Which of the following is an irrational number that lies between three and four? A) seven halves, B) square root seven, C) square root 13, D) square root 19, or E) 3.9.
So what exactly is an irrational number? An irrational number cannot be expressed as a fraction. In decimal form, it does not terminate, meaning it doesn’t end. And as a decimal, it doesn’t repeat.
So since we know it cannot be expressed as a fraction, that eliminates option A. And then we know as a decimal it cannot end, so we can eliminate option E. We have three options left, and they’re all square roots, so we need to make sure that none of these are perfect squares, meaning they simplify to be a rational number, something that can be written as a fraction.
The square root of seven does not simplify, the square root of 13 doesn’t simplify, and the square root of 19 doesn’t simplify. None of them are perfect squares. So all three of these are irrational.
So now we have to decide which one lies between three and four. So we could rewrite three as the square root of some number. Well, three squared is nine, so if we would take the square root of nine, we would get three, so we can rewrite three as the square root of nine.
Now four, okay, four squared is 16, so the square root of 16 is four, so our number has to be between the square root of nine and the square root of 16, so let’s place our options on a number line.
Square root seven would be below square root nine; square root 13 would be between these two square roots, so that’s most likely our answer, but let’s check D. And D is the square root of 19, which is larger than the square root of 16. And our number needed to lie between three and four, so our answer is square root 13.
However, we also could’ve completed this problem a different way. If we had a calculator handy, we could’ve plugged in the square root of seven, the square root of 13, and the square root of 19 into the calculator and decided which one would lie between three and four, so we would still get square root 13. However, if you didn’t have a calculator handy, using the number line would be a good way to solve this problem.