Week 05 – Rational and Non-rational numbers
Rational Numbers
A Rational Number can be made by dividing two integers. (An integer is a number with no fractional part.)
Example:
1.5 is a rational number because 1.5 = 3/2  (3 and 2 are both integers)
Most numbers we use in everyday life are Rational Numbers.
Here are some more examples:
| Number | As a Fraction | Rational? |
|---|---|---|
| 5 | 5/1 | Yes |
| 1.75 | 7/4 | Yes |
| .001 | 1/1000 | Yes |
| −0.1 | −1/10 | Yes |
| 0.111… | 1/9 | Yes |
| √2 (square root of 2) |
? | NO ! |
Oops! The square root of 2 cannot be written as a simple fraction! And there are many more such numbers, and because they are not rational they are called Irrational.
Another famous irrational number is Pi (π):
Irrational Numbers
An Irrational Number is a real number that cannot be written as a simple fraction.
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).
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 0.333… (3 repeating) is also rational, because it can be written as the ratio 1/3
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.
π = 3.1415926535897932384626433832795… (and more)
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.
So we can tell if it is Rational or Irrational by trying to write the number as a simple fraction.
Example:Â 9.5Â can be written as a simple fraction like this:
9.5 =Â 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 | 7/4 | Rational | ||
| .001 | 1/1000 | 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 numbers
… I explain why on the Is It Irrational? page,
… and so we know it is an irrational number
Famous Irrational Numbers
|
Pi is a famous irrational number. People have calculated Pi to over a quadrillion decimal places and still there is no pattern. The first few digits look like this:
3.1415926535897932384626433832795 (and more …) |
|||||
 |
The number e (Euler’s Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this:
2.7182818284590452353602874713527 (and more …) |
|||||
 |
The Golden Ratio is an irrational number. The first few digits look like this:
1.61803398874989484820… (and more …) |
|||||
 |
Many square roots, cube roots, etc are also irrational numbers. Examples:
|
But √4 = 2 (rational), and √9 = 3 (rational) …
… so not all roots are irrational.