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Negative Numbers, Opposites and Additive Inverses Additive Inverse In the History of Numbers presentation we talked about Brahmagupta who showed that subtracting a number from itself results in zero. We call that number the additive inverse. The additive inverse is the opposite or negative of a number and the sum of a number and its additive inverse is zero. Examples: 2+-2 = 0 -3+3 = 0 Negative Integers We know that each positive integer has a corresponding negative integer. These corresponding integers are called opposites. They are also called additive inverses. Examples: -2 is the opposite of 2 and -2 + 2 = 0 15 is the opposite of -15 and 15 + -15 = 0 We may say that 2 is the opposite of -2, -15 is the opposite of 15 and so on. They are opposites of each other. The words opposite, additive inverse, negative and minus are often used in the same sense. Example: -2 may ● ● ● ● Other examples? be read in any of the following ways: “negative 2” “minus 2” “the opposite of 2” “the additive inverse of 2” Uhhhh, so why do we care……. We care, because now we are going to be doing some work with positive and negative numbers and we need some rules. Three 1. 2. 3. rules we need are: Same-sign rule Mixed-sign rule Neighbor-sign rule Same-sign rule When you are combining numbers with the same sign, keep the sign and add the numbers. Again, we start with the number line…… Let’s add two positive numbers: 3 + 2 = + 3 + 2= +(3 + 2) = 5 Let’s add two negative numbers: -3 + -2 = -(3 + 2) = -5 Examples? Steps to the same-sign rule 1. Ask yourself: Is the sign of the numbers positive or negative? 2. Whichever sign the numbers have, put down that sign followed by a set of parentheses. 3. Inside the parentheses, place the numbers and stick a plus sign between them. 4. Add the numbers, take away the parentheses and there is your answer. Examples: 1. 18 + 20 2. - 15 - 5 3. 7 + 51 4. 100 + 256 5. - 18 - 5 6. - 8 - 9 Real life example! When working out this rule with positive numbers, imagine that you receive money from different people. Example: Suppose you receive $2 from one person and $3 from another person. Altogether you have $5, so the answer is +5 When working with negative numbers, imagine that you owe money to different people. Example: Suppose you owe $3 to one friend and you also owe $2 to another friend. Altogether you owe $5, so the answer is -5. Owing money is a pain, so it’s negative! Mixed-sign rule When you are combining numbers with mixed signs, ignore the signs and see which number is bigger. Take the sign of the bigger number and write it down in front of a set of parenthesis. Then, inside the parenthesis, subtract the smaller number from the larger number. Example: 5 - 9 = -(9 - 5) = -4 -3 + 10 = +(10 - 3) = +7 Examples? Steps to the Mixed-sign rule 1. Ignore the signs, and see which number is bigger. 2. Take the sign in front of that larger number and place it in front of a set of parentheses. 3. Inside the parentheses and still ignoring the signs, place the larger number first and subtract the smaller number from it. 4. Subtract the numbers, take away the parenthesis and there is your answer. Examples: 1. 18 - 2 2. -15 + 8 3. 22 - 24 4. -88 + 10 5. -9 + 15 6. -10 + 17 Neighbor-Sign Rule You use the neighbor-sign rule when two signs stand next to each other with no number between them. You might see it like this: 2 - (+3) or 2 - +3. The two signs merge to become one. Here is the pattern: + + turns into + + - turns into - + turns into - - turns into + Steps to the Neighbor-sign rule 1. Look at the two neighboring signs. 2. Using the patterns, determine which sign the two signs will become. 3. Change the signs into one sign. 4. Work out your answer using either the same-sign rule or the mixed-sign rule. Note: The Neighbor-sign rule has no bearing on what the final sign of the answer will be. Examples: 1. 13 - (-4) 2. 15 + (-3) 3. -5 + (+2) 4. -7 - (+4) 5. -10 - (+8) 6. 12 + (-9)