Operations with Integers:
Remembering the debit/credit
analogy for negative and positive numbers, it would make sense that debits
should add to a larger debit, and credits should add to a larger credit.
Integers work exactly that way. We're interested in the magnitudes of each,
these magnitudes are called the absolute values.
Even though as mentioned in the
previous section we maintain the “-” sign in front of the
negative number, when we speak of absolute values we are
referring to the magnitude of the negative or positive
number and for negative numbers in particular, the “-” is
dropped. For example, in speech one normally says “I have
a $200 debit,” not “I have a -$200 debit.” The symbols
used for absolute value are a pair of vertical bars placed
around the number.
Examples:
| -3 | =
3 | 5 |
= 5
(read: the absolute value of -3
is 3, and the absolute value of 5 is 5)
Handling negative numbers:
So, to add two negative numbers
together, think of their magnitudes
, add these magnitudes together, then put the “-” back in place.
Example:
-3 + -6 : magnitudes are 3, and 6, add these together
we get 9, place the “-” back and we get -9 for an answer. (One normally would
write this as -3 – 6 = - 9, but for now let's keep the “+”.)
Example:
-3 + -6 + -10 + -1: magnitudes are 3, 6, 10, and 1,
so we have 20, add the “-” and the answer is -20.
Handling positive numbers:
Example:
3 + 6 + 9 : magnitudes are the same as the positive
numbers, so we have 18 for the answer.
Handling mixed positive and
negative numbers:
Recall the debit/credit analogy. If I have a
$20 debit and a $20 credit then if I pay my debt, I'd have to use all of the
$20 credit leaving no debit and no credit and we use the number '0' for that
case. What we are saying is - $20 + $20 = $0. The same works for the Integers.
The number '0' is called the additive
identity, that is, adding 0 to any number does not change that number.
Examples:
0 + 4 = 4 5 + 0 = 5
-2 + 0 = -2 0 + -6
= -6.
Back to our analogy, if I have a $5 debt and $10 on
hand, if I pay my debt, that would leave $5 that I can spend on something else.
In symbols this would be:
- $5 + $10 = $5
Integers obey the same rule.
Examples:
5 + -1 =
4
- 4 + 6 = 2
Think of our example, if I have
$5 and spend $1 then I'm left with $4.
If I spend $4 from the $6 I had
then I'm left with $2.
Back to the number '0', what
number must I add to 5 to get 0?
Think of our analogy, if I have
$5, how much must I spend to be left with no money? Well the answer is $5 of
course. I spent $5, so that would be a debit and remember we write the debit
with a “-” sign, so the answer would be –5.
5 + - 5 =
0
-5 + 5 = 0
Examples:
10 + ? = 0 answer: -
10, ? + 15 = 0 answer: -15
? + - 4 = 0?
answer: 4
- 8 + ? = 0 answer: 8
Additive inverses are those two
numbers with the same magnitudes but with different signs, when added together
yield the additive identity, namely '0'.
Another
look at adding numbers with mixed signs:
Consider: - 4 + 6
Looking at the magnitudes of
these numbers we have 4 and 6. We know that additive inverses add to 0 and 0 +
any number is that number. Since 6 is larger than 4
let's write 6 as 4 + 2
now we have
- 4 + 4 +
2
usually written: -4 + ( 4 + 2 )
Now you can see the additive
inverses, 4 and -4 and we know they add to 0 so we have
0 + 2 =
2
2 is the answer.
Example:
7 + -10
Looking at the magnitudes we have 7 and 10. 10 is larger, so let's write 10 in
terms of 7 and 3 (remember we're dealing with negative 10, so 7 and 3,
both, must be negative)
7 + -7 +
-3
You can see the additive
inverses, 7 and -7, so we have
0 + -3 =
-3
-3 is the answer.
THIS IS THE PATTERN THE
STUDENT MUST RECOGNIZE, THE LARGER NUMBER MUST BE WRITTEN IN TERMS OF THE
SMALLER NUMBER!
And, be careful of that “-” sign!
6 – 10 is ? Well, subtraction is
really “adding a negative number,” (in fact, in advanced topics such as ring
theory, subtraction is not even mentioned, instead we talk about a set being
closed under addition, and additive inverses.)
In this case, we're adding -10
to 6 which is -4. It's paramount to “think” of subtraction this way when
it comes time to regroup terms when symplifying alebraic expressions.
So, 6 – 10 is actually 6 +
-10 (this is MOST important)
Opposites:
As a final note, we often refer
to the additive inverse as the “opposite of”, i.e., the opposite of 6 is
-6. The opposite of -10 is 10. We don't write the '+' sign since it is
understood for positive numbers.
Simple Rule for adding mixed
numbers:
Subract the magnitudes and keep
the sign of the larger one.
Example:
6 – 10 = 6 + -10; magnitudes, 6
and 10, subtract them we have 4, sign of largest is “-” so the answer is -4.
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