Lecture Note
University
Indiana University South BendCourse
MATH-A 100 | Fundamentals of AlgebraPages
6
Academic year
2020
Rose G
Views
11
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 So -|13| = -13 Example 2: -|-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 -|-7| = -7 Example 3: |-(-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 |-(-25)| = 25 Example 4: |-(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.
Dealing with Division Involving Zero: Special Study Cases
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