Welcome to a review lesson on sign numbers.
Looking at the number line, all of the positive real numbers
are located to the right of zero,
or all the numbers that are greater than zero
on the number line.
As we move right on the number line,
we approach positive infinity,
and all the negative real numbers are to the left of zero,
or all the numbers on the numbers line that are
less than zero in this direction here.
As you must of the number line,
we approach a negative infinity.
And because we determined whether a number is positive
or negative based upon whether it's less than
or greater than zero,
zero is neither positive or negative.
And now let's review absolute value.
The absolute value of a number
is the distance that number is from zero on the number line.
We indicate absolute value using vertical bars
as we see here.
So the absolute value of negative three
is equal to the distance negative three is from zero.
Looking at the number line, here's negative three
and here's zero.
This distance is equal to the absolute value
of negative three, which we can see
is one, two, three units.
So the absolute value of negative three is three.
Next we have the absolute value of positive three,
which is the distance three is from zero,
which would be this distance here,
which notice the distance from three to zero
is also one, two, three units.
So the absolute value of positive three
is also three.
Notice how both negative three and positive three
are the same distance from zero.
Next we have negative, absolute value of negative three.
Well if we want the opposite of the absolute value
of negative three,
and we already know the absolute value of negative three
is three, so this part
is equal to three,
but because we have this negative outside the absolute value
where we want the opposite of this absolute value,
this is equal to negative three.
And then finally we have the absolute value of zero,
and because zero is zero units from zero,
the absolute value of zero is zero.
Now let's review the addition, subtraction,
multiplication, division of sign numbers.
Before we look at the addition and subtraction though,
remember that adding negative is equivalent
to subtracting a positive,
and subtracting a negative is equivalent
to adding a positive.
And we explained why this is in a previous lesson.
So looking at 2-a,
notice how we we have three plus negative two,
which if it's helpful we can rewrite as three
minus positive two, which we know equals one.
But let's go and model these using the number line.
So to model three plus negative two, or three minus two,
we start by modeling positive three.
So starting at zero we move right three units,
and then whether we add negative two or
subtract positive two,
we would move left three units from three
in this direction here,
giving us the result of positive one.
Next we have negative three plus two.
Again, let's go ahead and model this on the number line.
So to model negative three, we start at zero
and move left three units.
So here's negative three,
but because we're adding positive two from here,
we'll move right two units,
which gives us negative one.
So negative three plus two equals negative one.
I also like to use the money analogy
to determine the sum or difference of integers
where we can think of negative three as a $3 loss,
and because we're adding positive two
this would be like a $2 gain.
So if we lose $3 and then we win $2,
we're still down $1,
giving us the sum of negative one.
For c, notice here we have negative three
minus negative two.
Let's write this as an addition problem.
Minus negative two is equivalent to plus two,
if we can write this as negative three plus two,
which we just did in example b,
we know the sum is negative one,
which is also modeled here on the number line.
And then finally for d, we have negative three
plus negative two,
which if we want too we can write as negative three
minus two.
Let's go ahead and model this on the number line.
So we first start with negative three,
so from zero we move left three units,
and then because we're adding negative two
or subtracting two from here,
we move left another two units,
which gives us negative five.
So negative three minus two equals negative five.
Thinking this in terms of money,
we can think of negative three as a loss of $3,
and then minus two or plus negative two
would be like losing two more dollars,
giving us a total loss of $5,
giving us negative five.
Now let's talk about multiplying and dividing.
The rules for multiplying and dividing sign numbers
are the same where if we multiply or divide
two numbers with the same sign,
as we see here
and here,
the result is positive.
And if we multiply or divide two numbers with
different signs, as we see here
and here,
the result is always negative.
So for example, 3-a, we have negative five
times negative six.
So because we have a negative times a negative,
this product is positive 30.
For b, we have three times negative four,
we have a positive times a negative,
and therefore this product is negative 12.
And then for c, we have negative 24 divided by eight,
we have a negative divided by a positive,
and therefore this quotient is negative three.
Remember, the fraction bar means division.
And then for d, we have two thirds times negative 1/5,
we have a positive times a negative,
so we know our product is going to be negative.
Then to multiply the fractions,
we multiply the numerators and multiply the denominators.
Two times one is two,
and three times five is equal to 15.
So notice how our product is negative 2/15.
Before we continue, let's talk about how we indicate
negative fractions.
Notice how here we have the negative
in front of the fraction 2/15,
which is the most simplified form of this product,
but it's also true that negative two
over positive 15,
or positive two over negative 15,
is equivalent to negative 2/15,
but this form here is considered the most simplified.
So this is the form we should use
when indicating a negative fraction.
For example four, when you evaluate
the following exponents,
looking at these first two examples, notice how here
we have a negative five in parentheses
raised to the second power,
and here we just have negative five squared.
It's important to recognize the difference between
these two.
In this first example, we're squaring negative five,
which means this is equal to negative five
times negative five,
which is equal to positive 25.
But here, negative five squared really means
the opposite of positive five squared,
so the negative is not attached to the exponent.
This is equal to negative five times five,
which is equal to negative 25.
So it's extremely important that we recognize
the difference between these two.
Here we have negative five cubed.
Again, because negative five is in the parentheses,
we have three factors of negative five.
So this gives us negative 125.
And here we have the opposite of positive five cubed,
or negative five cubed, which is negative,
and then five times five times five,
which does give us the same result of negative 125,
but for a different reason.
Here we only have one negative,
here we have three negatives.
And for our last example, we want to simplify
the given expression using the order of operations.
Following order of operations, we simplify inside
the parentheses first, which doesn't apply here.
So for the next step we want to simplify the exponents.
So we'll first simplify a negative two cubed
and positive five squared.
So we'll have negative eight
and then divided by
negative two cubed
is equal to negative two
times negative two times negative two,
which equals negative eight.
So we have negative eight divided by negative eight.
And then we have minus negative three,
and then we have minus positive five squared,
which is five times five or 25.
The next step is to multiply and divide from left to right.
So we'll find this quotient here.
Negative eight divided by negative eight
is equal to positive one.
Now we have positive one minus negative three
minus 25.
The last step is to add or subtract from left to right.
So notice here we have one minus negative three,
which we can also write as one plus three
minus 25.
So here we have one plus three, that's four.
Four minus 25 is equal to negative 21.
Okay, I hope you found this helpful.