Pretty straightforward so far, because you don’t need a placeholder. But what happens when you have one hundred and three single items? How do you write that without a placeholder?
✓ You can write 13, but that’s misleading and just plain wrong.
✓ You can try 100 + 3, but such a system of notation makes math opera- tions much tougher.
✓ You can try Roman numerals and write 103 as CIII, but the disadvantages of Roman numerals (with only seven symbols and no zero) are many and make the system a poor candidate for math.
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 103.
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.
There are vestiges of the base 12 system in today’s twelve-hour clocks. 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:
✓ Whenever you multiply anything by zero, the answer is always zero. For example, 3 × 0 = 0; 274,561 × 0 = 0; and so forth.
✓ When you add 0 to a number, the answer is the same number. For exam- ple, 2 + 0 = 2, 27 + 0 = 27, and so forth.
✓ Any number raised to the power of 0 is 1. For example 7560 = 1, and 70 = 1.
See Chapter 11 for more on powers and exponents.
See how nice zero can make your math life? Anytime you’re solving a math problem, look for zero. It doesn’t look like much, but it can help you.
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Chapter 3: Zero to One and Beyond
Going Backward: Negative Numbers
As we mention earlier in the chapter, a negative number is a number that’s less than zero. You represent negative numbers with a minus sign; for exam- ple, –1, –23, and –8,542. Zero isn’t negative (but it’s not positive, either).
Negative numbers may seem like a fantasy concept, but they’re very real in many lines of work.
Working with negative numbers
In mathematics, negative numbers are a concept. But concepts don’t put bread on the table (unless you’re a mathematician). In your daily activities, you work with negative numbers in the real sense, and they almost always represent a reduction or a deficit.
In some trades, when a positive quantity decreases, the math “stops” at zero.
In parts management, food management, or hospital stockroom management, when you have 0 of something, you’re all out. There’s no concept of negative widgits, negative eggs, or negative IV solutions. But the reason you get to 0 units is because of inventory draws (reductions), and each reduction in inven- tory 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. Low income and high expenses can occur in the construction trades, the automotive trades, and even in a doctor’s office.
Negative cash flow is a real and painful concept. And if the business checking account is overdrawn, that’s a very serious negative number.
Negative numbers aren’t always grim. For example, the countdown — the process of counting hours, minutes, and seconds backward until something happens — is a “positive” application of negative numbers. It’s part of NASA rocket launches, adds drama to action movies, and announces the start of each new year.
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 posi- tive numbers to the right of 0, and mark negative numbers to the left of zero.
Figure 3-1 shows a number line.
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Figure 3-1:
A number
line. –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9
The farther to the right of 0 on the number line you go, the larger the num- bers get in value. 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.
And although negative numbers may seem to be the opposite of positive numbers, they act the same, and you can do the same math operations with them.
Getting Between the Integers: Fractions, Decimals, and More
Life was simpler in the third grade with only integers to deal with. But then again, you didn’t get a paycheck for attending the third grade. 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 frac- tions, decimal fractions, rational numbers, and irrational numbers. Although integers are nice, clean numbers to work with, you can’t ignore the numbers in between them.
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:
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Chapter 3: Zero to One and Beyond
and decimal fractions, which look like 0.46, 0.375, or 0.87695. If you combine a whole number with a fractional number, the result is called a mixed number.
For example, 5.243, $14.95, or
Check out Chapters 8 and 9 for more on fractions.
Figure 3-2 shows how fractions on the number line fall between the integers.
Figure 3-2:
Fractions on the number
line. –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3
–1½ 3.557 7¾
4 5 6 7 8 9
The leap from 0 to 1 is “one small step for math, one giant step for mathkind.”
It looks small, but in between 0 and 1 are many fractions (an infinite number, as it turns out).
The rational numbers(and their irrational friends)
On the job and in your personal life, you have two kinds of friends: rational and irrational. 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: You’re better off knowing both.
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 0.75? This decimal number is really the frac- tion seventy-five one hundredths, and when shown as a common fraction:
you see that it’s a ratio.
Following are examples of rational numbers:
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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 deci- mals in a repeating sequence. For example:
An irrational number is always acting out. It won’t let you express it as a simple fraction, and as a decimal fraction the digits go on forever in no repeating sequence. As you see, the irrational numbers don’t behave in a rational way.
The most famous irrational number is π (pi, the Greek letter, pronounced pie). Pi the ratio of the diameter of a circle to its circumference. You were first exposed to π in grade school, and you may use it in work if you calculate circular areas or the volumes of cylinders.
Pi has been calculated to over one trillion (!) decimal places, and the calcula- tions still don’t come out evenly. 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 1897, the Indiana House of Representatives considered a bill that would have set pi to a value of 3.2.
Two other numbers, Euler’s Number (the number e) and the Golden Ratio (represented by the Greek letter phi, ϕ), are also famous irrational numbers.
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.
One type describes everything we’ve discussed in the chapter so far, two types are never used except by mathematicians, and (to finish on a positive note) you use the last two types every day.
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, neg- ative numbers, zero, rational numbers, and irrational numbers. They’re all real, meaning you can find any of these numbers somewhere on the number line.
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Chapter 3: 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. In real life, you can’t square a number and get –1, but in conceptual life, you can. 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. In the 1600s, mathematician René Descartes wrote about them. He used the term “imagi- nary,” and he didn’t mean it as a compliment.
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.
Complex numbers
These aren’t numbers with a psychological disorder. But they’re not simple numbers, either. A complex number is a combination of a real number and an imaginary number.
You write it in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit.
Nominal numbers
A nominal number (sometimes called a categorical number) is a number you use for identification only. It doesn’t matter what the value of the number is.
Here are some examples of nominal numbers:
✓ Social Security Number
✓ Vehicle Identification Number
✓ Drivers License Number
✓ Inventory part numbers
✓ Universal Product Code
✓ The combination for a lock or safe. (Some locks and safes even use letters instead of numbers.)
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Handling Numerical Story Problems
Story problems (sometimes called word problems or life problems) are math problems where the details are presented more as a story than with straight figures. Not to worry. Story problems aren’t hard to solve. Check Chapter 7 for all the details about story problems.
Example: Automotive tech — a slippery task
You work at a BMW motorcycle dealership. You hope to study more auto- motive 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 plas- tic containers of oil you have.