# Is It Irrational?

Here we look at whether a square root is irrational ... or not!

### Rational Numbers

A "Rational" Number can be written as a "Ratio", or fraction.

Example: **1.5** is rational, because it can be written as the ratio **3/2**

Example: **7** is rational, because it can be written as the ratio **7/1**

Example **0.317** is rational, because it can be written as the ratio **317/1000**

But some numbers **cannot** be written as a ratio!

They are called **irrational** (meaning "not rational" instead of "crazy!")

## The Square Root of 2

The square root of 2 is **irrational**. How do I know? Let me explain ...

### Squaring a Rational Number

First, let us see what happens when we **square** a rational number:

If the rational number is a/b, then that becomes a^{2}/b^{2} when squared.

### Example:

(\frac{3}{4})^{2} = \frac{3^{2}}{4^{2}}

Notice that the exponent is **2**, which is an **even number**.

But to do this properly we should really break the numbers down into their prime factors (any whole number above 1 is prime or can be made by multiplying prime numbers):

### Example:

(\frac{3}{4})^{2} = (\frac{3}{2×2})^{2} = \frac{3^{2}}{2^{4}}

Notice that the exponents are still even numbers. The 3 has an exponent of 2 (3^{2}) and the 2 has an exponent of 4 (2^{4}).

In some cases we may need to simplify the fraction:

### Example: (\frac{16}{90})^{2}

Firstly: **16** = 2×2×2×2 = 2^{4}, and **90** = 2×3×3×5 = 2×3^{2}×5

(\frac{16}{90})^{2} = (\frac{2^{4}}{2×3^{2}×5})^{2}

= (\frac{2^{3}}{3^{2}×5})^{2}

= \frac{2^{6}}{3^{4}×5^{2}}

But one thing becomes obvious: every exponent is an **even number**!

So we can see that when we square a rational number, the result is made up of prime numbers whose exponents are all **even** numbers.

When we square a rational number, each prime factor has an **even exponent**.

### Back to 2

Now, let us look at the number 2: could this have come about by squaring a rational number?

### As a fraction, 2 is **\frac{2}{1}**

Which is **\frac{2^{1}}{1^{1}}** that has **odd exponents**!

But we want **even exponents** (so its square root will be rational)

We could write 1 as 1^{2} (so it has an even exponent), and then we have:

2 = **\frac{2^{1}}{1^{2}}**

But there is still an odd exponent (on the 2).

We can simplify the whole thing to **2 ^{1}**, but still an odd exponent.

We could try things like 2 = \frac{4}{2} = **\frac{2^{2}}{2^{1}}** but we still cannot get rid of an odd exponent.

Oh no, there is always an **odd** exponent.

So 2 could **not** have been made by squaring a rational number!

So its square root must be irrational.

In other words: whatever value that was squared to make 2 (ie **the square root of 2**) cannot be a rational number, so must be **irrational**.

Note: for another proof check out Euclid's Proof that Square Root of 2 is Irrational.

### Try Some More Numbers

### How about square root of 3?

3 is 3/1 = 3^{1}

But the 3 has an exponent of 1, so 3 could not have been made by squaring a rational number, either.

The square root of 3 is **irrational**

### How about square root of 4?

4 is 4/1 = 2^{2}

**Yes!** The exponent is an even number! So 4 can be made by squaring a rational number.

The square root of 4 is **rational**

This idea can also be extended to cube roots, etc.

## Conclusion

To find if the square root of a number is irrational or not, check to see if its prime factors all have **even exponents**.

It also shows us there **must be** irrational numbers (such as the square root of two) ... in case we ever doubted it!