Video: Approximating the Value of Irrational Numbers

By considering square numbers, determine an approximation for √83 to the nearest whole number.

02:24

Video Transcript

By considering square numbers, determine an approximation for root 83 to the nearest whole number.

So in the question, it asked us to consider square numbers. So what are square numbers? Well, square numbers are the result of multiplying a number by itself. So, for example, one is a square number because one multiplied by one equals one. And four is a square number because two multiplied by two is equal to four.

So then, if we list the next square numbers, we’ve got nine, 16, 25, 36, 49, 64, 81, and 100. And I stopped at 100 because that’s over the number that we’re looking at. And I say that because the number we’re looking at is 83 because what we want to consider is the square numbers either side of 83. So less than 83, we have 81, which is nine multiplied by nine. And then, greater than 83, we have 100, which is 10 multiplied by 10.

Well, if we know that 83 is between 81 and 100, then therefore, root 83 must be between root 81 and root 100. And it must be greater than root 81 and less than root 100. And therefore, we can say that root 83 must be greater than nine. That’s because root 81 is equal to nine because nine multiplied by nine is 81. But it must be less than 10. And that’s because root 100 is equal to 10 because 10 multiplied by 10 is 100.

Well, the question asks for an approximation for root 83 to the nearest whole number. So we need to decide whether it’s closer to nine or 10. Well, let’s think about 83 for a second. Well, 83 is only two greater than 81. However, it’s 17 less than 100. So therefore, 83 is closer to 81. So therefore, root 83 must be closer to root 81.

So therefore, our approximation for root 83 to the nearest whole number is going to be nine.

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