2. 2. The number 2 is to the right of –13. So 2 has a greater value than –13. This is written 2 > –13.
3. – 0.003. The following number line shows that the number –0.003 is to the right of –0.03, which means –0.003 is bigger than –0.03. You can also rewrite –0.03 as –0.030 for easier comparison. The original statement is written –0.003 > –0.03.
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Operations in algebra are nothing like operations in hospitals. Well, you get to dissect things in both, but dissecting numbers is a whole lot easier (and a lot less messy) than dissecting things in a hospital.
Algebra is just a way of generalizing arithmetic, so the operations and rules used in arithmetic work the same for algebra. Some new operations do crop up in algebra, though, just to make things more interesting than adding, subtracting, multiplying, and dividing. I introduce three of those new operations after explaining the difference between a binary operation and a nonbinary operation.
Breaking into binary operations
Bi means two. A bicycle has two wheels. A bigamist has two spouses. A binary operation involves two numbers. Addition, subtraction, multiplication, and division are all binary operations because you need two numbers to perform them. You can add 3 + 4, but you can’t add 3 + if there’s nothing after the plus sign. You need another number.
Introducing nonbinary operations
A nonbinary operation needs just one number to accomplish what it does. A nonbinary operation performs a task and spits out the answer. Square roots are nonbinary operations. You find
Getting it absolutely right with absolute value
One of the most frequently used nonbinary operations is the one that finds the absolute value of a number – its value without a sign. The absolute value tells you how far a number is from 0. It doesn’t pay any attention to whether the number is less than or greater than 0; it just determines how far it is from 0.
Remember: The symbol for absolute value is two vertical bars:
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Here are some examples of the absolute-value operation:
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Basically, the absolute-value operation gives you an undirected distance – the distance from 0 without regard to direction.
Getting the facts straight with factorial
The factorial operation looks like someone took you by surprise. You indicate that you want to perform the operation by putting an exclamation point after a number. If you want 6 factorial, you write 6!. Okay, I’ve given you the symbol, but you need to know what to do with it.
Remember: To find the value of n!, you multiply that number by every positive integer smaller than n.
Here are some examples of the factorial operation:
✓ 3! = 3 · 2 · 1 = 6
✓ 6! = 6 · 5 · 4 · 3 · 2 · 1 = 720
✓ 7! = 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5,040
The value of 0! is 1. This result doesn’t really fit the rule for computing the factorial, but the mathematicians who first described the factorial operation designated that 0! is equal to 1 so that it worked with their formulas involving permutations, combinations, and probability.
Getting the most for your math with the greatest integer
You may have never used the greatest integer function before, but you’ve certainly been its victim. Utility and phone companies and sales tax schedules use this function to get rid of fractional values. Do the fractions get dropped off? Why, of course not. The amount is rounded up to the next greatest integer.
Remember: The greatest integer function takes any real number that isn’t an integer and changes it to the greatest integer it exceeds. If the number is already an integer, then it stays the same.
Here are some examples of the greatest integer function at work:
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You may have done a double-take for the result of using the function on –3.87. Just picture the number line. The number –3.87 is to the right of –4, so the greatest integer not exceeding –3.87 is –4. In fact, a good way to compute the greatest integer is to picture the value’s position on the number line and slide back to the closest integer to the left – if the value isn’t already an integer.
Practice Questions
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Practice Answers
1. 8. 8 > 0.
2. 6. –6 < 0 and 6 is the opposite of –6.
3. 3. 3 is the largest integer smaller than 3.25.
4. – 4. –4 is smaller than –3.25, is to the left of –3.25, and is the largest integer that’s smaller than –3.25.
If