Which of the following is an irrational number that lies between two and three? Square root seven, five halves, 2.6, square root three, or square root 10.
An irrational number cannot be written as a fraction. And if it’s in a decimal form, the decimal does not end and the decimal does not repeat. This means we can eliminate some of them. It cannot be written as a fraction. Therefore, five halves is not our answer. And when written in a decimal form, the decimal does not end. And 2.6 ends. So we can cross that out as well.
So we are left with square root seven, square root three, and square root 10. If we would punch these on our calculator, we would get these long decimals that do not end and do not repeat. So these are all are irrational. So now, we have to decide which one is between two and three.
If we would plug in square root seven into our calculator, we will get about 2.65. If we do the same for square root three, we get 1.73 — around that. And then square root 10 is around 3.16. So now to decide which one is between two and three, placing these on a number line is easier to tell which one is between two and three.
Since square root seven is around 2.65, that number is between two and three. Square root three is 1.73 and that’s less than two. And square root 10 is about 3.16 and that’s greater than three. Therefore, our final answer will be square root seven.