Math A-100 IUSB Sec.1.4 Notes – Week 1 Adding Signed numbers Try not to remember any “rules” when you are adding & subtracting signed numbers. That often leads to using the wrong “rule” in the wrong place. Here’s some tips students have told me they use: 1) think of it as money. Example 1: 7 + -5 = ? think of 7 + -5 as “you have 7 dollars and was charged 5 dollars” How much do you have left? 2 dollars So 7 + -5 = 2 Example 2: -7 + 5 = ? think of -7 + 5 as “you owe 7 dollars and to get paid 5 dollars” How much do you have left? You still owe 2 dollars So -7 + 5 = - 2 Example 3: -7 + -5 = ? think of -7 + -5 as “you owe 7 dollars and charge another 5 dollars” How much do you have left? You now owe 12 dollars So -7 + -5 = - 12 2) Look at what you have more of, positives or negatives. Whichever one you have more of will be the sign of your answer. Example 1: 7 + -5 = ? You have more “positives” than you do “negatives” so your answer is positive So 7 + -5 = 2 Example 2: -7 + 5 = ? You have more “negatives” here than you do “positives” so your answer is negative So -7 + 5 = - 2 Subtracting Signed numbers There are several ways to approach subtracting signed numbers. You can: A) Think of it as “opposite of” B) make any subtraction question into an addition question, by simply placing the + sign in front of the “-“ sign NOTE: MyMathLab (a.k.a. MML) puts ( ) around the number being subtracted, for example 15 – (-3). This is to highlight that fact that the second number is negative so you don’t miss that there are two “-“ signs. Example 3: 7 – (-5) = ? Option A: -(-5) means “the opposite of -5”, which is + 5, so you have 7 + 5 = 12 Option B: 7 – (-5) becomes 7 + -(-5) which is 7 + 5 = 12 Example 4: - 7 – (-5) = ? Option A: -(-5) means “the opposite of -5”, which is + 5, so you have -7 + 5 = -2 Option B: - 7 – (-5) becomes -7 + -(-5) which is -7 + 5 = -2 Example 5: 7 – 13 = ? Option A: - 13 can mean “the opposite of 13”, which is - 13, so you have 7 + -13 = -6 Option B: 7 – 13 becomes 7 + - 13 which is 7 + - 13 = -6 Example 6: - 7 – 13 = ? Option A: -(-5) means “the opposite of -5”, which is + 5, so you have -7 + 5 = -2 Option B: - 7 – 13 becomes -7 + - 13 which is -7 + - 13 = -20 NOTE: I have noticed that when students try to remember “rules” for adding & subtracting, they often think of the rules for multiplying and dividing instead!! That is why I recommend thinking of it as “money” instead. Multiplying & Dividing Signed Numbers Multiplying & Dividing Signed number involves one of more “-“ signs: 1 “-“ sign means “the opposite of” 2 “-“ signs means “the opposite of the opposite of”, in other words they cancel each other out 3 “-“ signs means “the opposite of the opposite of the opposite of”; two of them cancel out, but the third is still there, so you still have one 1 “-“ sign 4 “-“ signs means “the opposite of the opposite of the opposite of the opposite of”, in other words two sets of 2 “-“ signs Etc. Short cut: if you have an odd number of “-“ signs, all but one cancel each other out, but you still have 1 “-“ sign left, so the answer is negative if you have an even number of “-“ signs, they all cancel each other out, so the answer is positive Example 1: 8 x (-5) x (-2) Two “-“ signs, so they cancel out  answer is positive 8 x (-5) x (-2) = 80 Example 2: 8 x (-5) x (-2) x (-1/4) three “-“ signs, so they all cancel out except one “-“ sign  answer is negative 8 x (-5) x (-2) x (-1/4) = -20 Division questions that include 0 If 0 is included in a division question, then you have one of two special cases: 1) 0 divided by a number – dividing means breaking too smaller groups  If you start with 0 things, smaller groups of 0 are still 0.  The answer = 0 2) A number divided by 0 number – dividing means breaking too smaller groups  How do you break up a group into 0 groups?????? You can’t, so  The answer = “undefined”, since we don’t know how to process it Example1: 0/4 = 0 broken into 4 groups = 0 in each group = 0 Example 2: 0/-6 = 0 broken into 6 groups = 0 in each group = 0 Example 3: 4/0 = 4 broken into 0 groups = how do you have 0 groups? = undefined Example 4: -6/0 = -6 broken into 0 groups = how do you have 0 groups? = undefined “Opposite” vs Absolute value The “opposite” of a number is also called the “Additive Inverse”. It means “what do you add to your number to get 0 as the answer. We usually think of “opposites” as simply “having the other sign”. Examples: The opposite of 8 is -8 The opposite of -13 is +13 or just 13 We often use the “-“ sign to represent “opposite of”, especially if there are ( ) involved. Examples: -13 is “negative 13, but -(13) is “the opposite of” 13 - (-4) is “the opposite of” – 4 -[ - ( -6)] is “the opposite of” is “the opposite of” -6 Absolute Value The absolute value of a number is how many places the number is from 0 on the number line, in either direction (i.e. the direction does not matter). We represent the “absolute value of” a number by putting it in between two straight lines, i.e. | | Examples: |13| is “absolute value of 13. Since 13 is 13 places from 0 on the number line, then |13| = 13 |-13| is “absolute value of -13. Since -13 is also 13 places from 0 on the number line, then |-13| = 13 |4/5| is “absolute value of 4/5. Since 4/5 is 4/5 of a plac from 0 on the number line, then |4/5| = 4/5 |-2/3| is “absolute value of -2/3. Since -2/3 is 28/3 of a place from 0 on the number line, then |-2/3| = 2/3 NOTE: “distance”, i.e. the number of places away from 0 on the number line, cannot be negative (how do you travel -10 miles?). Since absolute value measures the distance (i.e. number of places) a number is from 0, then absolute value, i.e. the number coming out of the | |, is always positive. Combining “Opposite” with Absolute value Since “opposite of” and absolute value are NOT the same thing, we can combine them. Start from the inside and work your way out. Example 1: -|13| is the “opposite of” the absolute value of 13. Start with the inside: |13| |13| = 13 or maybe think of it as |13| = (13) Then do the “-“ out front” -(13) is -13 Example 2: So -|13| = -13 -|-7| is the “opposite of” the absolute value of -7. Start with the inside: |-7| |-7| = 7 or maybe think of it as |-7| = (7) Then do the “-“ out front” -(7) is -7 So Example 3: -|-7| = -7 |-(-25)| is the absolute value of the “opposite of” -25. Start with the inside: –(-25): -(-25) = 25 Then do the “| |“ out front” |25| is 25 So Example 4: |-(-25)| = 25 |-(10)| is the absolute value of the “opposite of” 10. Start with the inside: –(10): -(10) = -10 Then do the “| |“ out front” |-10| is 10 So |-(10)| = 10 NOTE: The Absolute Value does NOT mean “the opposite”. They are completely different calculations.