An **Irrational Number** is a real number that **cannot** be written as a simple fraction.

You are watching: A number that cannot be written as a ratio of two integers

Irrational means **not Rational**

Let"s look at what makes a number rational or irrational ...

### Rational Numbers

A **Rational** Number **can** be written as a **Ratio** of two integers (ie a simple fraction).

### Irrational Numbers

But some numbers **cannot** be written as a ratio of two integers ...

...they are called **Irrational Numbers**.

### Example: **π** **(Pi)** is a famous irrational number.

We **cannot** write down a simple fraction that equals Pi.

The popular approximation of 22/7 = 3.1428571428571... is close but **not accurate**.

Another clue is that the decimal goes on forever without repeating.

## Cannot Be Written as a Fraction

It is **irrational** because it cannot be written as a **ratio** (or fraction),**not because it is crazy!**

So we can tell if it is Rational or Irrational by trying to write the number as a simple fraction.

**Example: 9.5** can be written as a simple fraction like this:

9.5 = *19***2**

So it is a **rational number** (and so is **not irrational**)

Here are some more examples:

Number As a Fraction Rational or**Irrational?**

1.75 | 74 | Rational | ||

.001 | 11000 | Rational | ||

√2(square root of 2) | ? | Irrational ! |

## Square Root of 2

Let"s look at the square root of 2 more closely.

When we draw a square of size "1",what is the distance across the diagonal? |

**The answer is the square root of 2**, which is** 1.4142135623730950...(etc)**

But it is not a number like 3, or five-thirds, or anything like that ...

... in fact we **cannot** write the square root of 2 using a ratio of two number ...

... (you can learn **why** on the Is It Irrational? page) ...

... and so we know it is **an irrational number**.

## Famous Irrational Numbers

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The number | ||

The Golden Ratio is an irrational number. The first few digits look like this: | ||

Many square roots, cube roots, etc are also irrational numbers. Examples: But √4 = 2 is rational, and √9 = 3 is rational ... ... so ## Note on Multiplying Irrational NumbersHave a look at this: π × π = π2 is known to beirrational But √2 × √2 = 2 is rational So be mindful ... multiplying irrational numbers ## Fun Facts .... Apparently But followers of 434,435,1064,2022,3987,1065,3988,2023,2990,2991 Surds Square Roots Scientific Calculator Is It Irrational? Numbers Index |