zero we added in prior to multiplying the second time accountsfor the fact that we were multiplying by 40 rather than 4. If the bottomnumber has three digits, start as in the last example, but do not add thetwo products together. Instead, cancel or erase all of your previous carries,put a zero below the rightmost digit AND the ten’s digit of your previousproduct, multiply the hundred’s digit of the bottom number by the entiretop number, and add the three products together. For example:Example 13. Multiply 85 · 324.Solution 13. Scratch work: 28 53 2 43 4 0 18 3 23 41 7 05400 22331 75 57 51824004540000So 85 · 324 27, 540.Recall that multiplication is commutative, so some people may prefer toalways put the number with the least digits on bottom. In that case, thelast example becomes:Example 14. Multiply 85 · 324.Solution 14. 85 · 324 324 · 85Scratch work:1 23 2 4 8 51 6 2 0So 85 · 324 27, 540. 1 33 2 4 8 51 6 2 02 5 9 2 02 7 5 4 0

1.1. INTRODUCTION15DivisionDivision is represented by the symbols “ ” or “/”. The “/” is used mostoften in computer science, and we will not be using it in this book. We willsee later that the fraction bar also shows division.Just as subtraction is the inverse operation of addition, division is theinverse operation of multiplication. Thus, 15 5 3 because 3 · 5 15.This means that multiplication and division should be able to undo eachother, but here we must be careful, as this isn’t quite as straight-forward asit was with addition-subtraction. For example, while it is still true that ifyou multiply a number by 2, you could then get back the original numberby then dividing by 2:as 8 · 2 16 and 16 2 8.You must be more careful when 0 is involved. Let’s examine what happens when you are dividing into 0 (0 is in front of sign), when you aredividing by 0 (0 is after sign), and when you are doing both.Example 15. Find 0 8 by treating division as the inverse operation ofmultiplication.Solution 15. 0 8 · 8 0.What number can be placed in the blank to make this true? Well,0 · 8 0 (and this is the only number which could be placed inthe blank to make a true statement), so:0 8 0.This is only one example, but we will spoil the suspense and go aheadand state this more generally.Let a be any number other than 0, then: 0 a 0.This means that division into 0 (except by itself), is as trivially easy asmultiplication by 0 or 1 or division by 1. What about division by 0?

16CHAPTER 1. REAL NUMBERS AND THEIR OPERATIONSExample 16. Find 8 0 by treating division as the inverse operation ofmultiplication.Solution 16. 8 0 · 0 8.What number can be placed in the blank to make this true? Theanswer is no number, as any number times 0 is 0, not 8. Wewill call division by 0, undefined.Therefore, 8 0 undefined.Thus, division into 0 makes the division very easy; division by zero makesthe division impossible. What happens when these two rules collide?Example 17. Find 0 0 by treating division as the inverse operation ofmultiplication.Solution 17. 0 0 · 0 0.What number can be placed in the blank to make this true? Theanswer, of course, is 0! Or 1, .er, or 2, or 16, or 137,. Whilethis division is no longer impossible, it is actually too possible!Instead of getting one nice answer like we would desire whendividing two numbers, we get way too many options. Therefore,we will label this as undefined as well, but not for the same reasonas before.So, 0 0 undefined.Together, these last two examples illustrate another important point.For any number a: a 0 undefined.To do old-fashioned division of large numbers without a calculator, youneed to remember how to do long division.

1.1. INTRODUCTION17Example 18. Divide 74, 635 23.Solution 18.324523 7463569564610392115115023 goes into 74 three times23 · 3 69Subtract then bring down the 6 from the 7463523 goes into 56 two times; 23 · 2 46Subtract then bring down the 3 from the 7463523 goes into 103 four times; 23 · 4 92Subtract then bring down the 5 from the 7463523 goes into 115 five times; 23 · 5 115Subtracting there is a remainder of zero.Thus, 74, 635 23 3, 245.Don’t forget that whenever you place a number on top of the divisionbar, you immediately multiply this number by the divisor (number in front).At this point, you may check to make sure you put the proper number uptop. For example, had we thought that 23 went into 74 four times at thebeginning of the last example, we would have multiplied 23·4 92, and thenwe would have noticed that 92 is bigger than 74. That is a clear indicationthat the number you are trying is too big. On the other hand, had wethought 23 went into 74 two times, we would have multiplied 23 · 2 46and subtracted 74 46 28. This difference is larger than the divisor (the23), so we know that we are trying too small a number. Occasionally, afterbringing down the next number, the new number is still too small for thedivisor to divide into. This is fine, it just means that we need to enter a 0on top. To illustrate this, let us do a few more examples.

18CHAPTER 1. REAL NUMBERS AND THEIR OPERATIONSExample 19. Divide 8, 736 42.Solution 19.20842 873684330336336042 goes into 87 two times42 · 2 84Subtract then bring down the 3 from the 873642 goes into 33 zero times; 42 · 0 0Subtract then bring down the 6 from the 873642 goes into 336 eight times; 42 · 8 336Subtracting there is a remainder of zero.Thus, 8, 736 42 208.Example 20. Divide 68, 068 34.Solution 20.200234 68068680000606868034 goes into 68 two times34 · 2 68Subtract then bring down the 0 from the 6806834 goes into 0 zero times; 34 · 0 0Subtract then bring down the 6 from the 6806834 goes into 6 zero times; 34 · 0 0Subtract then bring down the 8 from the 6806834 goes into 68 two times; 34 · 2 68Subtracting there is a remainder of zero.Thus, 68, 068 34 2, 002.

1.2. INTEGERS, ABSOLUTE VALUES AND OPPOSITES1.219Integers, Absolute Values and OppositesNegative Integers - what are they?Many people dislike negative numbers, because they believe (mistakenly)that there is no good physical meaning for them. The idea being thatwhile you can picture what three pennies represent, and what five penniesrepresent, how do you picture negative two pennies? It doesn’t seem tohelp if we tell you that three minus five is equal to negative two, because ifyou have three pennies and try to take away five of them, nothing obvioussprings to mind. This just means, however, that we will have to develop anew way of doing subtraction (next section), and a new way to think aboutnumbers rather than just a collection of objects.One way in which negative numbers may arise is when you have assigneda zero, and it is possible to go below this number as in the following examples.Example 1. In America, we mostly use the Fahrenheit scale to measuretemperature, and we have thermometers which can measure the changes.Winter in Ohio can get very cold, so what do we do if the temperature is0 F , and then it gets colder? Wouldn’t it be confusing if we kept calling thetemperature 0 F just because we don’t feel as though we can take anymoreaway?Example 2. Elevation is measured in height above sea level. This wouldseem to make sense because any land next to the ocean which is lower wouldbe underwater (and hence not land). Inland, though, the elevation coulddrop lower (e.g. Death Valley, California).In both of these examples, we could have defined our scales to avoidnegative numbers, but then typical values would have become unwieldy. Forexample, there is a Rankine scale where a change of one degree Rankine is thesame as one degree Fahrenheit, yet whose zero is defined to be absolute zero(the coldest temperature possible). Some typical values you may encounter:water freezes at 492 R, water boils at 672 R, and a pleasant summer daywould be 540 R ( 80 F ). As for elevation, we could measure distance fromthe center of the Earth, but since the radius of the Earth is approximately4000 miles, the elevation of sea level would then become approximately21,000,000 feet!

20CHAPTER 1. REAL NUMBERS AND THEIR OPERATIONSNegative numbers can have a more physical meaning. Often there maybe other, equivalent, terminology used instead, but the idea is still there.Example 3. A company is keeping track of how much profit it makes eachmonth. One month, due to a large number of renovations being made, itsrevenue (amount of money taken in) is 2000 less than its expenditures.Therefore the company’s profit is negative 2000, or 2000 in the red.Example 4. You own a store which sells Item X. At the end of the month,you restock your shelves, and you like to order enough to have 50 Item Xin stock at the beginning of the next month. One month there is an unusualdemand for Item X, and 73 customers request one. Selling to the first fiftyis no problem, but what do you do then? You could just explain to thosecustomers that you are sold out, and to try back again next month after youget another shipment, but you might lose their business. Instead, you couldtake their order and either have Item X shipped directly to them or promiseto set aside their order and have them called when it arrives. If you choosethis second method, you are choosing to use negative numbers. After all, youcertainly do not want to order just 50 more Item X’s - you have essentiallyalready sold 23 of them, and even a more normal demand month could runyou out of your product. Rather, you should think of your inventory asnegative 23 so that when you reorder, you will order 23 for those alreadydemanded, and then another 50 so that you begin the next month as usual.It was in commerce, like this last example, where negative numbers firstappeared. Notice that in neither of these last two examples would redefiningzero make any sense - i.e. you would not want to define losing one milliondollars to be “ 0 profit”.Another physical meaning for negative numbers occur whenever you havetwo characteristics which cancel each other when combined. The most common example of this is electric charge.Example 5. A sodium ion (N a ) with a charge of positive one will forman ionic bond with a chlorine ion (Cl ) with a charge of negative one. Theresulting sodium chloride molecule (salt) will be neutral (charge equal tozero).

1.2. INTEGERS, ABSOLUTE VALUES AND OPPOSITES21Other examples of this type of characteristic in nature would be matterversus antimatter, and up versus down spin in elementary particles. It isuseful to remember that when you combine (i.e. add) a positive one with anegative one, they make zero.Finally, keep in mind that negative numbers may just mean that animplication or direction is the wrong way. This was the case in the previousexample involving the company with the negative 2000 profit. For thatmonth, the company did not make 2000, rather they had to pay it out.The following example also illustrates this point.Example 6. Slick and the Kid are playing poker. The Kid consistently losesso they are keeping track of how much he owes Slick every night, intendingto settle up at the end of the week. After one night, he owes Slick 40; aftertwo nights, 60; etc. After the sixth night, the Kid owes Slick 270. On theseventh and final night, Lady Luck smiles on the Kid, and he wins 300.Slick, getting ready to settle up, stacks 270 worth of chips on the table.“Well, Kid,” Slick says, “this represents the amount of money you owed megoing into tonight. Let me settle up with you by removing your winningsfrom this debt.” After he counts out all 270, Slick frowns. “Hmmm, theredoesn’t seem to be any more to take away, so I guess we are even.” The Kid,who only knows old-fashioned subtraction, nods and says “Yup.”, therebyensuring that he would always find people willing to play him in poker.Had that have been you instead of the Kid, you would surely have noticedthat Slick should still owe you 30. This goes to show that you already knowhow to do fancier subtraction than just the old-fashioned take-away kind!By the way, while you might have said that Slick owes the Kid 30, youcould also say that the Kid owes Slick negative 30. The negative herewould indicate that the implication (in this case, who owes whom) is thewrong way around.Negative Integers - notationTo represent a negative, we will use the “ ” sign. That is right, thesame sign we use for subtraction. Similarly, if we wanted to emphasize thata number is positive, we could place a “ ” before it, such as 8 for positiveeight, but the “ ” sign is traditionally left off. Therefore, in some of thefuture problems, we may temporarily put a “ ” sign to show positive in ourscratch work, but any final answer which is positive will not have the “ ”.

22CHAPTER 1. REAL NUMBERS AND THEIR OPERATIONSNow that we have negative integers, we can define the set of all integers:integers {. . . , 3, 2, 1, 0, 1, 2, 3, . . .}.The periods of ellipsis at the beginning show that there are an infinitenumber of numbers before the negative three. Sometimes people like toillustrate the set of integers by making a number line.¾ 12 10 8 6 4 2024681012Note that we only labeled every other tick mark, but this was done onlyfor space considerations - you could label every one if you would like. Alsonote that the number line does not stop at -13 and 13 but continues on inboth directions as shown by the arrows on the ends. The spacing betweenthe numbers should be uniform - although the units may be feet, inches,centimeters, miles, whatever. We will refer to the spacing as steps, markedoff by someone with a uniform stride length, but again any unit may besubstituted.The number line may be thought of as a straight railroad track whichruns east - west, with you standing at 0. We could give you directions to anypoint on the track by telling you how many steps to take to the east or howmany to take to the west, but the property of a negative sign of implyingthe “other” direction may be exploited. Instead of having two directions(east versus west), we only need one (in this case just east), and a negativenumber means go the other way. Thus, with you standing at 0, the number5 would be five steps to the east, while the number 7 is seven steps to thewest.Less than, Greater than, Less than or equal to, Greater than or equal toWhenever you are working with a set of numbers, it may be necessary tocompare two numbers to see which is the bigger or the smaller. With whole

1.2. INTEGERS, AB

Welcome to Core Mathematics I (Math 10021) or Core Mathematics I and II (Math 10006)! Beginnings are always di–cult, as you and your fellow students may have a wide range of backgrounds in previous math courses, and may difier gre