Addition and Subtraction of Real Numbers
Picturing Addition on the Number Line
The addition of real numbers can be visualized using the number line. Just one more reason this number line thingie is nice to have around.
Adding two positive real numbers is like starting at 0 and then going for two walks to the right. In the first walk, take as many steps as the first number says, and in the second walk take as many steps as the second number says. In between the two walks, make sure to hydrate.
For example, we already know that 3 + 5 = 8. That's some pretty basic addition. But picture it on the number line: go for a walk of 3 steps to the right of 0, then a second walk of 5 steps after that.
Adding two negative numbers is like starting at 0 and going for two walks to the left. Maybe because you're having a fight with the guy who just started off on a walk to the right. In the first walk, take as many steps to the left as the first number says, and in the second walk take as many steps to the left as the second number says. Watch your step and don't accidentally step on any negative signs. Those things can go right through your shoes and give you splinters.
For example, (-4) + (-3) = -7. We're starting at 0, then walking 4 steps to the left to get to -4 on the number line. Then we're walking another 3 steps left after that, which brings us to our final answer of -7.
Adding one positive and one negative number is, again, like going for two walks, only this time one walk will be to the right and one will be to the left. We apologize in advance, because we know how much you hate backtracking.
4 + (-5) = -1
See? We walked 4 steps to the right, then turned on our heel and walked backwards another 5 steps.
Notice that switching the order of the walks doesn't change the answer:
(-5) + 4 = -1
5 + __ = -17
This question is asking how many steps it takes to get from 5 to -17 on the number line, and in which direction those steps need to be taken. Picture the number line: you can see that if you walked off in a huff and got to the 5, but then your friend who you're having a spat with yells at you from the -17 and says he's sorry, it would take you 22 steps to the left to walk back over to him and hug it out.
5 + (-22) = -17