Real Numbers
Real Numbers are just numbers like:
1
12.38
0.8625
3/4
√2
198
In fact:
Nearly any number you can think of is a Real
Number
Real Numbers include:
Whole Numbers (like 0,1,2,3,4, etc)
Rational Numbers (like 3/4, 0.125, 0.333..., 1.1, etc )
Irrational Numbers (like
π, √3, etc )
Real Numbers can also be positive, negative or zero.
So ... what is NOT a Real Number?
Imaginary Numbers like √-1 (the square root of minus
1)
are not Real Numbers
Infinity is not a Real Number
And there are also some special numbers that
mathematicians play with that aren't Real Numbers.
Why are they called "Real" Numbers?
Because they are not Imaginary Numbers.
The Real Numbers did not have a name before Imaginary
Numbers were thought of. They got called "Real" because
they were not Imaginary. That is the actual answer!
Rational Numbers
A Rational Number is a real number that can be written as
a simple fraction (i.e. as a ratio).
Example:
1.5 is a rational number because 1.5 = 3/2 (it can be
written as a fraction)
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) :
Formal Definition of Rational Number
More formally we would say:
A rational number is a number that can be in the form p/q
where p and q are integers and q is not equal to zero.
So, a rational number can be:
p
q
Where q is not zero
Whole Numbers
Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5,
… (and so on)
No Fractions!
Counting Numbers
Counting Numbers are Whole Numbers, but without the
zero. Because you can't "count" zero .
So they are 1, 2, 3, 4, 5, … (and so on).
Natural Numbers
"Natural Numbers" can mean either "Counting Numbers" {1,
2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on
the subject.
Integers
Integers are like whole numbers, but they also include
negative numbers ... but still no fractions allowed!
So, integers can be negative {-1, -2,-3, -4, -5, … }, positive
{1, 2, 3, 4, 5, … }, or zero {0}
We can put that all together like this:
Integers = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...
}
Example, these are all integers:
-16, -3, 0, 1, 198
(But numbers like ½, 1.1 and 3.5 are not integers)
Irrational Numbers
An Irrational Number is a real number that cannot be
written as a simple fraction.
Irrational means not Rational
Examples:
Rational Numbers
OK. 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 ratio1/3
Irrational Numbers
But some numbers cannot be written as a ratio of two
integers ...
...they are called Irrational Numbers.
It is irrational because it cannot be written as a ratio (or
fraction),
not because it is crazy!
Example:
π
(Pi) is a famous irrational number.
π = 3.1415926535897932384626433832795 (and more...)
You cannot write down a simple fraction that equals Pi.
The popular approximation of
/7 = 3.1428571428571... is
22
close but not accurate.
Another clue is that the decimal goes on forever without
repeating.
Rational vs Irrational
So you 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
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.
If you 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 you 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:
√3
1.7320508075688772935274463415059
(etc)
√99
9.9498743710661995473447982100121
(etc)
But √4 = 2 (rational), and √9 = 3 (rational) ...
... so not all roots are irrational.