Algebra I All-in-One For Dummies. Mary Jane Sterling

, you can identify the coefficient and work on it to find a multiplicative inverse. The multiplicative inverse of is 9. So multiply both terms by 9.

      Notice how the distributive function works for you here!

      28 Use an additive identity to change the expression to one with only the variable term.

      29 Use an additive identity to change the expression to one with only the variable term.

      30 Use a multiplicative identity to change the term –7x to one with only the variable factor.

31 Use a multiplicative identity to change the expression to one with the variable factor having an integer for a coefficient.

Working with Factorial

      The nonbinary operation called factorial is important in problems involving probability, counting items, and, of course, is a basic function used in algebra. When you see n!, you know to take the number n and multiply it by every natural number smaller than it is:

      The number n must be a whole number. This means that 1! is rather redundant. There’s no natural number smaller, so . And then there’s 0!. There’s no natural number smaller than 0; in fact, 0 isn’t a natural number itself! By definition, . How can that be? Well, mathematicians decided it is so. And it wasn’t arbitrary or a flip-of-the-coin. By assigning 0! the value 1, it makes all sorts of formulas for counting and other applications work consistently.

      A particular challenge when working with factorials is to reduce fractions containing those functions. Basically, you find the common factors in the factorials in the numerator and denominator, eliminate them, and determine what’s left.

      

Q. What is ?

      A. If these were just the two numbers 4 and 3, you would either leave it as is, because 4 and 3 don’t have any common factors other than 1, or you would write this as a mixed number. It’s different with factorials. Rewrite the fraction after expanding the factorial values.

      Q. What is ?

      A. Write out the factorials and reduce the fraction.

      Q. What is ?

      A. The value of is defined to be 1. This is by design/definition.

      Q. What is ?

      A. “Yikes!”, you say. I have to write out the product of all the numbers from 100 down to 1 and then from 97 down to 1? “No,” is the answer. You can take a shortcut. In the numerator, instead of writing from 97 down to 1, just use 97!. Here’s how it works:

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Applying the Greatest Integer Function

      The greatest integer function is one of the nonbinary functions that is frequently used in algebra and its applications. This function is a method of rounding numbers. When you “round” a number to its nearest integer or tenth or thousandth or thousand, and so on, you move up or down to get to the closer value. With the greatest integer function, it doesn’t matter how close, it matters in which direction.

      When rounding numbers, you determine what digits need to be dropped and whether the target place value or number goes up by 1 or stays the same.

      Suppose you want to round 3.667 to the nearest integer. The target number is the 3. Because the next digit, the 6, is greater than 5, you round the 3 up to 4 and eliminate the rest of the digits. You can replace them with 0 or, in this case, just keep the number 4. So, in rounded form, or .

      What if you’re asked to round 1234.567 to the nearest hundred? The target number is the 2. The next digit down is 3, which is smaller than 5, so you leave the 2 as is and replace the 3 and the rest of the smaller digits with 0s. or .

      Note: When the digit below the target digit is exactly 5, and nothing else, you round up.

      The greatest integer function acts a bit differently than rounding. It’s similar in that the greatest integer function eliminates unwanted fractions or decimals, but the greatest integer function only goes in one direction: down. Or, if you already have an integer, the greatest integer leaves the number alone.

      

Q. Which is larger: 5.7 rounded to the nearest integer or [5.7]?

      A. Rounding 5.7 to the nearest integer, you home in on the 7, which will be eliminated. Because 7 is larger than 5, you round up: or . Using the greatest integer function, . So, rounding gives you the larger value.

      Q. Compare