# "There exists a ..., which..." or "There exists a ..., that..."

Suppose that I want to assert that there exists an object satisfying a certain property, which of these constructions would be correct?

There exists a number which squares to four.

or

There exists a number that squares to four.

My confusion comes from the fact that on the one hand, the clause seems to be restrictive, not just any number will do, on the other hand it seems to be non-restrictive, the number two and negative two both work....

Both of those sentences will work and the clause is non-restricitve as this will also work:

There exists a number that squares to nine.

An alternative sentence:

There exists a number, that when squared, it equals four

The clause that comes after the word "which" or "that" is what determines which one to use.

• If the clause is absolutely pertinent to the meaning of the sentence, you use "that".
• If the clause is not essential to the sentence, use "which."

So:

There exists a number that squares to four.

The main subject of this sentence is the fact that this number squares to four. It is absolutely pertinent, so you should use "that".

On the other hand:

There is a number which you know very well that squares to four.

The fact you know the number well is an interesting aside, but not pertinent to the sentence. For that, you should use "which".

If your sentence was actually leading to something else, it could be the fact that it squares to four that is non-essential:

There is a number which squares to four that is part of my password.

In this example, the fact it is part of your password bis the main clause of the sentence, and the fact it squares to four is something of an aside.