40 Part 1: The World of Numbers
Arithmetic with Integers .......................................................................................................
Arithmetic with integers may be clearer if you have a sense of owing and having, or gaining and
losing. You can use the directional sense of the number line, right and left or up and down, to
help as well. We’ll look at each operation and discuss the rules associated with that operation.
Before we begin, let’s address important point about notation. If you write negative numbers with
a sign in front, like -7, it would make sense to write positive numbers with a sign in front, like +6.
It’s perfectly correct to do that, and sometimes you will see it, but most times, you won’t. You can
assume that a number that doesn’t have a sign in front, like 9 or 3, is a positive number, but a
negative number, like -5 and -101, will always have a sign to tell you it’s negative. A number with
a negative sign is also called a signed number.Adding Signed Numbers
Adding a positive number to a positive number is nothing new. 4 + 7 = 11, and 73 + 65 = 138.
A gain plus a gain is a bigger gain, so a positive number plus a positive number is a positive
number. And a loss plus a loss is a loss, so a negative number plus a negative number is a negative
number. If you lose $5 and then you lose $8, you’ve lost a total of $13, so -5 + -8 = -13. To put
it in formal language, if you’re adding numbers that have the same sign, you add their absolute
values and give the answer the same sign as the original numbers.
But what happens when you add numbers with different signs, when you add a positive and a
negative? Unfortunately, that’s one of the times when the answer is “it depends.” If you’re playing
football and you lose 3 yards on the first play and gain 14 yards on the second, the big gain
cancels out the loss and still moves you forward.
Picture it on a number line (or a football field, if you’re so inclined). Call the line of scrimmage,
your starting point, the point we label 0. The first play is a loss of 3, taking you to -3.Your second play starts from -3 and moves you 14 spaces in the positive direction. The first 3 of
those 14 bring you back to 0, and then you continue for another 11 in the positive direction:
-3 + 14 = 11. A negative number plus a positive number gave you a positive number.-11-10-9-8-7 -6-5 -4 -3 1234567891011-3-2 -1 0-11-10-9-8-7 -6-5 -4 -3 1234567891011-3-2 -1 0+14