Video Transcript
In this video, we’re going to learn
how to use the properties of addition in a set of rational numbers and determine the
additive inverse. Let’s begin by recalling what we
actually mean by the set of rational numbers.
A rational number is a number that
can be written as a fraction 𝑎 over 𝑏, where 𝑎 and 𝑏 are integers; that’s whole
numbers. Examples of rational numbers are
0.7, which can be written as seven-tenths, 19, and 0.3 recurring, which is the same
as one-third. If a number is not rational, we say
it’s irrational. And some examples are 𝜋 and the
square root of two. The set of rational numbers is all
possible numbers that satisfy this criteria. We should also recall what we mean
by the inverse. Inverse means opposite. So, when we apply an inverse
operation, we perform the opposite operation. This has the effect of undoing the
previous operation.
So, how does that help us to define
the additive inverse? We say that the additive inverse of
a number 𝑎 is the number that when added to 𝑎 gives zero. In real terms, the additive inverse
is simply the negative of that number. So, the additive inverse of 𝑎 is
simply negative 𝑎. Let’s have a look at an
example.
Find the additive inverse of
0.7.
We say that the additive inverse of
a number is the number that when added to the original gives zero. Let’s define the additive inverse
of 0.7 as being equal to 𝑥. By the definition for the additive
inverse, we can then say that 0.7 plus 𝑥 is equal to zero. We can think about solving this
equation in the usual way. We would subtract 0.7 from both
sides. Zero minus 0.7 is negative 0.7.
And so, we find the value of 𝑥 and
thus the additive inverse of 0.7 to be negative 0.7. Now, in fact, this makes a lot of
sense because we know that the additive inverse of a number is simply the negative
of that number; we just change the sign. The additive inverse of 0.7 is
negative 0.7.
We’re now going to consider a new
property called the associative property. The associative property of
addition says that we can add three or more numbers regardless of how they are
grouped together. Let’s look at a number line and see
where this comes from. Let’s say we want to add three plus
four plus one. We could perform this calculation
left to right. We start at three on the number
line and we add four by moving one, two, three, four spaces right. We then add one by moving one
further space to the right. Alternatively, we could begin by
adding four and one. This time, we’d start at four and
move one space to the right. Then, we’d add the three by moving
one, two, three spaces to the right. In either case, we end up at
eight. It didn’t really matter the order
in which we added. Now, we just chose a bunch of
random numbers here. But when working with, say,
fractions and decimals, this property can be really helpful. Let’s see what that looks like.
Simplify five thirteenths plus
three-quarters plus one-quarter using the properties of addition.
According to the order of
operations, we’ll usually look to deal with the calculation inside the parentheses
first. That would involve creating a
common denominator and then adding the numerators. Alternatively, we recall that the
associative property for addition says that we can add three or more numbers
together, regardless of how they are grouped. Now, of course, if there were, for
example, exponents or multiplications in this problem, we’d need to be a little bit
more careful. But since we only have addition, we
can do this in any order.
Let’s get rid of the parentheses
first. We now know that we can easily add
three-quarters and one-quarter because their denominators are the same. We just add the numerators. Three-quarters plus one-quarter is
four-quarters. And of course, if we have
four-quarters, we essentially have one whole. So, three-quarters plus one-quarter
is equal to one. And we can now add five thirteenths
and one. Now, we could write this as a mixed
number. It would be one and five
thirteenths. Alternatively, let’s think about
how many thirteenths one whole must contain. One whole must be equal to thirteen
thirteenths. And so, five thirteenths plus one
is the same as five thirteenths plus thirteen thirteenths.
Now that their denominators are
equal, we simply add the numerators. And we get five thirteenths plus
thirteen thirteenths equals eighteen thirteenths. We’ve simplified five thirteenths
plus three-quarters plus one-quarter by using the associative property. And we’ve got eighteen
thirteenths.
In our next example, we’ll look at
how we can identify the associative property with rational numbers.
Which equation shows the
associative property of addition? Is it (A) a half plus negative a
half equals zero? (B) One-half plus two-thirds equals
two-thirds plus one-half. Is it (C) one-half plus two-thirds
equals seven-sixths? (D) One-half plus two-thirds plus
three-quarters equals one-half plus two-thirds plus three-quarters. Or (E) two-thirds plus zero equals
two-thirds.
Remember, the associative property
of addition says that the sum of three or more numbers remains the same regardless
of how the numbers are grouped. So, we’re going to go through each
of our equations and identify which of these satisfies the associative property. We’re going to disregard (A)
immediately. (A), in fact, is showing us an
example of the additive inverse. The additive inverse of one-half is
negative one-half because the result is zero when we add them. And what about (B)? Well, no, the associative property
does tell us we can perform the addition in any order but that there are going to be
three or more numbers, and it’s regardless of how those three or more numbers are
grouped. So (B) does not show the
associative property. (C) just gives us an equation. It tells us that a half plus
two-thirds equals seven-sixths, and so it’s not (C).
So, what about (D)? We do indeed have three numbers on
each side of our equation, and they’re grouped differently on both sides. Otherwise though, the numbers
remain unchanged. So, yes, (D) does show the
associative property of addition. If we perform a half plus
two-thirds first and then add three-quarters, that’s the same as adding a half to
the result of two-thirds plus three-quarters. And so the answer is (D). (D) shows the associative property
of addition. We can see quite clearly it’s not
(E). That’s just showing us that when we
add zero to a number, it remains unchanged.
Now, let’s focus on the equation
shown in part (E). Two-thirds plus zero equals
two-thirds. We said that this shows us that
when we add zero to a number, we keep that original number. This has a special name. Just as (A) is an example of the
additive inverse, (E) is an example of the additive identity property. We say that the additive identity
is zero, since adding zero to a number doesn’t change it.
What property of addition is
demonstrated by a half plus zero equals one-half?
Well, we know that when we add zero
to a number, the number itself remains unchanged. This is known as the additive
identity property. And so, the property of addition
demonstrated by the equation a half plus zero equals one-half is the additive
identity property.
Let’s have a look at one further
example of how to use the properties of addition of rational numbers to complete a
calculation.
Evaluate one-quarter plus
three-quarters plus negative one-quarter.
Let’s begin by recalling what we
mean by the associative property of addition. The associative property of
addition says that the sum of three or more numbers is the same regardless of how
those numbers are grouped. And so we don’t really need the
parentheses. We can write it as one-quarter plus
three-quarters plus negative one-quarter. And then, since we can perform the
additions in any order, we’re going to regroup one-quarter and negative
one-quarter. So, why is this useful? Well, here, we have an example of
the additive inverse. The additive inverse of a number 𝑎
is the number that when we add it to the original number 𝑎, we get zero. In reality, all it is is the
negative of that number.
In our parentheses here, we have
one-quarter and negative one-quarter. And so their sum must be zero. Negative one-quarter is the
additive inverse of one-quarter. And so, really, the sum that we’re
now doing is three-quarters plus zero, which is simply three-quarters. One-quarter plus three-quarters
plus negative one-quarter is three-quarters.
In this video, we’ve learned that
the additive inverse of a number is the number that when we add it to the original,
we get zero. Really, what we’re doing is we’re
adding the negative version of that number, so the additive inverse of 𝑎 is
negative 𝑎. We also saw that the associative
property tells us that we can add three or more numbers regardless of how they are
grouped. We’ll get the same results no
matter which order we do the calculation in. We also saw that the additive
identity is zero. When we add zero to a number, the
number remains unchanged.
