THE MATH DUDE'S WHY MATH ISN'T AN AWFUL NERD (Chapter One)Why Math Isn't an Awful Nerd
Pure mathematics is the world's best game. It is more absorbing than chess, more of a gamble than poker, and lasts longer than Monopoly. It's free. It can be played anywhere--Archimedes did it in a bathtub.

--Richard J. Trudeau, Dots and Lines


There's a giant conspiracy in the world to make people think that math is awful. And a lot of people have bought into it. To be honest, I don't blame them--or perhaps you--because, frankly, math can be awful when it's presented to you in a boring and awful way. But I also know that math, and particularly algebra, doesn't have to be that way--that there's another side to it that you've maybe never seen. Math is kind of like that quiet kid in class that you always assumed was a big nerd but was actually rockin' out with her band every weekend on her way to becoming a big-time star. Stick with me for a while and I'll convince you that even though you might think that math is that kid you never wanted to talk to, it's actually that creative, fun, and surprisingly cool rock star.

To give you an idea of what I'm talking about, let's take a minute before diving into the nitty-gritty details of algebra to play a little game of ye olde math. This will probably sound strange, but the objective of this game is to first try and figure out what happens when we add up sequences of positive odd numbers, and then to figure out why it happens! Sounds like fun, right? Oh, not so much? Okay, I totally understand that adding up a bunch of positive odd numbers sounds like a really odd (yes, pun intended) way to spend part of your day. So why would you want to pay any attention to me and play this game?

Because whether or not you've realized it yet, life is all about learning, exploring, and trying new things. In other words, discovering and solving problems you haven't seen before is good for you! Each and every time you do it, you open up a new door that will help you solve other problems that you'll eventually face in your life...some of which might even be algebra problems. And believe it or not, it turns out that playing games with math can be a lot of fun too. So hear me out, because I think you're going to end up enjoying this. And if you don't, you can always send a note to [email protected] and tell me I'm crazy!

Basic Number Properties

Since this is a book about algebra, I'm going to assume you're already familiar with the basic properties of numbers--including terms like positive, negative, even, and odd. But since this is the very beginning of the book, let's make sure we're all starting on the same page by quickly summarizing the properties we need to know for our game--namely, the properties of positive odd numbers.

Positive odd numbers are numbers like 1, 3, 5, 7, 9, and so on. Start at 1 and add 2 to get the next one. Add 2 more to get the next one. Do it again, and again, and so on, forever. The positive part means that the number is greater than 0, whereas negative numbers are less than 0. And the odd part means that the number is not evenly divisible by 2, whereas even numbers are always evenly divisible by 2...that's why they're "even"! We'll have more to say about number properties in chapter 3.

Pop Quiz: Name That Number Property

Use these Pop Quizzes to make sure you understand what we're talking about. If you get stuck or want to check your work, solutions are given at the end of the book.



Again, just to make sure we're all on the same page, here are a couple of quick quiz questions with which to double check yourself. Yes, these should be easy.

P.S. There's one "trick" question here. Can you find it?


Basic Arithmetic

I'm also going to assume you're comfortable with the big four basic operations from arithmetic: addition, subtraction, multiplication, and division. That means you should be comfortable with doing things like adding, subtracting, and multiplying a list of numbers, and doing long division. To keep everything as clear as possible, here's a rundown on the notation (in other words, the mathematical symbols) that we're going to use for these all-important arithmetic operations:

Addition will be indicated by a traditional "plus" sign like "+" (no big surprise). So a simple addition problem will look like 1 + 2 = 3.Subtraction will be indicated by a traditional "minus" sign like "-" (again, no big surprise). So a simple subtraction problem will look like 3 - 2 = 1.Multiplication will be indicated by a dot like "*" (which may be a surprise). So a simple multiplication problem will look like 3 * 2 = 6. Why do we use a dot instead of the usual "times" symbol as in 3 × 2 = 6? Well, let's just say that the letter "x" is going to play a big role in our algebraic story as what's called a variable, and that the letter "x" looks an awful lot like the "X" symbol traditionally used for multiplication in elementary math. And not to freak you out, but sometimes we'll even omit the "*" entirely! For example, x * y is the same thing as xy, 2 * x is the same thing as 2x, and (1 + 2) * (3 + 4) is the same thing as (1 + 2)(3 + 4). It looks like a lot to remember right now, but you'll get used to it quickly.Division will typically be indicated by a slash symbol like "/" (which again may be a surprise). So a simple division problem will look like 6 / 3 = 2. Why not just use the "÷" symbol for division that we used in elementary math? Well, for now, you'll just have to trust me that the traditional "÷" sign won't work very well for a lot of what we'll be doing in algebra. Exactly why will become clearer and clearer as we move through the book.

Let the Game Begin!

Okay, let's get going adding up those positive odd numbers we talked about. To get started, let's take a look at the following sequence of problems:



1 = 1 This one is really easy...there's nothing to do!

1 + 3 = ___? Now they're "tougher"...okay, still easy.

1 + 3 + 5 = ___? Still easy? Yeah, pretty easy.

1 + 3 + 5 + 7 = ___? How about now?

1 + 3 + 5 + 7 + 9 = ___? And now?



So, what amazing thing do you think will happen when you find all these sums? Will trees start walking and talking? Will the universe implode? Will you get hungry? Well, other than that last one I have serious doubts that any of these things are going to happen. But let's do the experiment anyway since we'll never know unless we try. And maybe...just maybe...we'll discover something that's mathematically interesting too.

Now, let's make a list of the sequence of answers to the addition problems from before. The first number on our list is 1...since it's the first positive odd number and we're not adding anything to it. To get the second number we need to add together the first two positive odd numbers. That's 1 + 3 = 4. Okay, we're on a roll and so far our list looks like

I should point out that the "..." symbol just means that there are more numbers yet to come. So, what's the next one? Well, we need to add together the first three positive odd numbers: 1 + 3 + 5 = 9. So now our list looks like

Then we add the first four positive odd numbers: 1 + 3 + 5 + 7 = 16, and then the first five, and so on. The pattern that emerges looks like

Again, the "..." at the end just shows that we could keep on doing this forever to create a longer and longer list. Although I'll spare you the pain of actually doing that since I'm sure we both have better things to do with our lives. Now, do you see anything interesting in that list? Perhaps there's a pattern developing? If you do see something, that's awesome! If not, don't worry. Just like learning to solve crossword, Sudoku, or KenKen puzzles, it takes a while to develop an eye for this sort of thing. Believe it or not, after a bit of practice, patterns will just start jumping out at you.

Secret Agent (in Training) Math-Lib

Is Math Boring?

Each chapter includes a "secret agent math-lib" like this one in which you get to solve spy-caliber puzzles using the math we're learning. As if that weren't cool enough, each math-lib is part of a big spy saga that continues throughout the book. Which means that you have to read the whole thing to figure out how the story ends! If you get stuck, hints and solutions are available at the end of the book.

Good evening, agent in training ___________ (name of fruit or vegetable). Your mission in life, if you choose to accept it, is to use your perceptive powers of reasoning to solve math puzzles that could save ___________ (name of a planet) from destruction! I know that's a lot of pressure, so we're going to start things off nice and easy...at least until you graduate from secret agent math academy.



Okay, it's time for your first training session--and it's an important one because we are once and for all answering the question: "Is math boring?" Here's how it's going to work. First, I need you to think of some whole numbers:

(A) Which number between 1 and 10 is written in English using the same number of letters as the number itself? __________

(B) What number between 1 and 10 rhymes with a number between 11 and 20? __________

(C) How many times do you use the numeral "1" when you write all the whole numbers from 10 to 20? __________

(D) Which number between 1 and 10 has the most occurrences of the letter "e" and has not yet been used in this list? __________


If you've got four whole numbers in the spots above, then you've passed your first test. Next, you need to fill in the following blank spaces with the numbers from above (the letters in parentheses show you where to put the corresponding numbers), and then you need to figure out if the answer to each of the following problems is either an even (E) or an odd (O) number. As we'll talk about in more detail later, when part of a problem is written inside parentheses, it means that you need to do that part first. For example, the problem 4 * (2 + 1) means first add 2 + 1 = 3, and then multiply this sum by 4 to get 4 * 3 = 12. Got it? Okay, here we go:

Now, you need to figure out if the answer to each of the following problems is either a positive (P) or a negative (N) number. Remember, positive numbers are bigger than zero, and negative numbers are smaller than zero.

Okay, but what about the big question we set out to answer at the beginning of this training session: Is math boring? Well, I'm sure you already know the correct answer ("No!"), but just in case, this should help clear it up for you. All you need to do is insert the letters "E," "O," "P," or "N" from the answers to questions (1) through (4) into the following:



Is math boring?

By the way, if you have trouble finishing any of the "Secret Agent Math-Libs" in the book (there's one in each chapter), the answers are available by chapter in Math Dude's Solutions. But no peeking...unless you really need help!


Looking for Patterns in Numbers

Let's now return to the list we made earlier by adding up positive odd numbers. Whether you saw a pattern in this list right off the bat or not, let's walk through the process of finding one. The first thing you might notice is that the numbers in the list 1, 4, 9, 16, 25, 36 keep getting bigger. Is that a pattern? It sure is. But is that an interesting pattern? Well, that's kind of a subjective question, but in this case I'd say not really. After all, the list was created by adding up larger and larger positive numbers, so the sum had to keep getting larger too.

Since I'm so picky about patterns, let's set our sights on finding something a little more insightful. To give us a fresh look at the problem, let's try coming at it from a different angle. Imagine we have several groups of blocks, each of which is connected together to form what we'll call a strand of blocks.

Besides having a solitary block, you're lucky enough to have three other groups of blocks too: a 4-block strand, a 9-block strand, and a 16-block strand. Do those four numbers--1, 4, 9, and 16--look familiar? They should by now--they're the first four numbers from our list. So how is it that these blocks are going to help us find a pattern...hopefully even an "insightful" pattern? Well, imagine that all the blocks in each strand are held together by a rubber band that both holds the blocks together but also allows you to move them around and change their position relative to one another. Now, what do you think of doing this?

That's pretty interesting, right? The strands of 4, 9, and 16 blocks can be rearranged to form a 2- by- 2, 3- by- 3, and a 4- by- 4 square. Of course, the single block also forms a 1- by- 1 square all by itself without doing anything to it. Which means that the first four numbers from our list can all be folded into perfect squares! Not only is that interesting, I'd say it's incredibly interesting since most numbers in existence can't be folded into squares like this. And yet every single one of ours can--which means we must have some pretty special numbers on our hands.

Exponents and Perfect Squares

Multiplying a number times itself is called squaring that number--which seems like a pretty appropriate name given what we've found out about making squares. In other words, "three squared" is just another way to say 3 * 3. But a big part of math is about coming up with better and more efficient ways to write things, so people have developed another bit of notation to describe squaring numbers that looks like this:

1 * 1 = 12 = 1

2 * 2 = 22 = 4

3 * 3 = 32 = 9

4 * 4 = 42 = 16


The superscript number is called the "exponent"--in this case it's 2--and it tells you how many times to multiply together the "base"--that is, the number right before the exponent. So,

2 * 2 = 22 = 4

2 * 2 * 2 = 23 = 8

2 * 2 * 2 * 2 = 24 = 16


and so on. If you continued this pattern and kept adding more and more lines, you'd be finding what are called higher and higher "powers of 2"...that is, 25, 26, 27, and so on.

On a related note, numbers formed by squaring whole numbers (those are numbers that don't have fractional parts) are known as "perfect squares." That means that 1, 4, 9, and 16 are all perfect squares since 12 = 1, 22 = 4, 32 = 9, and 42 = 16. And there are a lot more perfect squares too...an infinite number, in fact. Just pick any whole number you like, multiply it by itself, and you've found yet another perfect square. You may have noticed that 1, 4, 9, and 16 are also the numbers that we've written on our list so far...and they're all perfect squares! If you're looking for more information on exponents right now (and you just can't wait), you'll want to check out chapter 4.

Pop Quiz: It's Hip to be a Perfect Square

Which of these numbers are perfect squares?
55 Am I a perfect square? ___ If yes, then I am equal to ___ * ___100 Am I a perfect square? ___ If yes, then I am equal to ___ * ___81 Am I a perfect square? ___ If yes, then I am equal to ___ * ___68 Am I a perfect square? ___ If yes, then I am equal to ___ * ___121 Am I a perfect square? ___ If yes, then I am equal to ___ * ___
Bonus question: If 24 = 16, then what does 26 equal? Is that a perfect square?


A Surprising Sequence of Numbers

Okay, let's get back to looking at that pattern of perfect squares we've discovered. The first thing we should think about is whether or not the pattern keeps on going forever. In other words, will every number we ever write on our list be the square of some other number? So far, we've verified that 1, 4, 9, and 16 are all perfect squares. Let's now look at a few more numbers and see if the pattern holds up. The next number on the list is 25, and 25 is equal to 5 * 5 = 52.

So, yes, that one works too. And the number after that is 36, which is equal to 6 * 6 = 62. Again, that one works. And if we keep going, we'll find that no matter how big a number we get to, the pattern will continue! For example, here's the pattern written all the way out to the tenth number on the list:

1) 1 = 1 = 12

2) 1 + 3 = 4 = 22

3) 1 + 3 + 5 = 9 = 32

4) 1 + 3 + 5 + 7 = 16 = 42

5) 1 + 3 + 5 + 7 + 9 = 25 = 52

6) 1 + 3 + 5 + 7 + 9 + 11 = 36 = 62

7) 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 = 72

8) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 82

9) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81 = 92

10) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100 = 102


Take a look at this pattern of numbers. Not only is each sum the square of some number, but each sum is actually the square of the number that gives its order in the list. In other words, to find the value of what we'll call the "nth" number on our list (we're just using the letter "n" to represent whichever number on the list we're interested in at the time...it could be 1, 2, 3, 4, or what ever), all you have to do is calculate the value n2. For example, the first number on the list is represented by n = 1, and therefore has a value of 12 = 1; the second number on the list is given by n = 2 and has a value of 22 = 4; and so on until we get to the tenth number on the list, represented by n = 10, which must have a value of 102 = 100. And although we haven't written them, we could go on and on to as high a value of n as we'd like.

Okay, so that's all pretty cool, but I've got an even more impressive way to look at the numbers on our list. Check this out:

It kind of looks like an awesome game board, right? Well, it sort of is...because it's your dear old friend the multiplication table. But notice the numbers running from top to bottom right down the middle of the table. Those numbers are all the perfect squares. In other words, those are all the numbers on our list! Each and every one of them, somewhat magically, turns out to fall right smack-dab on the line that goes down the middle of the multiplication table.

Recap

Are you starting to agree with me that our list of numbers is pretty amazing...or at least that it's pretty surprising? To really see why I'm so enthusiastic about it, let's take a step back and think about what we've actually done so far. We started by adding up a bunch of odd numbers. Not a big deal...yet. Because it turned out that when we did this several times in a row, we ended up creating a list of very special numbers known as perfect squares--1, 4, 9, 16, 25, 36, and so on. And, if you think about it for a minute, this seems pretty strange. After all, why should adding up odd numbers have anything to do with creating perfect squares? They're totally different beasts. And we're not talking horses versus ponies different--we're talking mosquitoes versus unicorns!

So, have we discovered an ancient secret soon to bring us all fame, fortune, and glory? Well, sorry to disappoint you but although our discovery is awesome, it isn't going to make us rich. Bummer, I know. But there is a silver lining here because although the solution may not directly bring you wealth, it will bring you something equally as important: knowledge (which, of course, could eventually turn into money). I know, I know, you'd rather have the cash...sorry! But let's not lose track of what we're really doing here--that is, trying to understand how and why the puzzle we've looked at works. Believe it or not, the solution is surprisingly simple. But don't be fooled by the word "simple" here, because the solution is also extremely elegant and, to my eyes at least, beautiful. So, let's figure out how it works.



A Positively Odd Puzzle

Each chapter includes a "math brain game" section like this one that's designed to put your brain into overdrive and get you thinking about algebra in new and unexpected ways. And they're usually pretty fun too!


Step 1--The Game Plan

Okay, we're going to solve our positively odd puzzle using strands of blocks...exactly like the ones we "folded" into perfect squares earlier. Let's use five strands: a single block and four made up of groups of 3, 5, 7, and 9 connected blocks. Do you recognize those numbers? We've seen them enough at this point that I bet you're starting to get tired of them. Of course, they're just the first five positive odd numbers. Believe it or not, using nothing more than these five strands of blocks, we can show why the incredible relationship we've discovered works. So, your job is to figure out how to arrange the strands of blocks on the grid below to prove that the sums of sequences of positive odd numbers, such as 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, and so on, must give numbers that are equal to perfect squares.

Here's a hint: The first block is in a good spot...so don't move it! And remember that besides placing the connected strands on the grid, you can also rearrange their shapes like we did before when we first discovered that the strands of 4, 9, and 16 blocks could be "folded" into perfect squares. So, give it a shot, and then read on when you either figure it out or get stuck and need some help!

Step 2--Can you Take a Hint?

Okay, I'm now going to put the strand of blocks for the next number on our list, 3, onto the grid. And to help out a little more, I'm also going to rearrange the shapes of the remaining 5, 7, and 9 block strands. Does that help you see the solution to the puzzle?

Remember, the goal is to come up with a way to show that whenever you start with 1 and add any number of the subsequent sequence of odd numbers to it, you get a number that's a perfect square. If you think about it, you'll see that this means that all the puzzle pieces (that is, all the strands of blocks) need to combine to form perfect squares within the large square grid (since the grid is a square!). Do you see how after we put the 1- by- 1 and 2- by- 2 squares on the board, the resulting shape is a perfect square? Can you use the remaining shapes to make 3- by- 3, 4- by- 4, and 5- by- 5 perfect squares too? At this point, I'm close to giving the answer away...so I'll stop talking for now. In the meantime, think about it some more and then move on when you're ready for the answer.

Step 3--The Solution!

Okay, it's the big moment we've all been waiting for--here's the solution to our positively odd puzzle:

It really is a thing of beauty, right? This drawing shows that adding up a sequence of odd numbers and getting perfect squares is, somewhat surprisingly, not a mystery...not at all, in fact. It's simply a result of the way that shapes fit together. That is, each successive odd number perfectly "wraps around" two sides of the square formed by the sum of the previous odd numbers. For example, the three blocks from the 3- block strand perfectly wrap around the single block, the five blocks from the 5- block strand perfectly wrap around the three blocks from the 3- block strand, and so on. Do you see why I've called it an elegant solution?

Wrap-Up

And with that we've arrived at a critical junction in our relationship--the point at which you decide whether or not to stick around and learn algebra with me. Some of you will see the elegance in the solution to our positively odd puzzle. Some will not. And that's okay! After all, we all have different tastes and preferences about food, cars, clothes, art, movies, music, and yes even math. For some, seeing a solution to a puzzle like the one we've looked at is genuinely exciting. Others are left wondering about the sanity of those who are now excited. But just because you're in the skeptical camp right now doesn't mean you won't learn to love math...at least a bit. I should know since, once upon a time, I was one of you! Honestly, I didn't always love math. But once I was introduced to "real" math--the kind that can be used to discover and explain things--I was hooked.

So, really, what's the point of all this? This puzzle, this prologue, this book? Well, first of all, we're going to learn algebra, and we're going to learn it well. And when I say learn, I'm not talking about memorizing sequences of steps and formulas for the sole purpose of acing your next test. No, I'm talking about learning algebra so that you really understand it. And that will not only help you ace your next test--with flying colors--but it will also help you ace what ever other challenges you face in your life. But that's still not enough for me! I'm not satisfied with just helping you "do" and understand algebra. I want to show you the bigger picture too--the picture that makes some people look at the puzzle we solved with a sense of delight. It may sound ambitious, but in addition to teaching you algebra, I want to help you discover the world's oldest, cheapest, most practical, powerful, fun, and undeniably amazing game.

Hello, world! It's me, math.

* Final Exam*

Each chapter ends with a summary of the main topics covered in that chapter, and a series of "exam" questions about those topics. This gives you a chance to review the main ideas from the chapter and make sure you understand them. If you don't, go back and review the topic before moving on...or else you're begging for confusion! If you get stuck, or you want to check your work, answers are available in Math Dude's Solutions.

* Number Properties


1. Think of an even number: _______.
Now think of another even number: ________.

Add them together: ________.
Is the answer even or odd? _______.Subtract the first from the second: ________.
Is the answer positive or negative? ________.
Is the answer even or odd? __________.Multiply them together ________.
Is the answer even or odd? _______.

2. Think of an odd number: ________.
Now think of another odd number: ________.

Add them together: _______.
Is the answer even or odd? _______.Subtract the first from the second: ________.
Is the answer positive or negative? _______.
Is the answer even or odd? ________.Multiply them together ________.
Is the answer even or odd? ________.

3. Think of an even number: ________.
Now think of an odd number: ________.

Add them together: ________.
Is the answer even or odd? ________.Subtract the first from the second: ________.
Is the answer positive or negative? ________.
Is the answer even or odd? ________.Multiply them together ________.
Is the answer even or odd? ________.

Do you see any patterns in these solutions? Perhaps something to do with combinations of pairs of even and odd numbers?

* Notation for Arithmetic

Translate the following sentences into arithmetic problems. And be sure to use our fancy algebra-ready notation for addition, subtraction, multiplication, and division.

4. "Thirty-two plus twenty-eight minus fifty-nine.".................... 32 + 28 - 59

5. "Eighteen times two divided by nine times two.".................... __________

6. "Eighteen times two, all divided by nine times two."..............._________


* Squaring Numbers

7. Pick a number between 1 and 10:. Let's call this number a.

Square that number a2 =. Let's call this number b.Multiply your original number by 2 2 * a =. Let's call this number c.Subtract c from b b - c =. Let's call this number d.Add 1 to d d + 1 =. Let's call this number e.

8. Now start again with the same number, a, that you picked be fore.

Subtract 1 from that number a - 1 =. Let's call this number f.Square this number f 2 =. Let's call this number g.

The numbers e and g are the same, right? Surprise! Want to know why? Good, I'm glad. Here's a hint: It's not magic, it's algebra. And we'll find out exactly why it works by the time we get to the end of the book. Sorry, there's no shortcut to learning.

* Searching for Patterns

Can you find the patterns in the following lists of numbers. In other words, can you find the key thing that changes from number to number in each list that allows you to predict what the next number in the list will be? Some of these are pretty tough, so don't be discouraged if you can't figure them all out. It's okay to peek at the solutions.

9. 1, 2, 4, 8, 16, 32,...What's the next number? ___

10. 2, 5, 8, 11, 14, 17,...What's the next number? ___

11. 1, 3, 7, 15, 31, 63,...What's the next number? ___

12. 1, 1, 2, 3, 5, 8, 13,...What's the next number? ___


THE MATH DUDE'S WHY MATH ISN'T AN AWFUL NERD Copyright 2011 Jason Marshall