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Going in for Operations

      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.

      Sorting out types of operations

      Operations in mathematics come in all shapes and sizes. There are the basic operations that you first ran into when you started school, and then you have the operations that are special to one branch of mathematics or another. The operations are universal; they work in all languages and at all levels of math.

      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 , 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 by performing this operation on just one number (see Chapter 6 for more on square roots). Another important nonbinary operation is absolute value. It will be used in the upcoming sections, where you subtract numbers. And two other important nonbinary operations are factorial and greatest integer. It gets better and better!

      Getting it absolutely right with absolute value

      The absolute value operation, indicated by two vertical bars around a number, , is greatly related to the number line, because it tells you how far a number is from 0 without any regard to the sign of the number. The absolute value of a number is its value without a sign. The absolute value 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.

      The formal definition of the absolute value operation is:

      So, essentially, if a number is positive or 0, then its absolute value is exactly that number. If the number you’re evaluating is negative, then you find its opposite — or you make it a positive number.

      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.

      To find the value of n!, you multiply that number by every positive integer smaller than n.

      There’s one special rule when using factorial: . This is by definition. The value of 0! is designated as being 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.

      

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.

      The symbol for the greatest integer function is a set of brackets, . You put your number in question in the brackets, evaluate it, and out pops the answer.

      

Q. Find the absolute value:

      A. . The distance from –4 to 0 is 4 units.

      Q. Evaluate:

      A. . You perform the operation inside the absolute value bars before evaluating.

      Q. Evaluate 3!

      A.

      Q. Evaluate 6!

      A.

      Q. Evaluate:

      A. . The number 6 is the biggest integer that is not larger than .

      Q. Evaluate:

      A. . 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.

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