Addition and Subtraction Starting with a negative number Intro Let us look at how NORMAL addition and subtraction with counting numbers works: Start with addition: 3 + 4 = 7 8 + 2 = 10 4 + 7 = 11 1 + 2 = 3 In each case, we see that the first number is a STARTING POINT The second number is HOW MUCH we move up the number line. We move UP because + (plus) means we get MORE. Real World Examples It was 3 degrees. It went up 4 degrees. It’s now 7 degrees. I had $8. I found $2. I now have $10. I was 4 km east from home (zero). I walked a further 7 km east away from home. I am now 11 km east from home. And now subtraction Let us represent the following problems on a number line 8 – 5 = 3 10 – 2 = 8 11 – 4 = 7 3 – 1 = 2 In each case, we see that the first number is again our STARTING POINT. The second number is HOW MUCH we move down the number line. We move DOWN because – means we have LESS. Real World Examples It was 8 degrees. It got 5 degrees colder. It is now 3 degrees. I was 10 km east from home. I walked 2 km west (closer to home). I am now 8 km from home. I had $11. I spent $4. I now have $7. Now… Sometimes addition and subtraction problems start with a negative number. Note the following problems for addition -‐3 + 6 = 3 (Start at -‐3. Move up 6. You end up at 3. Of the 6 to move up, 3 were used getting to 0. Then there was an extra 3 to still move up.) -‐2 + 8 = 6. (Start at -‐2. Move up 8. You end up at 6. Of the 8 to move up, 2 were used getting to 0. Then there were 6 left to move up.) -‐5 + 10 = 5. (Start at -‐5. Move up 10. You end up at 5). Worded questions that relate to these problems could be: I was $3 in debt. I then won $6. After paying my debt back, I had $3 spare. The temperature was -‐2. It went up 8 degrees. The new temperature is 6 degrees. I started off 5 km west from home. I moved 10 km east. I am now 5 km east of home. Notice that these problems are no different to normal positive numbers, except our starting point is in negative territory. But the idea of + moving UP still applies. There is one thing we should note: IF we add a counting number greater than the magnitude of the starting number, we will go beyond zero into positive territory. Examples: -‐2 + 5 = 3. -‐5 + 7 = 2. IF, however we add a counting number less than the magnitude of the starting number, we will simply move CLOSER to zero, but not reach it. -‐9 + 5 = -‐4. -‐3 + 2 = -‐1 -‐5 + 2 = -‐3. Note how we can represent these equations I started $9 in debt. I won $5. I used this to pay off some of the debt. I am now only $4 in debt. The temperature was -‐3. It went up 2 degrees. It is now a little warmer, only -‐1. I started 5 km west from home. I move 2 km eastwards. I am now only 3 km west of home. Now let’s look at subtracting, but starting with a negative number. -‐4 – 8 = -‐12 Start at -‐4. Subtract 8 means to move 8 places down, so this gets me all the way down to -‐12. -‐8 – 5 = -‐13 Start at -‐8. Subtract 5 means move down 5 places, so this gets me to -‐13. -‐2 – 3 = -‐5 Start at -‐2. Take 3 more and you end up at -‐5. Real World Applications I was $4 in debt. I then got an extra loan of $8. I am now $12 in debt. The temperature was -‐8. It went down 5 degrees. It is now -‐13 degrees. I was 2 km west from home. I move another 3km west. I am now 5 km west from home. It should make sense, that subtracting from a negative will always make you even further from zero. So the magnitude of the number will be greater.