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