I The How,When,and Why of Mathematics 3 2."Devising a plan."A cipher text may have weak points.What are these?How about the short words?Looking at the short words,in some sense,substitutes an easier problem for the one we have. 3.“Carrying out the plan.”The short words are: W: EO(which appears twice): EP; PK. Try using the most common one-and two-letter words.For each guess,check the beginning of the cipher text to see if it makes sense.It shouldn't take long for you to come up with the message. 4."Looking back."If your solution makes sense,then it is highly unlikely that a different replacement is also possible.So the solution is (with high probability) the only one. Would there have been other solution methods?Sure.For instance,not all letters have the same frequency in the English language.One analysis of English texts showed the letter e occurring most frequently,followed by (in this order)t,a,o, i,n,s,h,and r.(See [99,p.19].)We could have used this information to guess the assignment of letters. We also could have simply tried one value of n after another until the message made sense. Have you now solved the problem?If you know what the message says,then the answer to this question is yes.Are you done?Unless you solved the problem and wrote up a clear,complete solution,the answer to this second question is no. A solution consists of a report that tells the reader how you solved the problem and what the answer is.This needs to be done in clear English sentences.As you write up your solution,try to keep the reader in mind.You should explain things clearly and logically,so that the reader doesn't have to spend time filling in gaps. ○ We now move on to a very different kind of example.Consider the set of points in three-space.In case you haven't seen this before,these points are easily described. We take the familiar xy-plane,and place it parallel to the floor.The z-axis is the vertical line perpendicular to the xy-plane and passing through the origin of the xy- plane (see Figure 1.1).We'll review the important concepts before we begin our example. To locate a point,we will give three coordinates.The first coordinate is the x- coordinate and tells us the number of units to walk in the x-direction.The second is the y-coordinate,telling us how to move in the y-direction,and the third is the z-coordinate,telling us how far,up or down,to move.So a point in three-space is denoted by (x,y,)It is important to make sure you understand this.Try to think of how you would plot points.The point (1,0,0)(plotted in Figure 1.2)would ap- pear one unit in the positive direction on the x-axis (since it doesn't move in the y-direction or z-direction at all).The point (-1,1,0)would appear in the xy-plane
1 The How, When, and Why of Mathematics 3 2. “Devising a plan.” A cipher text may have weak points. What are these? How about the short words? Looking at the short words, in some sense, substitutes an easier problem for the one we have. 3. “Carrying out the plan.” The short words are: W; EO (which appears twice); EP; PK. Try using the most common one- and two-letter words. For each guess, check the beginning of the cipher text to see if it makes sense. It shouldn’t take long for you to come up with the message. 4. “Looking back.” If your solution makes sense, then it is highly unlikely that a different replacement is also possible. So the solution is (with high probability) the only one. Would there have been other solution methods? Sure. For instance, not all letters have the same frequency in the English language. One analysis of English texts showed the letter e occurring most frequently, followed by (in this order) t, a, o, i, n, s, h, and r. (See [99, p. 19].) We could have used this information to guess the assignment of letters. We also could have simply tried one value of n after another until the message made sense. Have you now solved the problem? If you know what the message says, then the answer to this question is yes. Are you done? Unless you solved the problem and wrote up a clear, complete solution, the answer to this second question is no. A solution consists of a report that tells the reader how you solved the problem and what the answer is. This needs to be done in clear English sentences. As you write up your solution, try to keep the reader in mind. You should explain things clearly and logically, so that the reader doesn’t have to spend time filling in gaps. We now move on to a very different kind of example. Consider the set of points in three-space. In case you haven’t seen this before, these points are easily described. We take the familiar xy-plane, and place it parallel to the floor. The z-axis is the vertical line perpendicular to the xy-plane and passing through the origin of the xyplane (see Figure 1.1). We’ll review the important concepts before we begin our example. To locate a point, we will give three coordinates. The first coordinate is the xcoordinate and tells us the number of units to walk in the x-direction. The second is the y-coordinate, telling us how to move in the y-direction, and the third is the z-coordinate, telling us how far, up or down, to move. So a point in three-space is denoted by (x, y,z). It is important to make sure you understand this. Try to think of how you would plot points. The point (1,0,0) (plotted in Figure 1.2) would appear one unit in the positive direction on the x-axis (since it doesn’t move in the y-direction or z-direction at all). The point (−1,1,0) would appear in the xy-plane
4 I The How,When,and Why of Mathematics Fg.1.1 one unit back on the x-axis and one unit in the positive y-direction.Finally the point (2,-1,3)is plotted in Figure 1.2. Let's go a bit further here.In two-space,what wasx=0?Since y does not appear in that equation,it is unrestricted and can be any real number.That's why x=0 in two-space is the y-axis.What is x=3?It is a line parallel to the y-axis through the point(3,0).So,let's try to generalize this to the situation in three-space.What's the planez=0?Recall that if a variable doesn't appear,then it may assume any value. So this means that z is fixed at 0 while x can take any value,as can y.Thus,the plane =0 is the xy-plane.Similarly,the yz-plane is the planex=0 and the xz-plane is the plane y =0.These three planes are called the coordinate planes.What's the plane z=3?x=2?y=yo?There's plenty to think about here,but let's start by asking what the distance is between two points in three-space. Example 1.2.Given two points (xo,yo,zo)and (x1,y1,1)in three-space,what is the distance between the two points? (2,-1,3) (-1,1,0) 1,0,0) Fig.1.2
4 1 The How, When, and Why of Mathematics x y z Fig. 1.1 one unit back on the x-axis and one unit in the positive y-direction. Finally the point (2,−1,3) is plotted in Figure 1.2. Let’s go a bit further here. In two-space, what was x = 0? Since y does not appear in that equation, it is unrestricted and can be any real number. That’s why x = 0 in two-space is the y-axis. What is x = 3? It is a line parallel to the y-axis through the point (3,0). So, let’s try to generalize this to the situation in three-space. What’s the plane z = 0? Recall that if a variable doesn’t appear, then it may assume any value. So this means that z is fixed at 0 while x can take any value, as can y. Thus, the plane z = 0 is the xy-plane. Similarly, the yz-plane is the plane x = 0 and the xz-plane is the plane y = 0. These three planes are called the coordinate planes. What’s the plane z = 3? x = 2? y = y0? There’s plenty to think about here, but let’s start by asking what the distance is between two points in three-space. Example 1.2. Given two points (x0,y0,z0) and (x1,y1,z1) in three-space, what is the distance between the two points? (1,0,0) (-1,1,0) (2,-1,3) x y z Fig. 1.2
I The How,When,and Why of Mathematics 5 (X1,y1,21p （X1,y1,z0） (xo,Yo,Zo) Fig.1.3 We follow Polya's method to find the solution. 1."Understanding the problem."Before we begin,we make sure we really under- stand the meaning of each word and symbol above.We spent the last few para- graphs making sure we all understand the symbols,and all the words are familiar ones that appear in a standard English dictionary.But,wait-has"distance be- tween two points"really been defined?We need to be sure that everyone means the same thing by this.The distance between these two arbitrary points would mean the length of the straight line segment joining the two points.That's what we need to find.What were we given?Two points and their coordinates. 2."Devising a plan."How do we solve something like this?We haven't covered anything yet,so what can the authors be thinking?If you have no idea how to get started,try thinking about finding the distance between two specific points.Of course,(and this is very important)this won't give us a general formula because it is much too specific,but maybe we'll get some ideas. So what's the distance between the two points(1,0,0)and(-1,0,0)?That ques- tion is easier to answer-it's two.What's the distance between (1,1,0)and (-1,-2,0)?This seems to be just the distance between two points in the fa- miliar xy-plane.We saw a formula for that at some point.It was obtained using the Pythagorean Theorem.What was it?If you can't recall the formula,look it up or(better,yet)try to derive it again. Our reasoning now brings us to a simpler,similar question.As you recall,this is precisely where Polya suggested we look for a plan.So far,it seems we can find the distance between two points as long as they lie in a plane parallel to one of the coordinate planes.But in this problem,if we look at the two points,they need not lie in such a plane.We can try to insert a third point that helps us to reduce the problem to one we can already solve.Which point?A picture will help here, so we draw one in Figure 1.3. We see that (xo,yo,zo)and (xi,y1,zo)lie in the plane z=zo.while (x1,y1,) and (x1,y1,1)lie on the same vertical line,in the intersection of the two planes
1 The How, When, and Why of Mathematics 5 x y z (x 0 ,y 0 ,z 0 ) (x 1 ,y 1 ,z 0 ) (x 1 ,y 1 ,z 1 ) Fig. 1.3 We follow Polya’s method to find the solution. ´ 1. “Understanding the problem.” Before we begin, we make sure we really understand the meaning of each word and symbol above. We spent the last few paragraphs making sure we all understand the symbols, and all the words are familiar ones that appear in a standard English dictionary. But, wait—has “distance between two points” really been defined? We need to be sure that everyone means the same thing by this. The distance between these two arbitrary points would mean the length of the straight line segment joining the two points. That’s what we need to find. What were we given? Two points and their coordinates. 2. “Devising a plan.” How do we solve something like this? We haven’t covered anything yet, so what can the authors be thinking? If you have no idea how to get started, try thinking about finding the distance between two specific points. Of course, (and this is very important) this won’t give us a general formula because it is much too specific, but maybe we’ll get some ideas. So what’s the distance between the two points (1,0,0) and (−1,0,0)? That question is easier to answer—it’s two. What’s the distance between (1,1,0) and (−1,−2,0)? This seems to be just the distance between two points in the familiar xy-plane. We saw a formula for that at some point. It was obtained using the Pythagorean Theorem. What was it? If you can’t recall the formula, look it up or (better, yet) try to derive it again. Our reasoning now brings us to a simpler, similar question. As you recall, this is precisely where Polya suggested we look for a plan. So far, it seems we can find ´ the distance between two points as long as they lie in a plane parallel to one of the coordinate planes. But in this problem, if we look at the two points, they need not lie in such a plane. We can try to insert a third point that helps us to reduce the problem to one we can already solve. Which point? A picture will help here, so we draw one in Figure 1.3. We see that (x0,y0,z0) and (x1,y1,z0) lie in the plane z = z0, while (x1,y1,z0) and (x1, y1,z1) lie on the same vertical line, in the intersection of the two planes
6 I The How,When,and Why of Mathematics Fig.1.4 x=x and y=y.We "devise our plan"using these three points.Can we get the distance we are looking for from these three points?Look at Figure 1.3 and see if you can guess the rest before going on to Step 3.You probably noticed that the vertical line makes a right angle with every line in the plane z=zo.This should suggest something to you-something like the Pythagorean Theorem. 3."Carrying out the plan."This is the only thing the reader will see.Everything that preceded this was to assist us in obtaining this solution.That means the reader doesn't know what the points are;we have to tell him or her that.We should make sure we say why a sentence follows from the previous one and we should use equal signs between equal objects.When we think we are done,we should tell the reader that too. Solution.Let P=(xo,yo,zo)and =(x1,y1,z1)be two points in space.We claim that the distance between these two points,denoted by d(P,O),is d(P0)=V(o-x)2+0o-yn)2+(o-z12 Proof.We introduce a third point with coordinates R=(x1,y1,z0).Since (xo,yo,zo) and (x1,y1,zo)both lie in the plane z=zo,we can use the distance formula for two points in a plane to find the distance between them.Thus,the distance is given by d(P,R)=V(xo-x1)2+(vo-yI)2 Now look at the distance between the two points (xi,y1,zo)and (xi,y1,z1).Since these points lie on the same vertical line,the distance is given by d(R,Q)=o-z1 Now,the distance we are looking for is the length of the line segment PO,which is the hypotenuse of the right triangle POR (see Figure 1.4)
6 1 The How, When, and Why of Mathematics x y z P R Q Fig. 1.4 x = x1 and y = y1. We “devise our plan” using these three points. Can we get the distance we are looking for from these three points? Look at Figure 1.3 and see if you can guess the rest before going on to Step 3. You probably noticed that the vertical line makes a right angle with every line in the plane z = z0. This should suggest something to you—something like the Pythagorean Theorem. 3. “Carrying out the plan.” This is the only thing the reader will see. Everything that preceded this was to assist us in obtaining this solution. That means the reader doesn’t know what the points are; we have to tell him or her that. We should make sure we say why a sentence follows from the previous one and we should use equal signs between equal objects. When we think we are done, we should tell the reader that too. Solution. Let P = (x0, y0,z0) and Q = (x1,y1,z1) be two points in space. We claim that the distance between these two points, denoted by d(P,Q), is d(P,Q) = q (x0 −x1)2 + (y0 −y1)2 + (z0 −z1)2 . Proof. We introduce a third point with coordinates R = (x1,y1,z0). Since (x0,y0,z0) and (x1,y1,z0) both lie in the plane z = z0, we can use the distance formula for two points in a plane to find the distance between them. Thus, the distance is given by d(P,R) = q (x0 −x1)2 + (y0 −y1)2 . Now look at the distance between the two points (x1,y1,z0) and (x1,y1,z1). Since these points lie on the same vertical line, the distance is given by d(R,Q) = |z0 −z1| . Now, the distance we are looking for is the length of the line segment PQ, which is the hypotenuse of the right triangle PQR (see Figure 1.4)
I The How,When,and Why of Mathematics 7 This is a right triangle,so we can obtain the length using the Pythagorean Theo- rem.So,we get d(P,Q)=Vd(P.R)2+d(R.Q)2 Substituting in what we found above,we obtain d(P0)=V(o-x)2+0o-y)2+(0-z1)2. This completes the proof. 4."Looking back."What we have presented is our version of the proof.You may find that you need to include more details.By all means,go ahead.If you had to stop and say,"where did that come from?"make sure you answer yourself. Write it in the text(you aren't going to sell this book back anyway,right?),or keep a notebook of"proofs with commentary.Note that though we used pictures to illustrate the ideas in our argument,a picture will not,in general,substitute for a proof.However,it can really clarify an idea.Don't rely on a picture,but don't be afraid to use one either. 0 Solutions to Exercises Solution (1.1).We are given that this code was created through a shift of the alpha- bet.Thus once we determine one letter,the other letters are easily found.Since we have a one-letter word,we'll start with it.Thus"W"must represent the letter"I"or the letter"A."Checking both shifts of the alphabet (W→I,X→J,Y→K,Z→L,A+M,B→N,C→O,etc W→A,X→B,Y→C,Z→D,A→E,B→F,C→G,et.) we find that if“W”represents the letter"I,”then"E”must represent the letter“Q.” The fact that"EO"appears as a word and"O"would represent the letter "A"in our coded text implies that“EO”would be the word“QA,”which is an interesting combination of letters,but hardly a word.Thus,"W"cannot represent the letter"T" and therefore“W'represents“A.” Using the shift described above and replacing the corresponding letters,we find that the code says the following. "THIS ENCODING ALGORITHM IS CALLED A CAESAR CIPHER.IT IS VERY EASY TO BREAK,RIGHT?" In fact,the Caesar cipher is quite easy to break.If this interests you,a very readable history of coding theory is presented by S.Singh in The Code Book,[99]
1 The How, When, and Why of Mathematics 7 This is a right triangle, so we can obtain the length using the Pythagorean Theorem. So, we get d(P,Q) = q d(P,R)2 +d(R,Q)2 . Substituting in what we found above, we obtain d(P,Q) = q (x0 −x1)2 + (y0 −y1)2 + (z0 −z1)2 . This completes the proof. 4. “Looking back.” What we have presented is our version of the proof. You may find that you need to include more details. By all means, go ahead. If you had to stop and say, “where did that come from?” make sure you answer yourself. Write it in the text (you aren’t going to sell this book back anyway, right?), or keep a notebook of “proofs with commentary.” Note that though we used pictures to illustrate the ideas in our argument, a picture will not, in general, substitute for a proof. However, it can really clarify an idea. Don’t rely on a picture, but don’t be afraid to use one either. Solutions to Exercises Solution (1.1). We are given that this code was created through a shift of the alphabet. Thus once we determine one letter, the other letters are easily found. Since we have a one-letter word, we’ll start with it. Thus “W” must represent the letter “I” or the letter “A.” Checking both shifts of the alphabet (W → I,X → J,Y → K,Z → L,A → M,B → N,C → O, etc. W → A,X → B,Y → C,Z → D,A → E,B → F,C → G, etc.) we find that if “W” represents the letter “I,” then “E” must represent the letter “Q.” The fact that “EO” appears as a word and “O” would represent the letter “A” in our coded text implies that “EO” would be the word “QA,” which is an interesting combination of letters, but hardly a word. Thus, “W” cannot represent the letter “I” and therefore “W” represents “A.” Using the shift described above and replacing the corresponding letters, we find that the code says the following. “THIS ENCODING ALGORITHM IS CALLED A CAESAR CIPHER. IT IS VERY EASY TO BREAK, RIGHT?” In fact, the Caesar cipher is quite easy to break. If this interests you, a very readable history of coding theory is presented by S. Singh in The Code Book, [99]