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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 you’re on an elevator in a building with four floors above the ground floor and five floors below ground level, you can have a grand time riding the elevator all day, pushing buttons, and actually “operating” with signed numbers. If you want to go up five floors from the third sub-basement, you end up on the second floor above ground level.
You’re probably too young to remember this, but people actually used to get paid to ride elevators and push buttons all day. I wonder if these people had to understand algebra first…
Adding like to like: Same-signed numbers
When your first-grade teacher taught you that 1 + 1 = 2, she probably didn’t tell you that this was just one part of the whole big addition story. She didn’t mention that adding one positive number to another positive number is really a special case. If she had told you this big-story stuff – that you can add positive and negative numbers together or add any combination of positive and negative numbers together – you might have packed up your little school bag and sack lunch and left the room right then and there.
Adding positive numbers to positive numbers is just a small part of the whole addition story, but it was enough to get you started at that time. This section gives you the big story – all the information you need to add numbers of any sign. The first thing to consider in adding signed numbers is to start with the easiest situation – when the numbers have the same sign. Look at what happens:
✓ You have three CDs and your friend gives you four new CDs:
(+3) + (+4) = +7
You now have seven CDs.
✓ You owed Jon $8 and had to borrow $2 more from him:
(–8) + (–2) = –10
Now you’re $10 in debt.
Tip: There’s a nice S rule for addition of positives to positives and negatives to negatives. See if you can say it quickly three times in a row: When the signs are the same, you find the sum, and the sign of the sum is the same as the signs. This rule holds when a and b represent any two real numbers:
I wish I had something as alliterative for all the rules, but this is math, not poetry!
Say you’re adding –3 and –2. The signs are the same; so you find the sum of 3 and 2, which is 5. The sign of this sum is the same as the signs of –3 and –2, so the sum is also a negative.
Here are some examples of finding the sums of same-signed numbers:
✓ (+8) + (+11) = +19: The signs are all positive.
✓ (–14) + (–100) = –114: The sign of the sum is the same as the signs.
