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Numbers, mathematics, algebra and geometry give everyone a warm feeling inside. Right? C’mon, don’t you want to curl up with a good math book on a cold winter evening? Hot cocoa, popcorn and long division! No? Hmm… Ok, so maybe you are one of the millions of people who shudder at the very word.
Most people agree about the necessity of math, but enjoying it is not as easy. This is understandable. From day one in school, people are thrown to the wolves and told they must learn math. For the most part, all training in the manipulation of numbers is done the same way. You are assigned a book and have chapter after chapter of problems to work out. As soon as you learn how to count, some fool always wants to know the answer when you take two sets of those numbers and add them. And it’s not enough that you learn how. They inundate you with page after page of numbers to add to prove it. After a few chapters of this, the people at the front of the room reverse themselves and want you to subtract numbers. They repeat this with multiplication and division.
It’s unfortunate that the process is so mind-numbing because the simple fact of the matter is that life as it is now would not exist without mathematics. Numbers do indeed form the foundation upon which technology and, indeed, all of life is built. Math can actually be, amazing as it seems, fun. Well okay, maybe not fun, but at least not the equivalent of being pecked to death by a duck.
This process is repeated for all mathematical operations. The bottom line is that it gets tedious. It can get to be so mechanical that a persons mind gets turned off. No excitement, just row after row of numbers. This is supposed to be remedied by the famous or rather infamous, story problem. However, story problems do not just give the numbers and say do this or do that. Embedded within the paragraph are key words that indicate what operation or operations are to be performed. The key is to understand what these words are. I will give examples and show how to determine what word is used to mean what operation. For example, one half of something is an expression used to indicate division. Story problems have scared off more students than anything that could be thrown at them in a history class.
You graduate from high school and apply for college. In many cases you are given a set of tests to determine the classes you need. The math test tells you what grade level your skills are. Where do you fit in? Maybe you know ahead of time that your math skills are not as strong as they should be, so you decide to work on raising them. Where do you start? The first thing you notice is that all math books tend to be divided by grade level. Maybe you take a pretest and find out that your math skills are lower than you think. It is hard for a person to pick up a book designed for grade seven when you are much older than that and realize you are unable to solve many of the given problems. This book is about skills, not grade levels. As you age, many of the techniques learned earlier tend to get rusty. This book is designed to be steel wool for the basics.
There are a few assumptions being applied. The first is that the basic operations are understood. How to add, subtract, divide, and multiply is known by most post high school people, but there are complications of these that will be pursued here. The second assumption is that people reading this want a book that does not over or under whelm. Start at chapter one, but if that is easy and well understood, skip and go to the next chapter. A suggestion is that when you find the chapter that is not understood; start reading at the previous chapter.
Chapter one will be an overview of the basic operations. What will make this different is the approach. There are many books on the market that give problem after problem. One learns by rote repetition. There won’t be any problems like traditional math books have. These can be made up by the reader or found in other books. The idea of this book is to show what and how the various operations work.
Chapter two will deal with the advanced form of addition and subtraction, that is, multiplication and division. These seem like they are totally different, but in reality they are very closely related. One good example that shows the close relationship is to examine how a computer multiplies and divides. To perform multiplication requires a series of logic devices called adders. In both chapters I will include decimals. There will be a few stray algebraic equations in this chapter just to show how they look.
We have graduated to chapter three, multiplication and division. The reader at this point will be familiar with the operations but that does not mean the work is done. We need to discuss units. Although the problem is stated in inches, once we get over twelve, it needs to be broken down to the lowest terms. This will of course imply more division, and subtraction. All units are English, metric will be described and a brief explanation will be given, but nothing detailed.
This will be in chapter four, devoted to the conversion between metric and what is commonly referred to as English. Actually, English, or the feet, inches, pound, units used in the United States, is called Imperial. Each measuring system by itself is relatively easy to convert, say from inches to feet or meters to kilometers. The problem comes in when converting from inches to meters or the other way around. There will be a conversion table and instructions on how it is used to select the correct units.
Chapter five will deal with story problems. Most complaints about math stem from trying to figure out to solve story problems. A novel approach will ease some of the pain when these are encountered. Most math books take situations that are easy to set up in story fashion but are not realistic. Most people don’t go down to the train station and try to judge when one train will arrive as compared to another. In the modern world we calculate how long it takes to get to work based on certain situations. We go to the store and need to ascertain how much clothes will cost based on the sale prices. These are real world story problems.
Algebra, in chapter six, is almost as scary. Looking at an algebra book for the first time you suddenly see that you will be doing operations with letters. Now how in the heck did they ever come up with that crazy idea? Numbers are bad enough. Closer to the back of the book it gets worse, not only are they using letters, but now some idiot has thrown in a bunch of Greek letters. We don’t have enough, now they start using foreign alphabets. Besides, algebra isn’t used in the ordinary real life day-to-day existence, is it? It is and I intend to show where and how.
Chapter seven will present a new angle. It will also show squares and rectangles. Here is where we enter the world of geometry. Some of the basic equations for area and circumference could have been covered earlier, but it is easier to understand these concepts after a basic understanding of working equations is established. It is also better to go from the simple to the more complex in the same location without having to constantly refer back to earlier chapters or other books. By the end of the geometry chapter, we will enter the modern digital world.
There are more advanced forms of math but to go from addition to calculus, a higher form of algebra, in one book is a little much. Each chapter will have a summary and at the end of the book will be an appendix that contains abbreviations and the Greek alphabet. This book is for entertainment first and learning second. So let’s round up those numbers and get them corralled, we have mathematical operations to perform!
Numbers and operations
Numbers have two forms, word and symbol. The word form spells out the word, like seven or eight million. Numbers are universal, but their spelling is not. The English number one becomes un in French and uno in Spanish. They may be spelled different but each represents a singular unit. Symbols include such things as 7 or 3543. Universally the Arabic set of symbols is used. Thus 1 apple in the English world is un apple (or pomme in French). However, there are many other numeric symbols, the most common of which is roman numerals. Arabic is used primarily because it is easier and saves room.
Using a symbol to represent a number goes back a long ways. However, up until the 1600’s, one also had to be careful how the symbols were used. Writing a list of sheep for sale could be a bit rough. In the real old days writing the number 20 would be written as 2(space) or 2x. The symbol for zero did not exist. Even when the 0 was suggested there were a lot of problems. For example, what is the sign of zero? Is it positive or negative? This is the only number that is always unsigned. Even some basic operations with zero have caused trouble. These basic operations will be covered in the chapters ahead.
Why do we even have numbers? They were invented to keep track of things. For many early civilizations and cultures counting consisted of nothing more than putting one rock for one object. If there were more rocks than objects, then some objects were missing. If there were more objects than rocks, then a gain was made. Our early civilization guy knew what he had by the size of the pile of rocks. It took many years for the art of counting to catch on. In the early years, it was just a matter of the pile of rocks; my pile is bigger than yours, etc. As civilization advanced, keeping track of the exact number of cows or sheep became important.
It was also important to keep track of dates and other numeric pieces of information. A good example is who owns the herd of sheep about to be taxed? Pity our tax collector who comes to collect from Joseph and finds out the sheepherder didn’t make it through last winter. The herd is now owned by his son, Joseph the second (or third). Our harried taxman tells the family that he has to go back to the office to change the records but he will be back on the 5th of the month to collect. By calling his son the second, he has given him a position in the linage. The 5th of the month is the 5th day in order. Numbers that stand for position or order are called ‘ordinals’ The numbers 0, 1, 2, and on to infinity are the counting numbers or ‘whole numbers’.
The word symbol will be used later so from this point on, numbers will be referred to as notations. At this point, we can tell that Octavia has more sheep than Remus, but how much more? How many sheep will there be if both herds are put together? Up to now, the only way was to count each herd, then count the herd after they were merged into one. There has to be an easier way. What happens when Octavia sells some sheep? How many are left? These are simple problems, but when it comes time to pay taxes, how does the government know how many sheep there are in his area? Ok, I agree, he could send someone around to count them. But this presents three problems. First, how can you be sure you have them all counted? While you are checking the herd on one side of the country, the herd you had counted on the other side has had a bunch of newborns. The second problem is there a way to write a number of the correct size. The total number then is called the ‘cardinal’ number. Cardinals tell how many, either in words or numerals. The third problem is keeping track of who has how many at the same time as keeping a total count.
The way that was invented to allow numbers to increase in size was placeholder. This sounds complicated. To make it simple, we will use the decimal system. Decimal means base ten or based on ten digits. Up to this point they are I, II, III, IV, V, VI, VII, VIII, IX, X. This is Roman numerals, so let’s do some quick translating and make them English digits, 1, 2, 3, 4, 5, 6, 7, 8, 9, hmm, we are out of digits and no one has invented the 0 yet. Ok, for 0 we’ll use x. The Romans used a special mark, or cone to mark where a zero would go. Otherwise, some careless tax counter could write down that Neros had II or 2 sheep. He really meant that he had I I or 101 sheep, but the space was not very obvious. Thus, instead of writing I I, our tax counter would write IxI.
An easier way had to be found! By the time our poor tax counter got back to the office, it was time to start on next years count. Operations were invented. Add, subtract, multiply, and divide. Once these came along a few rules had to be installed also but that will be dealt with in the following chapters. So how is it done? Does our local accountant say or write, add this number, then add this number, etc? Clearly there had to be a way to show what to do with the numbers.
A plus sign (+) was to become the symbol for addition. It’s simple, and straightforward.
A minus sign (-) denoted subtraction. A minus B or A take away B is also simple and easily understood. But, now we start to complicate things. How do we indicate multiplication and division? They could have used ++ for multiple additions and – – for multiple divisions, but that would have been too confusing.
Hence we have the letter x and the asterisk (*) symbol used for multiplication. As time went on, even placement indicates to multiply two numbers. But, how does one tell if 35 is the number thirty-five or three times five? The only way one could tell is by the introduction of another symbol, the parentheses (). Thus, 3(5) meant 3 times 5. This becomes very important when the discussion of algebra comes up. Now we will see expressions like 3x=8 or 3(x+1) = 10. How are these read? 3x=8 means that 3 x’s equals 8 or 3 times (something) equals 8. We are in the realm of algebra, so we know that the x is a variable, not the symbol for times. This in turn led to defining the order the operations must be done to attain the correct answer. For example:
3(5) + 1 or lets rewrite that as 3 x 5 + 1. We now have 3 times 5 plus one or 3 times 5 plus 1. Which answer is correct, 16 or 18. Once the order of operations is applied, the correct answer is 16.
Division is nothing more than multiple subtractions. But until it was defined, accountants and other number crunchers would have to take the original number, subtract the smaller amount, the from that answer, subtract again, until there was nothing left to subtract from. It would be tedious to say the least. Our most common symbols were /, ¸, )——– . Again, the hierarchy or which operation comes first also comes into play again. But, division also was presented with a unique problem. Nothing, or zero could be divided into, but not the other way around. Zero, divided by anything is still zero, but anything divided by zero is undefined. In other words, it is mathematically illegal to attempt to divide by zero.
These are the basic operations, add, subtract, multiply, and divide. With what is known to this point, our Roman tax collectors can count the head of sheep and assess the taxes. Since subtraction is now known, the poor sheepherders could get back the correct change (assuming they knew how to do the math). Once the annual taxes were collected, the refunds could even be sent back. It was a rather unusual situation though. It seemed no one was due a refund. Oh well, that’s politics.
Using what operations are available to this point, fences, usually just rocked off areas, could be computed. In fact, almost anything that was square or rectangular could be figured out. But, those pesky circles presented a problem. How does a person really know how big to make it, or how much area there was? Since it would be a few years before hardware stores opened up, how could a person be sure that walls met at right angles? It’s beginning to look like we don’t have enough symbols yet.
Enter the world of algebra and geometry.
Many a student opening an algebra book for the first time has been heard to exclaim, “It’s Greek to me”. Although the student meant it looked very complicated, The Greek alphabet plays a big part in algebra. As we progress on into geometry, more of the Greek alphabet will come into play. The full Greek alphabet will be listed in Appendix I. But the big question is, “What the heck do we need algebra for?” You don’t need it, if all you plan to do is balance your checkbook and figure out the monthly budget. However, you will fall into a few algebraic equations quite by accident. A good example is at the grocery store. If corn is three cans for a dollar forty-nine, how much is one can? Of course the solution is obvious, you divide 1.49 by 3 but the equation is 3x=1.49. There are a lot more uses for this more advanced form of math as shown in the chapter devoted to algebra.
A trip to the local hardware store can eliminate much of the need for geometry. But, if you are building a shed or tree house and need to make braces, a solid background with the Pythagorean theorem is needed. This is not a need to know branch of mathematics. There are plenty of squares, t-squares, and measuring tapes to do most of the work. However, if you are making your own blueprints and need to calculate how much wood you’ll need, the study of geometry is important. Angles, and areas of odd shaped figures are calculated with geometric formulae.
There are many ways to represent the entity we call numbers. Using words, they are written in every language on earth. In notation they are called numerals. Universally the Arabic numerals are used for the majority of numeric notation. However, Roman numerals are quite common, especially on clock and watch faces.
Cardinal numbers tell how many, like 5 rocks or fifty sheep. Ordinal numbers designate position or place as in the fourth of the month or Ramses II. These can be either written out in text form or the actual numerals or digits can be used.
The most fundamental arithmetic operations are; add, subtract, multiply and divide. Once these are mastered, they can be used alone or in combination to perform most of the mathematical work. Life is not that simple though. Many situations occur that require using more advanced forms of math. In everyday life, most problems can be solved using only algebra and geometry, as well as the basic arithmetic functions.
Our mathematical rodeo is just getting started, so, let’s add ’em up and subtract ’em out.
[From Laughs from Corn Country]
The first story:
“Two and a half bucks a gallon for gas! We may be sultans of our domain, but we don’t have their money.”
Jack had just pulled up to the pump in his American made, foreign car. You know the kind I mean, it has a Japanese name and is built in some factory in the boondocks of Nebraska. You can tell it was built in the good old United State of Automobile because the steering wheel is on the left.
“Ya know Mabel, these here A Rabs (he pronounced it A Rabs, two separate words), has got us in a danged stranglehold. They done wrapped their gas hose right around America’s carburetor and are pulling it tight. There’s gotta be a better way.”
Jack, being a farmer, was a self-taught mechanic. It was Friday night and Him and Mabel had just finished supper. Spring was in the air and tonight the wind was in the right direction so they could just sit on the back porch (they raised hogs and cattle also). They had no more than sat down when the wind shifted direction. Now you could tell they raised livestock! Jack smelled money, Mabel smelled what the cows left behind.
The breeze out of the southwest wafted over the barnyard. Because there was more money in beef than milk (and the overhead was less also), Jack’s herd consisted entirely of Herefords.
Although Mabel was accustomed to the aromas that come with a farm, she still preferred the indoorsy smell of incense and vases full of fresh flowers. At least in the winter the snow helped mask the outdoors.
“Mabel, tell that writer to shut up and answer your phone” Came the voice from the answering machine.
“Just a minute, I’ll get him.”
Laying the phone down, Mabel walks to the back door and yells at Jack who is walking towards the barn.
“Max is on the phone for ya”
“Ok. Yeah he asked me if I could help him get that old tractor of his running”
We’ll leave Jack while he’s on the phone and see what Mabel is up to out in the kitchen. I smell food. Must be a bake sale coming up because the table is covered with homemade pies and other pastries. On the counter is a bowl of Gooseberries and Mabel is getting ready to boil them to use in still another pie. Oh, here comes Jack and he looks puzzled.
“Just got off the phone. Max said he looked at that old tractor and thinks he has a better use for it. I’m going into town tomorrow to meet him over coffee. They got this new coffee joint I’ve been wanting to check out.”
Yeah, I know it’s bad English, but Jack is a farmer, not a writer. Let’s jump ahead a bit here and try to get to the coffee shop before Jack. Ah, we’re in luck, a booth right next to them. Ok, back to the story.
“Three bucks for this here Café Latte. Not bad fer one o them Italian drinks.”
“Max, that latte drink of your is just coffee with milk. They call it Latte to get a higher price for the fancy name. I got the same size as you but it’s regular coffee and only cost a buck and a half.”
It’s amazing how excited a person can get paying three dollars for a sixteen ounce cup of flavored water, but yet so angry at paying fifty cents less and getting four times the quantity for fuel. Just an observation, has little to do with the story.
“Jack, Smokey, our gas man was over yesterday to fill the barrels. I don’t believe those prices! I use diesel for the machinery of course. But then when I was in town yesterday morning, I filled up the car. At those prices, we could pay off the national debt in a week. That got me thinking. Remember a couple of years ago how that manure pile burned on ole Rubins place?
I’ve got this old pumphouse that I was going to tear down, but I wonder if we couldn’t build a small plant in there to convert the manure into usable methane? That old tractor runs but since finding the parts needed for the wheels is getting harder every day. That could be used as the power source.”
Over the next few weeks, Max and Jack spent all their free time converting that pumphouse into a gas station. Once they learned what was needed to convert manure into usable methane, it didn’t take much to modify some old junk equipment into what was needed. The idea was that they would dump the manure down a chute on one end and run it through the machinery and have a tank to store the methane.
While the boys were doing that, Mabel was researching how to convert a gas burning engine to be able to use methane. As it turned out, the conversion was relatively simple, at least to a mechanic.
Jack drove his tractor over to Max’s and finished installing the needed parts. He then pulled the gas tank off and installed the methane one. Jack was not only the mechanic, but was cautious. He replaced the ignition with a remote control device so he could start it from a safe distance. He pushed the button.
The old tractor chugged to life! Success!!
Now, if this story sounds like a lot of bull, You’re right.
k, this is going to be a long one but it’s time for some blatant advertising! I have three books for sale. Laughs from Corn Country is a collection of my stories. That one is pretty safe. But I have a math book and that usually turns people off. My book is not like other math books so I put the intro and first chapter here to show what it’s like (and to encourage sales of course).