Positive and Negative Integers – Part 2 To quickly review, here were the major ideas from part 1: •
An integer is a number without any fraction or decimal part to it. It can be positive, negative, or zero. (…‐3, ‐2, ‐1, 0, 1, 2, 3, …)
•
Every integer has an opposite which is the same distance away from zero. (3 and ‐3, 17 and ‐17, etc. )
•
Subtracting any value is the same thing as adding it’s opposite. a – b = a + (‐b)
•
Adding and subtracting integers can be done by using a number line or by memorizing a set of steps. There are lots of different ways to combine integers, so pick the one that is easiest and makes the most sense to you.
In part 2 we are going to address the following two things: absolute value and multiplication/division. We will start first with absolute value since it fits in so nicely with what we have already talked about. Consider the following number line with two points labeled A and B.
A
B
What is the value of A? What is the value of B? I hope you said A was negative 4 and B was positive 4. Now if I asked you what these had in common, the answer would be pretty simple. They are both 4’s. But here’s whats more important…how far away from zero are they? If you count the number of spaces from 0 to point B, you will get 4 units. If you count the number of spaces from 0 to point A, you will also get 4 units. This value is what we call the absolute value. The absolute value of an integer is equal to it’s distance from zero on the number line. We denote absolute value by two vertical lines like this: A = 4 or B = 4 You probably noticed that A and B are the same number. That is OK since they are both the same distance away from zero. Any number and it’s opposite will be the same distance from zero and therefore have the same absolute value.
Think about it this way. If you drew a line between 0 and point A then pulled out a ruler and measured it, how long would that line be? Do the same with point B. They’re the same length right? When you measure distance, you usually just use positive lengths…the direction doesn’t really have any impact on the length of the line. It doesn’t matter if you draw it to the left of 0 or to the right of 0, you get 4 units either way. This brings up an important point. When you think about absolute value, think about the words length, distance, and magnitude. Absolute value has to do with the “size” of a value or the length between two points. The reason to mention this now is that in the far future you will see new operations that use that same
symbol but will do something a little different. Even so, these operations will still be related to this idea of size and length. So let’s look at some more examples. What if I asked you to find the absolute value of ‐23? Well, you could draw a number line, put a point at ‐23 then count how many units away from zero it is. Or…you could just take off the negative sign. We already know how far ‐23 is from zero. IT’S 23 UNITS AWAY! Remember, the direction doesn’t matter. All we care about with absolute value is the distance from zero. So in this case we would say the absolute value of ‐23 is 23. Or, using our symbols: −23 = 23 . Here are some more examples. Read through these and make sure they all make sense.
8 =8
345 = 345
−57 = 57 1 1 − = 2 2
−1924 = 1924
−87.3 = 87.3
Notice anything in the last two? Absolute value is not just for integers. It works with fractions and decimals too! Just like you can measure half an inch or 87.3 meters, you can find the distance from zero for these values. And the last thing to watch out for is this: ABSOLUTE IS ALWAYS POSITIVE! It doesn’t matter what number you put inside the bars, it will always come out positive. When you put in a negative, it turns positive. When you put in a positive, it stays positive.
Now we move on to the next big category in positive and negative numbers: multiplication and division. The nice thing is that this is extremely easy. And, just like with the addition and subtraction rules, it works for any value on the number line, not just with integers. Here it is, the rule for multiplication and division with negatives: If the signs are the same, the answer is positive. If the signs are different, the answer is negative. That’s it. Seriously. You do all the same things you know how to do with multiplication and division and then just figure out if the answer should be positive or negative. Example: 5 ⋅ (−6) So we multiply 5 and 6 to give us 30 then we decide whether its positive or negative. The two numbers we have are +5 and ‐6 so the signs are different. Therefore, using our rule, the answer should be negative. So 5 ⋅ (−6) = −30 Let’s try another with division. How about −84 ÷ (−21) Well, we do the division first so 84 divided by 21 is 4. But is it positive 4 or negative 4? The numbers we are dealing with are ‐84 and ‐21 so their signs are the same. The rule says the answer is positive. So −84 ÷ (−21) = 4 Yes, it is really that easy. If the signs are the same, the result is positive. If the signs are different, the result is negative. Here are some more examples to look at (remember we said this works for fractions and decimals too):
−3 ⋅ 7 = −21
− 8 ÷ 2 = −4
(−5) ⋅ (−8) = 40
(−3.7) ⋅ (2.1) = −7.77
28 ÷ (−4) = −7
(−63) ÷ (−9) = 7
100 ⋅ (−3) = −300
−150 =5 −30
61 ÷ (−75) = −0.8133
36 = −6 −6
−35 2 =1 −21 3
That’s really all there is to it. Though we need to mention the same caution as in part 1. Now that you have seen the rules for adding/subtracting and the rules for multiplying/dividing with negatives you can see how easy it is to confuse the two. When you are practicing, try to practice all the operations together so you are not doing a worksheet of only addition/subtraction or a sheet of only multiplication/division. Practice with all of them together so you can really feel comfortable going back and forth between the different rules without confusion. Be careful, check your work, and practice, practice, practice! Good luck! www.mathmadesimple.org