Video: Identifying the Irrational Number That Lies between Two Given Rational Numbers

Which of the following is an irrational number that lies between −7 and −3? [A] −√5 [B] −√53 [C] −5 [D] −√29 [E] −11/2.

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Video Transcript

Which of the following is an irrational number that lies between negative seven and negative three?

Well, to be able to solve this problem, the first thing we have to ask ourselves is what is an irrational number? Well one definition or it’s a definition that’s gonna be helpful for this question is that an irrational number is a number that cannot be written as a simple fraction. So we can use that to eliminate some answers straight away.

If we look across the E, we can see that E is equal to negative 11 over two. But this is already actually a simple fraction. So we can eliminate this as our answer. And then if we take a look at C, C is equal to negative five. Well, negative five is actually not an irrational number. And actually you could write it as a simple fraction because it could be written as negative five over one. So in that case, we can also eliminate answer C.

Okay, so we’re now left with three possible answers. All three of these are irrational numbers. So they can all be considered. So to actually solve it now, we can just stop and think, well, right, what are actually gonna be our parameters for our values?

First of all, I am gonna start with the lowest possible value. So what could the lowest possible value be? Well, we know that seven squared is equal to 49. So therefore, we know that our lowest possible value must be negative root 49 as it says that our irrational number must lie between negative seven and negative three. But if we look at any of the answers actually lower than negative 49, it means that yes, again, we can actually eliminate one of our answers because negative root 53 would be too low a value. So we can now eliminate this answer.

Great! So we’ve now got two answers left, either negative root five or negative root 29. So let’s see what the highest possible value could be. Well, we know that three squared is equal to nine. So therefore, the highest possible value is negative root nine. So again, we can take a look at our last two answers. And so actually at close inspection, we can see that we actually can’t have answer A, negative root five, because that is in fact too big an answer cause negative root five is actually greater than negative root nine. So therefore, we can eliminate answer A. And we can say that our answer is gonna be D, negative root 29.

And we can just check that. And in order to check it, we can actually just look at root 29. And actually we have a look I think. Are there two square numbers that 29 would be in between? And actually, we see yes, there are because 29 is between 25 and 36. So therefore, we can say that root 29 is greater than root 25 and less than root 36. So therefore, we can say that root 29 is greater than five, but less than six. So therefore, we can say that negative root 29 is gonna be greater than negative six and less than negative five. So therefore, it would lie between our values negative seven and negative three. So therefore, we can say with confidence that negative root 29 is an irrational number that lies between negative seven and negative three.

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