Recall **Corollary 4 **from this post:

Corollary 4.If is a subgroup of , then the following are equivalent:(i) is well-ordered;

(ii) is not dense;

(iii) is cyclic.

Let be a non-zero real number and take to be the subgroup of generated by and . This is a cyclic group if and only if

for some and

, i.e. is rational.

Now by the above corollary, is dense iff it is not well-ordered, i.e. iff

.

So we have the following criterion for irrationality:

**Criterion 1. ** is irrational if and only if holds.

Note that is dense in iff is dense in . So we deduce:

**Criterion 2. ** is irrational .

We demonstrate how this may be useful by proving that certain types of numbers are irrational.

**Proposition 1. **Let be a sequence of non-zero integers such that

exists, and

as .

Then is irrational.

*Proof. *We have

.

Multiplying by if necessary, we can take the expression in brackets on the right to be positive and it tends to as . Moreover, if

then

which cannot happen infinitely often as the left hand side tends to zero. So the conclusion follows by **Criterion 2**.

**Proposition 2. **If is a sequence of non-zero integers such that and , then

exists and is irrational.

*Proof. *It suffices to show that and above hold.

Convergence follows easily from the ratio test, so holds. Now

as ,

i.e. holds.

Some special cases of **Proposition 2 **are particularly interesting:

**Corollary 1. ** is irrational for all positive integers .

*Proof.* Take . .

**Corollary 2. ** is irrational.

*Proof. *Take in **Corollary 1**.

**Corollary 3. ** and are irrational.

*Proof.** *Take and for , for sine, for cosine.

**Corollary 4. ** and are irrational, where is the modified Bessel function of the first kind.

*Proof.* Taking in **Corollary 1 **shows that is irrational. Taking shows that is irrational.

**Corollary 5. ** is irrational.

*Proof.* If it is rational, then so is , and so is

.

Taking in the above shows that this is false.

**Corollary 6. **Let be an integer and the Fibonacci sequence. Then

(i) is irrational.

(ii) is irrational.

*Proof.* (i) Take and use .

(ii) Take .