Which of the following is an irrational number that lies between three and four?
(a) Seven halves, (b) square root seven, (c) square root thirteen, (d) square root nineteen, and
(e) three point nine.
An irrational number is a number that can’t be expressed as a fraction, also meaning it can’t be expressed as a ratio. If you were to write an irrational
number as a decimal, it would never end, nor would the decimal ever repeat.
Let’s first go through all of our options and decide which numbers are not
irrational. Since option (a) is seven halves, that is written as a fraction. Therefore, option (a) is not an irrational number. (b) Square root seven, (c) Square
root thirteen, and (d) square root nineteen are all three square roots that aren’t perfect squares,
meaning they can’t be written as a nice fraction using integers. And in finally option (e)
three point nine, that’s a decimal that ends. An irrational number can be written as a decimal,
but that decimal will never end and won’t repeat.
So (e) is not an option as an irrational number. So now we’re gonna be looking
between (b), (c), and (d) and we wanna know which of the following is an irrational number that
lies between three and four. So let’s place these on a number line.
Now that we’ve placed our options on this number line, let’s go ahead and write
some perfect squares that we know that are close to these. Nine is a perfect square because three squared is nine, so the square root of nine is
three. Sixteen is also a perfect square because four squared is sixteen; the square root of
sixteen is four.
So we need to know which irrational number lies between three and four. Well
square root seven isn’t between three and four and square root nineteen isn’t between three
and four. The only number that is between three and four is the square root of thirteen.
So the square root of thirteen is the irrational number that lies between three and
four. So our answer is (c).