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      Making a Difference with Signed Numbers

      Subtracting signed numbers is really easy to do: You don’t! Instead of inventing a new set of rules for subtracting signed numbers, mathematicians determined that it’s easier to change the subtraction problems to addition problems and use the rules that you find in the previous section. Think of it as an original form of recycling.

      Consider the method for subtracting signed numbers for a moment. Just change the subtraction problem into an addition problem? It doesn’t make much sense, does it? Everybody knows that you can’t just change an arithmetic operation and expect to get the same or right answer. You found out a long time ago that isn’t the same as . You can’t just change the operation and expect it to come out correctly.

      So, to make this work, you really change two things. (It almost seems to fly in the face of two wrongs don’t make a right, doesn’t it?)

When subtracting signed numbers, change the minus sign to a plus sign and change the number that the minus sign was in front of (the second number) to its opposite. Then just add the numbers using the rules for adding signed numbers.

       

       

       

       

      

These first examples put the process of subtracting signed numbers into real-life terms:

      Q. The submarine was 60 feet below the surface when the skipper shouted, “Dive!” It went down another 40 feet. What is the submarine’s depth now?

      A. . Change from subtraction to addition. Change the 40 to its opposite, –40. Then use the addition rule. The submarine is now 100 feet below the surface.

      Q. Some kids are pretending that they’re on a reality-TV program and clinging to some footholds on a climbing wall. One team challenges the position of the opposing team’s player. “You were supposed to go down 3 feet, then up 8 feet, then down 4 feet. You shouldn’t be 1 foot higher than where you started!” The referee decides to check by having the player go backward, by making the player do the opposite, or subtracting the moves. What was the result?

      A. Putting a negative sign in front of each assigned move, you have: . The player ended up 1 foot lower than where they started, so they had moved correctly in the first place.

      And now here are some examples of subtracting signed numbers:

      Q. Solve:

      A. . The subtraction becomes addition, and the +4 becomes negative. Then, because you’re adding two signed numbers with the same sign, you find the sum and attach their common negative sign.

      Q. Solve:

      A. . The subtraction becomes addition, and the –5 becomes positive. When adding numbers with opposite signs, you find their difference. The 2 is positive because the +5 is farther from 0.

      

To subtract two signed numbers:

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      Multiplying Signed Numbers

      When you multiply two or more signed numbers, you just multiply them without worrying about the sign of the answer until the end. Then, to assign the sign, just count the number of negative signs in the problem. If the number of negative signs is an even number, the answer is positive. If the number of negative signs is odd, the answer is negative.

      

The product of two signed numbers:

      The product of more than two signed numbers:

        has a positive answer because there are an even number of negative factors.

        has a negative answer because there are an odd number of negative factors.

      

Q.

      A. Multiply the two factors without their signs, and you get 6. There are two negative signs in the problem, so the result is positive. The answer is +6.

      Q.

      A. There are three negative signs

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