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In this video, we’re going to learn how to distinguish between rational and irrational numbers and represent real numbers on number lines.
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We’ll simply begin by defining what it means for a number to be real and for a number to be rational.
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Nearly any number you can think of is likely to be a real number.
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In fact, any number that you can list on your usual number line is real.
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These include integers.
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That’s whole numbers, like the number four.
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A number like three over two is a real number.
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We could add 𝜋 to our number line is being approximately equal to 3.14.
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And we’d also include negative numbers, such as negative three.
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Any number that we can add to this number line belongs in the set of real numbers.
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The numbers we don’t include in this set are ∞ or negative ∞ nor imaginary numbers.
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Now these might sound silly, but they are a group of numbers based on the square root of negative one.
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Then we can split the set of real numbers itself up into several groups.
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The two subsets we’re interested in are rational numbers and irrational numbers.
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These types of numbers are all real, but they’re also mutually exclusive.
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That is to say, a number can’t be both rational and irrational.
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It’s just one or the other.
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A diagram representing this might look a little bit as shown.
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We have the set of real numbers, which contains both rational and irrational numbers.
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Integers, which are whole numbers, are included in the set of rational numbers, and then included within those are the natural numbers.
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Those are the counting numbers, one, two, three, and so on.
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The definition of a rational number is a real number that can be written as 𝑎 over 𝑏 where 𝑎 and 𝑏 are integers, for example, two-fifths or 0.3 recurring, which is, of course, one-third.
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There will be some numbers which might not feel like they should be rational, for example, 0.142857 recurring.
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This is actually the same as one-seventh.
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And since this is a fraction made up of two integers, it’s a rational number.
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In fact, any recurring decimal can be written as the quotient of two integers.
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So recurring decimals are examples of rational numbers.
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An irrational number is simply the opposite of this.
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It’s a number which is not rational.
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Some key examples of this are 𝜋 and the square root of two.
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Now that we have our definitions, let’s look on how to apply them.
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Is 0.456 a rational or irrational number?
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Let’s begin by recalling the definition of a rational number.
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A rational number is one that can be written in the form 𝑎 over 𝑏, where 𝑎 and 𝑏 are integers.
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They’re whole numbers.
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It follows then that an irrational number is one that can’t be written in this form.
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So let’s have a look at the number we’ve been given.
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0.456 is a terminating decimal.
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So let’s see if we’re able to write it as a fraction.
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We spot that the number has four tenths, five hundredths, and six thousandths.
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This means we can write it as the sum of four-tenths, five hundredths, and six one thousandths.
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And then we remember that we can add fractions when their denominators are the same.
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Let’s create a common denominator of 1000.
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To achieve this, we’re going to multiply the numerator and denominator of our first fraction by 100 and of our second fraction by 10.
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Four-tenths is equivalent to four hundred one thousandth.
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And five one hundredths is equal to fifty one thousandths.
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And once their denominators are equal, we simply add the numerators.
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400 plus 50 plus six is equal to 456.
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And in turn, 0.456 is equal to four hundred and fifty-six thousandths.
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We could also simplify this to 57 over 125.
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Though, this isn’t entirely necessary.
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All we really needed to show was that we could write our number as a fraction whose denominator and numerator are both integers.
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Since we’ve shown that 0.456 can be written as the quotient of two whole numbers, it’s 456 over 1000, we can say that 0.456 must indeed be a rational number.
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The answer is yes.
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Now, in fact, we can generalize this and say that any terminating decimal can be written in this form.
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So all terminating decimals are rational.
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So we now have that terminating decimals and recurring decimals are rational.
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We’re now going to look at what happens if we combine rational and irrational numbers.
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Is 𝜋 over three a rational or an irrational number?
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We begin by recalling the definition of a rational number.
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It’s one that can be written in the form 𝑎 over 𝑏 where 𝑎 and 𝑏 are integers.
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They’re whole numbers.
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Once we have this definition, we can say that an irrational number is one that’s not rational.
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In other words, it can’t be written in this form.
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We’re looking to identify whether 𝜋 divided by three is rational or irrational.
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We actually know that 𝜋 is an example of an irrational number.
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It’s one of the ones we’re most familiar with.
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So, actually, we need to ask ourselves what happens if we divide an irrational number by an integer?
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Well, it’s still irrational.
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There’s no way to write 𝜋 divided by three in the form 𝑎 over 𝑏 where 𝑎 and 𝑏 are both integers.
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Three is an integer, 𝜋 is not.
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So this is irrational.
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𝜋 divided by three is an irrational number.
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Now, in fact, we can generalize this.
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We can say that multiplying or dividing an irrational number by a rational number not equal to zero will give us an irrational number.
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Of course, if the rational number was equal to zero and we were to multiply by it, we’d get zero, which is rational.
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And so it’s really important that we specify that we’re multiplying our irrational number by a nonzero rational.
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In our next example, we’ll practice how to identify an irrational number from a list.
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Which of the following is an irrational number?
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Is it (A) the cube root of 70, (B) the cube root of 64, (C) 59, (D) the square root of 144 over 81, or (E) 109.5?
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Let’s begin by recalling what we mean when we say a number is irrational.
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For a number to be rational, we must be able to write it in the form 𝑎 over 𝑏 where 𝑎 and 𝑏 are integers.
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They’re whole numbers.
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Then if a number is not able to be written in this form, it’s not rational.
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And in fact, we say it’s irrational.
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So to work out which of our numbers is irrational, we’re going to go through them in turn.
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Let’s begin by looking at the cube root of 70.
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If we list out the cube numbers we know by heart, we see that we have three cubed is 27, four cubed is 64, and five cubed is 125.
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None of the numbers on the right-hand side are equal to 70.
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And this certainly tells us that the cube root of 70 doesn’t have a whole number in integer solution.
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It will be somewhere between four and five.
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It’s likely to be much closer to four since 70 is only a little bit bigger than 64.
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So rather than trying to work out exactly what the cube root of 70 is equal to, we’ll look at the remaining four numbers.
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The cube root of 64, that’s (B), is actually in our list of cubic numbers.
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The cube root of 64 is equal to four.
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So actually, we can write it as four or four over one, meaning it is a rational number.
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And we can disregard (B) from our list.
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Let’s now look at (C).
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59 is the same as 59 over one.
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Once again, both 59 and one are whole numbers.
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They’re integers.
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So we disregard (C).
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It’s also rational.
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But what about (D)?
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Well, one of the rules we have for working with square roots is that we can find the square root of a fraction by finding individually the square root of the numerator and the square root of the denominator.
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The square root of 144 is 12, and the square root of 81 is nine.
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This means root 144 over 81 is rational.
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It is written in the form 𝑎 over 𝑏.
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𝑎 is 12, and 𝑏 is nine.
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They’re both whole numbers.
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So what about (E)?
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Well, this is a terminating decimal.
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And in fact, we know that all terminating decimals are examples of rational numbers.
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It, therefore, cannot be irrational.
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And we’re going to disregard this one.
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And so, by a combination of looking at the cube numbers and disregarding the other options, we see that the answer must be (A).
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The number that is irrational is the cube root of 70.
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We briefly looked at how we might estimate solutions to roots.
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Let’s now have a look at an example that involves this process.
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Which of the following is an irrational number that lies between three and four?
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We know that a rational number can be written in the form 𝑎 over 𝑏 where 𝑎 and 𝑏 are integers.
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They’re whole numbers.
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An irrational number then can’t be written in this form.
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And so we’re going to begin by working through the numbers in our list and identifying which are rational.
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Let’s begin with (A), the number 3.9.
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3.9 is a terminating decimal.
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And actually, all terminating decimals are rational.
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In this example, 3.9 can be written as 39 over 10.
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Both 39 and 10 are whole numbers, so 3.9 must be rational.
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We’ll consider the square root of 19, the square root of 13, and the square root of seven at the same time.
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If we list out the first few square numbers, we have one squared is one, we have four, nine, 16, and 25.
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None of these are in this list.
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And that’s a good indicator to us that when we find the square root of 19, 13, and seven, not only will we get a nonwhole number, we’ll get a decimal of some description.
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That decimal will also be irrational.
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And so we’ll come back to these in a moment.
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Before we do, let’s look at (D).
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Seven over two is already in the form that we identified.
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It’s in the form 𝑎 over 𝑏.
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𝑎 is seven, and 𝑏 is two.
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And they’re both whole numbers.
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And so we can disregard (D) from our list.
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So we know both (A) and (D) are rational numbers.
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So let’s go back to our square roots.
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A use of a number line here might help us.
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We have one squared, which is one.
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And so the square root of one is one.
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We know that two squared is equal to four, so the square root of four is equal to two.
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We see that three squared is equal to nine, so the square root of nine is three.
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Similarly, the squares of 16 is four.
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And the square root of 25 is five.
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We can also conversely say that one squared is one, two squared is four, and so on.
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We have the square root of 19.
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So let’s find that on our number line.
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19 is here.
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Notice that 19 lies between the result of four squared and five squared.
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We can say that the square root of 19 is greater than the square root of 16 but less than the square root of 25.
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Or we can also alternatively say that the square root of 19 is greater than four and less than five.
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Let’s repeat this process with the square root of 13.
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13 is here.
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It lies between the result of three squared and four squared, which means, of course, that the square root of 13 must be greater than three and less than four.
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We’ll do this one more time for the square root of seven.
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Now seven is all the way down here.
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It’s between the result of two squared and three squared.
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And of course, this means that the square root of seven must be between two and three.
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It’s greater than two and less than three.
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We want to find the irrational number that lies between three and four.
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And of course, we’ve just identified that that’s the square root of 13.
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And so out of our list, the irrational number that lies between three and four is the square root of 13.
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It’s (C).
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In our final example, we’ll look at how to use the laws of radicals to compare the size of irrational numbers.
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Determine which has a larger value: Is it two root three or three root two?
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There are two ways to answer this.
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One method is to estimate the value of the square root part first and then multiply them by two and three, respectively.
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It’s much easier, though, to undo this simplification of the surd.
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Let’s see what that looks like.
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We begin by recalling that two is equal to the square of four.
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And so, if we take our first number, two root three, we can then write that as the square to four times the square root of three.
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But of course, for real numbers 𝑎 and 𝑏, the square root of 𝑎 times the square root of 𝑏 is the same as the square root of 𝑎𝑏.
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This means that two root three is the same as the square root of four times three, which is 12.
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Let’s now repeat this process with our second number.
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That’s three root two.
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This time, we recall that three is equal to the square root of nine.
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And so we’re going to rewrite three root two as the square root of nine times the square root of two.
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That is, of course, equal to the square root of nine times two, which is the square root of 18.
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And so let’s compare the sizes of these two numbers.
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We have the square root of 12 and the square root of 18.
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Now it follows that since 18 is bigger than 12, the square root of 18 must be bigger than the square root of 12.
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And so the square root of 18 has a larger value than the square root of 12, which, of course, means that three root two must have a larger value than two root three.
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The answer is three root two.
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In this video, we learned that a rational number is a real number that can be written as 𝑎 over 𝑏 where 𝑎 and 𝑏 are integers.
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These include all recurring and terminating decimals.
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Those are examples of rational numbers.
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We then say that an irrational number is the opposite of this.
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It’s a number which is not rational.
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And some key examples of this are 𝜋 or any radical or surd where the number inside the surd is not a square number.
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We saw that if we were going to represent these sets as a Venn diagram, they are mutually exclusive.
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There is no overlap.
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A number cannot be rational and irrational at the same time.
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And then all of these numbers make up the set of real numbers.
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We saw that multiplying or dividing an irrational number by nonzero rational number gives an irrational number.
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And we saw that we can estimate the value of irrational numbers which are radicals or surds by considering the square numbers that lie around them.