We are asked how we to compare and order rational and irrational numbers. Let's take a look at an example. We will try to compare two irrational numbers

23

and

4.205639 \dots

and a rational number

4.3 .

Let's begin by finding an approximation of

23 .

We will do this using perfect squares. The two closest perfect squares to the number

23

are

16

and

25 .

16 < 23 < 25

Now we can rewrite the above inequality with square roots. We will also calculate the square roots of the perfect squares.

16 < 23 < 25 ⇕ 4 < 23 < 5

We found that

23

is somewhere between

4

and

5 .

However, the two other numbers we want to compare it with are also between

4

and

5,

which means that we need a better approximation. Let's use decimals. Let's see if

23

is greater than

4.5 .

4.5 \cdot 4.5 = 20.25 4.5 < 23 < 5

Now we know that

23

is somewhere between

4.5

and

5,

which is a better approximation. Let's approximate

4.205639 \dots

as a rational number by rounding it to the nearest tenth.

4.205639 \dots \approx 4.2

Finally, we can plot each number on a number line to compare them. We know that

23

is somewhere between

4.5

and

5,

so we will plot it between these numbers.

As we can see, to compare rational and irrational numbers, we must first find rational approximations of the irrational numbers. We can do this using perfect squares or by rounding.

Exercise