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      16 Determine which is greater: or .

      17 Determine which is greater: 5! or .

18 Determine which is greater: or .

Tackling the Basic Binary Operations

      What is a binary operation? A bicycle has two wheels. A biannual term lasts two years. And a binary operation requires two numbers. These operations are performed on two numbers — one written before the operation symbol and one after. Addition and subtraction are pretty familiar, but the multiplication and division symbols come in several varieties.

      Adding signed numbers

      If you’re on an elevator in a building that has 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 be elevator operators 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 , they probably didn’t tell you that this was just one part of the whole big addition story. They didn’t mention that adding one positive number to another positive number is really a special case. If they 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:You now have seven CDs.

       You owed Jon $8 and had to borrow $2 more from him:Now you’re $10 in debt.

      

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:

       : The signs are both positive, and so is the sum.

       : The sign of the sum is the same as the signs.

       : Because all the numbers are positive, add them and make the sum positive, too.

       : This time all the numbers are negative, so add them and give the sum a minus sign.

      Adding same-signed numbers is a snap! (A little more alliteration for you.)

      Adding different signs

      Can a relationship between a Leo and a Gemini ever add up to anything? I don’t know the answer to that question, but I do know that numbers with different signs add up very nicely. You just have to know how to do the computation, and, in this section, I tell you.

      

When the signs of two numbers are different, forget the signs for a while and find the difference between the numbers. This is the difference between their absolute values (see the “Getting it absolutely right with absolute value” section, earlier in this chapter). The number farther from 0 determines the sign of the answer.

        if the positive a is farther from 0.

        if the negative b is farther from 0.

      Look what happens when you add numbers with different signs:

       You had $20 in your wallet and spent $12 for your movie ticket:After settling up, you have $8 left. You knew the answer would be positive, because +20 is farther from 0 than –12, and the difference between 20 and 12 is 8.

       I have $20, but it costs $32 to fill my car’s gas tank:I’ll have to borrow $12 to fill the tank. This time the answer will be negative, because –32 is farther from 0 than +20. The difference between the two numbers is 12.

Here’s how to solve these two situations using the rules for adding signed numbers.

       : Find the difference between 20 and 12: . Because 20 is farther from 0 than 12, the result is positive, so .

       : Find the difference between 20 and 32: . Because 32 is farther from 0 than 20 and is a negative number, the result is negative, so .

      Here are some more examples of finding the sums of numbers with different signs:

       : The difference between 6 and 7 is 1. Because 7 is farther from 0 than 6 is, and 7 is negative, the answer is –1.

       : This time the 7 is positive. It’s still farther from 0 than 6 is, and so the answer is +1.

       : If you take these operations in order from left to right (although you can add in any order you like), you add the first two together to get –1. Add –1 to the +7 to get +6. Then add +6 to –5, the last number, to get +1.

      

Q.

      A. The signs are the same, so you find the sum and apply the common sign. The answer is –10.

      Q.

      A. The signs are different, so you find the difference and use the sign of the number with the larger absolute value. The answer is –7.

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