Hi this is Rob, welcome to Math Antics.
In this lesson, we’re gonna learn about a math concept called Absolute Value.
That sounds pretty intense doesn’t it?
But don’t worry, it’s actually pretty simple.
…so simple in fact, that it might seem kinda boring.
At least that’s the way I remember it when I was in school…
Ok class,
this is an Absolute Value sign.
And it’s like magic!
If I put in a positive number,
it comes out unchanged.
But if I put in a negative number,
then just like magic,
it comes out positive.
Any questions?
[sound: crickets chirping]
Well, maybe it wasn’t quite that bad.
But Absolute Value sure seemed a lot less exciting than the name suggested.
It seemed like just a way to turn a negative number into a positive one.
For example, the Absolute value of 2 is 2
But the Absolute Value of negative 2 is also 2
The Absolute value of 5 is 5
But the Absolute Value of negative 5 is also 5
See those vertical lines on either side of the numbers?
That’s the symbol used for Absolute Value.
So when you see something in between those vertical lines,
it means to find the Absolute Value of it.
But what does Absolute Value even mean?
From these examples, it seems like it’s just
the positive version of any number.
Well… sort of.
But it’s a little more involved than that.
Absolute Value is a more general concept in math
that actually has more to do with the idea of ‘distance’
than it does with the idea of positive and negative numbers.
It’s usually introduced in basic math
right after you learn about negative numbers,
but Absolute Value is a concept that’s even more useful
and more interesting in advanced math.
Because of that, I’m going to teach you about Absolute Value
using an idea that you usually don’t encounter
until a little later on in your math journey.
And that idea is Vectors.
The term “vector” might sound kinda technical,
but they’re actually really simple.
Vectors are basically just arrows.
In really life, an arrow can have all sorts of different properties.
For example, this arrow is made of wood and has feathers,
while this arrow is red plastic with a round thingy at the end.
But in math, the arrows we call vectors only have two properties.
They have a direction and a magnitude.
What do those two properties mean?
Well, direction is pretty obvious.
It’s just which way the arrow (or vector) is pointing.
It could be up, down, left right,
or just about any direction you can think of.
But what does magnitude mean?
Well, that word might make you think of an Earthquake!!
[sound: rumbling]
Phew…
Actually, magnitude is just a fancy word for
the amount, extent or strength of something.
…like how strong an earthquake is.
…or how bright a star is
…or how heavy an object is.
In the case of a vector,
you can think of magnitude as being the ‘length’ of the arrow.
To see what I mean, let’s use a number line to measure some vectors.
This vector has a magnitude (or length) of 2
because it starts at zero and ends at 2.
This vector has a magnitude (or length) of 5
because it starts at zero and goes to 5.
Okay, so we know the magnitudes of these vectors,
but what about their directions?
Well, they are both pointing to the right on your screen.
And since we’re using the number lines as our reference,
they’re pointing in the positive direction, right?
But what about this vector here?
It starts at zero, like our other vectors do,
but it ends at NEGATIVE 2.
And the arrow indicates that it’s pointing
in the exact opposite direction from the other two vectors.
It’s pointing to the left or in the negative direction of the number line.
So what do you think its magnitude is?
For those of you that said ‘2’, you’re right!
Even though the vector is pointing in the negative direction,
its length is still a positive number.
Its length is 2, just like this vector
that’s pointing in the positive direction.
They’re pointing in opposite directions,
but if you rotate one vector around,
you can see that they really do have the same length or magnitude…
and another way to say that is that they have the same “Absolute Value”.
Ah… see why I said that Absolute Value has more to do with distance?
In fact, when it comes to any number that you’d find on the number line,
you can think of its Absolute Value as its distance from zero.
That explains why Absolute Value is
a little bit boring when you first learn about it.
Because the number line is a 1 dimensional space,
there are only two possible directions: positive or negative.
If you ask for the Absolute Value of any positive number along that line,
you’re asking for the distance that number is from zero,
which is just the number itself.
And since the negative numbers are a mirror image of their positive counterparts,
when you ask for the Absolute Value of any negative number along the line,
the only difference is the direction or ‘sign’ of that number.
And that’s why the Absolute Value of a negative number is just its positive counterpart.
Hopefully that seems a little more interesting
than just thinking about Absolute Value as a way to
turn negative numbers into positive ones.
In advanced math, Absolute Value gets even more interesting
and it actually gets kinda complex.
Vectors can point in all sorts of directions besides just positive or negative.
But hopefully thinking about Absolute Value as
the magnitude (or length) of a vector makes it a little more interesting
and it shows you that Absolute Value isn’t just
some silly rule that someone made up to make math even harder.
And now that you know that Absolute Value is related to distance,
it will help you understand a useful application
of Absolute Value in the realm of basic math.
To see what I mean,
suppose that you and your best friend each have a certain amount of money in your pockets
and one of you has more than the other.
Now, it doesn’t really matter to you who has more money,
but you want to know the difference between the amounts.
How do you find the difference between two amounts?
Yep… you subtract them.
And the answer you get from subtracting depends on the order of the numbers
because subtracting doesn’t have the commutative property, right?
If we subtract the amounts in this order:
“SEVEN” minus “FOUR”, we’ll get the answer 3.
But if we subtract the amounts in this order:
“FOUR” minus “SEVEN”, we’ll get the answer NEGATIVE 3.
Do you notice something about these answers?
Yep… even though the sign (or direction) is different,
the magnitudes are the same (they’re both 3)
That means the Absolute Values of the answers would be the same.
And like I said, we don’t really care who has more money,
we just want to now the how much that difference is.
So in this problem,
we only need the magnitude (or Absolute Value) of the difference.
No matter which way we do the subtraction,
we just take the absolute value of the answer to get want we want.
This idea can be really helpful when you’re entering numbers
into your calculator to solve math problems like this one…
Let’s say an airplane (we’ll call it Plane A) is flying at an altitude of 7,328 meters.
And another plane (Plane B) is flying at an altitude of 9,150 meters.
And the problem asks you to find the difference in their altitudes.
You know that means you need to subtract,
so you quickly get out your calculator and start typing in the first number.
But as soon as you get the 7,328 entered and hit the subtract button,
you realize that the second altitude is bigger.
That means you’ll have a negative number as your answer.
Should you start over and type in the bigger number first?
Thanks to Absolute Value, you don’t have to.
No matter which order you subtract the numbers in,
the magnitude of the answer will be the same.
…just the direction (or sign) of the answer will be different.
So if you continue on and enter 9,150 and then hit the equals sign,
the answer you get is negative 1,822.
Now all you have to do is mentally think of that number
as an Absolute Value and ignore the minus sign.
The difference in the altitudes of the two planes is 1,822 meters.
If you’re not quite convinced of that,
try the problem for yourself subtracting both ways and see what answers you get.
In one case you’ll get 1,822
and in the other you’ll get -1,822.
So now you know that in its most basic form,
Absolute Value is just the distance between a number and zero on the number line.
And you’ve also seen how it can be helpful
when you want to find the difference between two different numbers,
regardless of which is greater.
In that case, the Absolute Value represents
the distance between those two numbers.
The last thing I want to show you in this videos is
how to handle a couple situations involving Absolute Value
that you might encounter on tests when evaluating mathematical expressions.
For example,
what if you’re asked to evaluate this expression involving Absolute Values?
We learned how to multiply integers in the last video,
but now these integers are inside Absolute Value signs.
So what so we do?
Well, when it comes to Order of Operations,
Absolute Values signs are similar to parentheses
which means that you need to take care of them first
before you start working on the other arithmetic operations.
So in this problem, before we can multiply the integers,
we need to take the Absolute Value of the numbers first.
The Absolute Value of -3 is 3
and the Absolute Value of 5 is 5.
So the problem simplifies to 3 times 5
which is 15.
That example was pretty easy, but what about this one:
Negative Absolute Value of negative 8?
Why is there an extra negative sign outside of the Absolute Value signs?
Well, whenever you see a negative sign
immediately outside and to the left of a group
like parentheses, braces, or the Absolute Value signs,
it means that you need to “negate” that group.
That means you need to multiply that group by negative 1.
So if it helps, you can think of this problem like this:
-1 times the Absolute Value of -8.
And since Absolute Value signs are groups like parentheses,
to simplify, you would first need to do the Absolute Value.
The Absolute Value of -8 is 8
and then we multiply that by -1 and we get the answer -8.
That seems pretty simple too,
but I want to use this example to point out an important difference
between parentheses and Absolute Value signs.
It’s a difference that can trick you on a test if you’re not careful.
Let’s see the expression we just simplified side-by-side
with a similar expression that has parentheses instead of Absolute Value signs.
In both of these cases,
the negative sign outside the group is telling us to negate the group,
that means multiplying it by -1.
But there’s a very important difference between the two expressions.
The Order of Operations rules tell us to
do things that are inside of groups first, right?
But there’s nothing to DO inside the parentheses…
it’s just the a -8 hanging out inside of them.
And the parentheses themselves don’t tell us to do anything,
so this expression is just asking us to multiply -1 times -8,
which is positive 8.
But, in the case of the Absolute Value signs,
while they do function like groups,
they aren’t JUST for grouping like the parentheses are.
The are also telling us to DO something to whatever is inside of them.
What are they telling us to do?
They tell us to just use the ‘magnitude’ of the number inside,
which would be positive 8.
So in this expression we multiply -1 times +8
which simplifies to -8.
Do you see how it would be easy to confuse these examples on a test?
They look similar at first glance, but they simplify to different answers.
The key is to always remember that Absolute Value signs
are not just there to group things.
They’re also asking you to find the Absolute Value
of whatever number or expression is inside the group.
Alright… as you can see,
Absolute Value is a lot more than just a way to turn negative numbers into positive ones,
although that’s basically what it does when you’re just dealing with numbers on the number line.
But it has lots of other application in math too…
And look…
This student has a negative attitude about math class.
But not to worry…
I put him in a absolute value sign,
and presto…
He comes out with a positive attitude about math class.
Any questions?
Remember… you can’t get good at math just by watching videos about it.
You actually have to do practice problems
so the ideas you’re learning really sink in.
So be sure to do some Absolute Value problems on your own.
As always, thanks for watching Math Antics and I’ll see ya next time.
Learn more at www.mathantics.com