Trademarks: Wiley, the Wiley Publishing logo, For Dummies, the Dummies Man logo, A Reference for . Part I: Arming Yourself with the Basics of Basic Math 9. DUMmIES ‰ Technical Math FOR DUMmIES ‰ by Barry Schoenborn and Bradley Simkins Technical Math For Dummies® Published by Wiley Publishing, Inc. Trademarks: Wiley, the Wiley Publishing logo, For Dummies, the Dummies Man logo, Mark Zegarelli is the author of Basic Math & Pre-Algebra For Dummies.
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For Pubic Release: Distribution Unlimited. The Air Force Research Laboratory. The Handbook of. Essential Mathematics. Formulas, Processes, and Tables. Trademarks: Wiley, the Wiley logo, For Dummies, the Dummies Man logo, A Reference for the Barry's the coauthor of Technical Math For Dummies, Medical. The Differences between Pre-Calculus and Calculus. Every good thing must come to an end, and for pre-calculus, the end is actually the beginn In Pre-.
See Chapter 5 for details about multiplication and division. A chalk line. If Og found some mastodons for dinner, he went to his tribe, made the sign for a mastodon, and pointed in the direction where he saw them. Some careers are more number-intensive than others, but every trade uses numbers. This ability is what separates human beings from the lower-order creatures, such as oysters and fire hydrants. The beauty of numbers in counting their simplest application is that answers come with no skills besides counting.
However, even counting requires careful administration. And for speed and efficiency, you can go beyond counting to arithmetic, as we show you in later chapters.
In this chapter, you review the common types of numbers you work with and some uncommon, strange, and unbelievable numbers, too. You also explore the secrets of zero. All this requires no more than a set of fingers and toes to count with. Natural Numbers Natural numbers are basic numbers, which are also called counting numbers.
Most people just call them numbers. Natural numbers have a familiar look: They serve two purposes: Counting is the technique you use for inventory and all stock keeping. Natural numbers are also the fundamental unit of downloading, no matter what your line of business is. Using natural numbers takes on a personal meaning after work. You use natural numbers for ordering, describing things in a certain order. In your personal life, using numbers for ordering becomes painfully clear at the Department of Motor Vehicles.
Zero is part of a larger group of numbers. However, an exception exists in the field of computing: Zero becomes the first counting number and takes the first position in arrays and other data structures. Counting numbers with extras Integers are like counting numbers, but there are more of them. The set not only includes the counting numbers 1, 2, 3, 4, 5, 6, and so forth but includes zero 0 and negative numbers —1, —2, —3, —4, —5, —6, and so forth.
You can Chapter 3: Zero to One and Beyond also call these numbers whole numbers. Say the word integer with a soft g. Integers can be positive or negative, odd or even. Of course, a positive number is greater than zero. An even number can be divided by 2 with no remainder. Zero is an even number.
These integers look as orderly and evenly spaced as the chorus line in a Broadway musical. As you can imagine, if you have an infinitely wide stage, the negative integers at the left extend forever.
The positive integers at the right do the same thing. Integers are the numbers you use to perform all simple math. You can do all arithmetic operations addition, subtraction, multiplication, and most division with integers. Integers are also useful for plotting the points on a graph or chart.
Where do they get these names? The word integer comes from Latin and means untouched. Speaking of untouchable, integer is a relative of the word integrity. Making math easier What is zero 0?
Zero may look like nothing, but it represents something — it appears in numbers and in calculations where digits ought to be. In a three-digit number, such as , those digits are more than just a 1, a 2, and a 3: But what happens when you have one hundred and three single items?
How do you write that without a placeholder? Ladies and gentlemen, boys and girls, what you need is a placeholder. In the number one hundred and three, let the 1 show one hundred, use a 0 show no tens, and have the 3 show three units, giving you Zero and the decimal system made most other math systems obsolete. Mathematicians point out that the decimal system is a base 10 system.
The Maya of Central America used a base 20 system, and they used zero, too. And for the nerdy band of brothers, the computer age brought forth the base 2 binary and base 16 hexadecimal systems.
Zero can be your biggest friend in mathematics because it makes for quick work: See Chapter 11 for more on powers and exponents.
See how nice zero can make your math life? Where did zero really come from? Historians cite many different civilizations that might have developed decimals and zero, but the system may have evolved in the Indus Valley near the western edge of modern-day India , and the Indians may have gotten techniques from China. Then the word spread to the Middle East. In , Muhammad ibn Musa al-Khwarizmi said if there was no number in a place, you should use a little circle.
The Arabic word for that little circle is safira meaning it was empty or sifr meaning nothing. That led to the modern English word cipher. Sifr also leads us to the Italian word zefiro meaning zephyr or zephyrum and the Venetian contraction zero. Chapter 3: Zero to One and Beyond Going Backward: You represent negative numbers with a minus sign; for example, —1, —23, and —8, Working with negative numbers In mathematics, negative numbers are a concept.
In your daily activities, you work with negative numbers in the real sense, and they almost always represent a reduction or a deficit. But the reason you get to 0 units is because of inventory draws reductions , and each reduction in inventory is the application of a negative number. Stock on hand minus the amount of the draw results in a new, lower amount of stock on hand.
In virtually all trades, accounting transactions can result in amounts lower than zero. For example, when a cosmetologist is sick, she has no clients no inflow of cash , but the rent is due on the station at the salon outflow of cash.
Not so good. Negative cash flow is a real and painful concept. Traveling down the number line In mathematics, negative numbers are part of series of numbers.
One way to visualize the series is to draw a number line.
Put 0 in the center, mark positive numbers to the right of 0, and mark negative numbers to the left of zero. Figure shows a number line. Basic Math, Basic Tools Figure The farther to the left of 0 you go, the more the numbers decrease in value. Looks can be deceiving. For example, the 9 in —9 is has a larger magnitude than the 8 in —8, but the minus sign the negative sign, — makes a difference. A larger negative number has less value than a smaller negative number.
Getting Between the Integers: Fractions, Decimals, and More Life was simpler in the third grade with only integers to deal with.
So there comes a time in your career when you must also know about other types of numbers. In between the integers are many other numbers, known as common fractions, decimal fractions, rational numbers, and irrational numbers. Our fractional friends Fractions are the most common numbers in the technical careers. A fraction is part of a number, more than zero but less than one. The word comes from the Latin fractus or frangere, which means broken or to break, as opposed to integers, which are unbroken whole numbers.
The two kinds of fractions are common fractions, which look like this: Zero to One and Beyond and decimal fractions, which look like 0. If you combine a whole number with a fractional number, the result is called a mixed number. For example, 5. Figure shows how fractions on the number line fall between the integers. Fractions on the number line. The rational numbers and their irrational friends On the job and in your personal life, you have two kinds of friends: Both kinds are valuable to know except maybe the one who puts bean sprouts and peanut butter on pizza.
The same is true with rational and irrational numbers: A rational number can be expressed as a ratio, the quotient of two integers. Any common fraction fills the bill, as it shows the ratio of the top number to the bottom number. What about 0. This decimal number is really the fraction seventy-five one hundredths, and when shown as a common fraction: Following are examples of rational numbers: Basic Math, Basic Tools Like the integers and like the fractions, mathematicians have proved that an infinite number of rational numbers exist.
You can express some numbers as fractions, but they produce infinite decimals in a repeating sequence. For example: An irrational number is always acting out. Pi the ratio of the diameter of a circle to its circumference. Pi has been calculated to over one trillion! And they never will. At least once, the government has tried to legislate the value of pi. Oh, how we authors wish we made these things up. In , the Indiana House of Representatives considered a bill that would have set pi to a value of 3.
Taking a Look at the Lesser-Known Numbers The numbers you encountered so far in this chapter are the numbers you use in your work and at home. Here is the lightning round of other number types. Real numbers Real numbers is the name for all the numbers covered in this chapter to this point. That includes natural numbers, integers, fractions, positive numbers, negative numbers, zero, rational numbers, and irrational numbers. Zero to One and Beyond Imaginary numbers An imaginary number is a number that includes the square root of —1.
This value is supposedly impossible. The math expression is: The symbol for the imaginary unit is i. A number that includes i for example, i or 7i or —3i is imaginary. Early mathematicians thought imaginary numbers were useless. But the world of mathematics evolved, and in time mathematicians found the concept of imaginary numbers to be very useful. You use imaginary numbers in engineering disciplines like signal processing and vibration analysis. A complex number is a combination of a real number and an imaginary number.
Nominal numbers A nominal number sometimes called a categorical number is a number you use for identification only. Here are some examples of nominal numbers: Some locks and safes even use letters instead of numbers. No, not so. Humans are an infinitely clever species. They created the pet rock, the Furby, and the iPod.
And indefinite numbers. The following are just a few indefinite numbers. You know, just a handful. Not to worry. Check Chapter 7 for all the details about story problems. Automotive tech — a slippery task You work at a BMW motorcycle dealership. You hope to study more automotive technology in school and open your own shop some day.
But at the dealership, you have a pretty basic starter job, and your boss asks you to determine the on-hand quantity of BMW motor oil. Determine how many plastic containers of oil you have. Zero to One and Beyond 1. Take a look at the entire quantity to be counted. The following figure shows the number of containers to be counted: Count the containers. The answer is 48 containers. The boss is pleased. He takes you to another stockroom and gives you a similar problem.
The following figure shows a new group of containers to be counted: Simple counting solves the problem. In the figure above, you have containers. Basic Math, Basic Tools Arithmetic, however is more efficient. With arithmetic, you would have figured out there are 16 rows of containers, with 28 cans in each row.
Then you would have used multiplication to multiply the number of rows by 28 to get the total number of cans. The moral of the story and actually there are two: The difference between that and oil? Getting the order right You and a co-worker are in the drive-through at Taco Palace.
Basic maths for dummies
Together, your order is three tacos, two burritos, and two soft drinks. Use counting to confirm its correctness. Count the soft drinks. Either speak the number or use your fingers. Count the burritos. Again, either speak the number or use your fingers. Count the tacos. You ordered three tacos and you only got two. The order is short. Let the person at the pick-up window know.
Notice that for this problem only counting was required. The beauty of numbers in counting their simplest application is that answers come with no skills beside how to count. Although this example is extremely simple, it shows that counting is sometimes the fastest, most accurate way to solve a problem. Chapter 4 Easy Come, Easy Go: In carpentry, you begin by hammering a nail and making a simple cut with a saw.
In cooking, you begin with simple recipes. Even in sports, such as boxing, baseball, and the martial arts, you start with a basic stance. Math is the same way. The basics are addition and subtraction. Counting is well and good see Chapter 3 , but eventually you run out of fingers and toes.
That is to say, eventually counting is tedious, and you must go on to addition and subtraction. Basic Math, Basic Tools In this chapter, you review exactly what addition is and the parts of the addition operation.
You do the same thing with subtraction. You also see the easiest ways to check your work. Making Everything Add Up Addition is the process of combining quantities.
You probably knew this, because addition is an operation you grow up with. Each item to be added is an addend. Addition works with all kinds of numbers — integers, zero, rational numbers, fractions, and irrational numbers. That is: Chapter 4: Addition and Subtraction The same is true with decimals. For example, 3. To add in a column, simply arrange the numbers so that they are all aligned on the right side, and begin the addition process.
Because these applications are cell-based, vertical columns are the logical approach to addition. Figure shows addition in Excel. A Figure Adding numbers in Microsoft Excel. This feature is handy when you want to verify addition of a large number of items in several rows and columns. The technique is called downfooting and crossfooting, and you can find details on the Internet. Basically, you sum the columns and the rows and compare the results for accuracy. Basic Math, Basic Tools Be careful with certain tools.
So does a printing calculator yes, they still make them. And of course, adding 0 to 0 gets you 0. Adding negative numbers If you had the wisdom, taste, and discernment to read Chapter 3, you know that negative numbers have a minus sign — in front of them and are numbers less than 0.
You can add negative numbers. No problem. The result of adding negative numbers is a larger negative number. In algebra, you express the idea as: Addition and Subtraction Carrying the extra Adding in any digit position for example, the ones column, the tens column or the hundreds column is easy.
But what do you do when the result it more than ten? You carry. The term carry means that when the results of adding a column are higher than 9, you record the right-hand digit the ones number and add the lefthand digit the tens number to the next column. Just add. But what about adding and ? Not so simple. Well, to be fair to the poor ones column, no column can hold more than a single digit.
What to do? Record the 4 and carry the 1 to the next column. This process is simple, but we describe it fully here anyway for clarity. Look at the addition of 6 and 8 in the ones column. The sum of 6 and 8 is 14, so you write the 4 as part of the sum and carry the 1 to the top of the tens column. So, you write the 11 as 1 in the tens column and carry the one to the top of the hundreds column. Now, one more addition completes the work. Checking your work This tried-and-true tip is old as the hills and twice as dusty.
To check addition, add the column in reverse order. You should get the same sum.
Important Formulas in Algebra
The answer should be the same. Addition and Subtraction Subtraction: Just Another Kind of Addition Subtraction is the process of removing a quantity from a quantity.
Subtraction is the second math operation you grew up with besides addition. The words essentially mean the same thing and you may see either of them in subtraction problems. A remainder is the balance of a quantity left after it has been reduced by subtraction. A difference is a numeric comparison between two quantities. The number you have left is the remainder. The number you subtract from which is usually the larger number is called the minuend.
The number you are subtracting is the subtrahend. When you subtract, use a minus sign —. A subtraction problem in a column looks like this: Like addition, subtraction works with all kinds of numbers — integers, zero, rational numbers, fractions, and irrational numbers.
Basic Math, Basic Tools Subtracting a positive is the same as adding a negative What does it mean to say that subtraction is the inverse of addition? This may not seem like a big deal now, but it becomes important when you get to algebra see Chapter For more info about positive and negative numbers, flip to Chapter 3.
Subtracting negative numbers You can subtract negative numbers. The peculiar thing is that subtracting a negative is like adding a positive. In algebra, you express the idea as —a — —b The catch is that subtracting a negative number changes the sign. Subtracting 0 from 0 is equal to 0. Subtracting multiple items You can subtract multiple items all at once, but be careful — doing so may be a little confusing.
Addition and Subtraction This column of subtractions really represents multiple individual subtractions. Subtracting multiple items is more obvious in a spreadsheet. Figure shows subtraction in Microsoft Excel. Subtracting numbers in Microsoft Excel.
Spreadsheets and printing calculators tell you the whole story, but pocket calculators and smartphones only show the results so far. You can take shortcuts, such as adding all the negatives first.
In the example you can first combine the negatives. Basic Math, Basic Tools The subtractions are lumped-together negatives, and you subtract them from the positive amount: The answer is still 3, just as before.
In subtraction, you do it all the time to make subtraction easier. The term borrowing refers to converting one unit from the next position at the left into the units you are working with. As you know, the positions of the numbers are the ones column, the tens column, the hundreds column and so forth. You can freely borrow from the column at the left of the column you are working in. To illustrate this, say you want to do this subtraction: Look at the ones column.
If you were subtracting 3 from 6, you could easily do it in your head. But how can you take 6 from 3? But where do you come up with 13?
Look at the subtraction problem this way: To make the 3 in the ones column into 13, just borrow a single 10 from the tens column. Addition and Subtraction Now you have an answer for the ones column. How do you take 50 from 10? That reduces the hundreds column by , and the problem looks like this: The hundreds column is a no-problem column. Just subtract. The problem now is When you lump the numbers together again, you see the answer is Or to put it another way: Polonius gives some good advice to his son, Laertes, before the boy goes off to college in Paris.
Just add the difference back to the subtrahend to get the minuend. It should be the same number you started out to subtract from. For example, in the problem: The answer is , the number you started out with. Basic Math, Basic Tools Example: Flour Power You have a job at the Berkeley Artisan Bakery, a large commercial bakery despite its name with many exotic types of flour in stock, including Flour Amount Almond flour kilograms kg Amaranth flour 50 kilograms Atta flour 25 kilograms Bean flour 25 kilograms Brown rice flour kilograms Buckwheat flour kilograms Cassava flour 10 kilograms How much flour do you have?
This calculation is simple, as are most addition problems. Form a column with the flour amounts. Add the amounts. Nothing to it! The answer is kilograms. Addition and Subtraction Example: This morning, you counted 1, sheep in the pens, and now you have a smaller number.
Your supervisor asks you how many trucks picked up the missing sheep today. Can you tell him? But you can get there from here. Subtract the number of sheep left from the number of sheep you had at the start of the day. The difference is the number of sheep trucked away. Figure out how many trucks hauled the sheep away.
Do this part in your head. If sheep are gone and each truck holds sheep, how many trucks picked up sheep? The answer is nine trucks. They used black sheep as markers, and the herder would only count the black ones. If they were all with the herd, chances were all of the sheep were together. If a black sheep was missing, the herder and his dogs would set out in search of the missing black sheep and whatever other sheep had gone with this marker. Chapter 5 Multiplication and Division: The good news is that just about everybody learned multiplication and division in elementary school.
The bad news is that many people have forgotten how to do these operations. If you had problems with multiplication and division in elementary, middle, or high school, be troubled no more. And the conversion of all weights and measures requires multiplication and division. Check out Chapter 6 for more on measurement and conversion. Basic Math, Basic Tools In this chapter, you discover the names of the parts of multiplication and division equations as well as a couple of different ways to multiply and divide.
Go Forth and Multiply! You can mumble something about forgetting something and make a mad dash to your car, or you can use the skills in the big carbon-based calculator in your head, your brain. The following sections show you how to take the latter route. Multiplication is just a form of repeated adding. Every art and craft has its special words, and multiplication is no different.
In multiplication, the number to be multiplied is called the multiplicand, and the number doing the multiplying is the multiplier. The result is the product. Here, is the multiplicand, is the multiplier, and , is the product. The numbers and are factors of , One popular online source says that multiplication was documented in the Egyptian, Greek, Babylonian, Indus valley, and Chinese civilizations.
And the Ishango bone, found in the then-Belgian Congo in and dated to about 18, to 20, BC, has marks that may suggest knowledge of addition and multiplication. Just speculation about the use, but the marks are real. You run across a variety of math symbols that represent multiplication. They all mean the same thing: Sometimes, depending on the problem, you can more easily and cleanly show multiplication by using one symbol rather than another. Chapter 5: The asterisk is used mainly on computers, adding machines, and on some calculators.
Here are these signs in action: This situation occurs in algebra when numbers and letters appear together, such as in the term 3ab.
We cover algebraic variables more thoroughly in Chapter Sometimes multiplication is represented as a grid, so you can see the numbers represented as rows and columns. Multiplication shown as a grid of objects. Memorizing multiplication tables: Faster than a calculator Sources say that the Chinese invented the multiplication table. But regardless of who came up with it, you should commit the multiplication table to memory. Look at that last point. Basic Math, Basic Tools Figure shows a classic multiplication table.
You can also find 12 x 12 and 20 x 20 tables. Classic 9 x 9 multiplication table. There are full explanations of intermediate statistical ideas. Throughout the book are many examples that use statistical software to analyze the data. In each case, the computer output is given as well as an explanation of how the output was achieved and what it means. There are an extensive number of examples to cover the many different types of problems you will face.
Lots of tips, strategies, and warnings.
The setup of the book allows you to skip around and still have easy access and understanding of any given topic. The book uses understandable language and tries to keep things conversational to help you understand, remember, and practise statistical definitions, techniques, and processes.
Finally there are clear and concise step-by-step procedures. For technical support, please visit www. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. He has been a carpenter for the movies, a stage electrician, a movie theater manager, a shipping clerk, an insurance clerk, and a library clerk.
Recently, his company worked with the California Integrated Waste Management Board to teach scientists and administrators how to write clearly. He was a movie reviewer for the L. Herald-Dispatch and wrote a monthly political newspaper column for The Union of Grass Valley, California, for seven years. Bradley Simkins was born and raised in Sacramento, California, and became a sixth-generation journeyman plasterer.
He and his family live in Sacramento, where he owns Book Lovers Bookstore, an independent bookstore. Norm Andersen math ; Mrs. Eada Silverthorne English ; Ms. Susan A. Schwarz English ; Mr. Norman E.Smaller dividends take less time. Like addition, subtraction works with all kinds of numbers — integers, zero, rational numbers, fractions, and irrational numbers.
Well, being a hunter-gatherer is all very fine, but to tweak an old song , we can show you a better time. Driving a car is hard, but most people can do it. Eliminate excess information So really, having a math phobia is even more reason to do math.
He has been a carpenter for the movies, a stage electrician, a movie theater manager, a shipping clerk, an insurance clerk, and a library clerk.