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What really IS a real number? | What really IS a real number? | ||
| − | + | A real number is a number that belongs to the set ℝ. | |
| − | + | The set ℝ is the set of all numbers on the number line, even the ones which you cannot exactly pinpoint(e.g. π.) | |
| − | + | To be more precise the ℝ is comprised of the union of 4 smaller subsets. | |
| − | + | Starting form the smallest... | |
| − | + | N -> Natural Numbers - which is the set of natural numbers: 1, 2, 3, 4, ...., n | |
| − | + | W -> Whole Numbers - simply the Natural Numbers + {0}: 0, 1, 2, 3, 4, ...., n | |
| + | Z -> Integer Numbers - this is the Whole Numbers + {-Natural Numbers}: -n, ..., -3, -2, -1, 0, 1, 2, 3, ..., n | ||
| + | |||
| + | Q -> Rational Numbers - this is the set of all numbers that can be written as ( a / b ) where a and b belong to Z, and b is not 0: 3/4, 19/31, ... a/b | ||
| + | |||
| + | Q' -> Irrational Numbers - this is the set off all numbers that cannot be written as a fraction of any numbers belonging to Z: sqrt(2), sqrt(3), π, e, ... | ||
| + | This does not include numbers like 0.333333.... since it can be written as 1/3. | ||
| + | |||
| + | Below is a diagram showing the sets included in ℝ. | ||
[[Image:image030.gif]] | [[Image:image030.gif]] | ||
| + | |||
| + | To get a better idea of what a real number IS, it may be useful to determine what a real number IS NOT. | ||
| + | |||
| + | To be blunt and real number is NOT and imaginary number. But what is an imaginary number? | ||
| + | |||
| + | Imaginary numbers are those written in the form: | ||
| + | a + b''i'' | ||
| + | |||
| + | Where ''i'' is the imaginary unit, or sqrt(-1). | ||
| + | |||
| + | This number does exist, and is by no means 'imaginary' it is just called imaginary because we cannot represent it in our universe. | ||
| + | It has a value, but does not have a representation. Can you show you sqrt(-1) bananas? Can you even approximate sqrt(-1) bananas? | ||
| + | |||
| + | This idea of being able to be 'represented' in our universe helps to explain what a real number is. | ||
| + | |||
| + | Some real numbers may be impossible to find their exact value, but an approximation can be made and we can perceive values such as pi as being half the distance around the unit circle. | ||
| + | Or when we see e, the golden ratio all over nature. | ||
| + | |||
[[Image:cartoon]] | [[Image:cartoon]] | ||
Revision as of 13:34, 1 May 2012
What really IS a real number?
A real number is a number that belongs to the set ℝ.
The set ℝ is the set of all numbers on the number line, even the ones which you cannot exactly pinpoint(e.g. π.)
To be more precise the ℝ is comprised of the union of 4 smaller subsets.
Starting form the smallest...
N -> Natural Numbers - which is the set of natural numbers: 1, 2, 3, 4, ...., n
W -> Whole Numbers - simply the Natural Numbers + {0}: 0, 1, 2, 3, 4, ...., n
Z -> Integer Numbers - this is the Whole Numbers + {-Natural Numbers}: -n, ..., -3, -2, -1, 0, 1, 2, 3, ..., n
Q -> Rational Numbers - this is the set of all numbers that can be written as ( a / b ) where a and b belong to Z, and b is not 0: 3/4, 19/31, ... a/b
Q' -> Irrational Numbers - this is the set off all numbers that cannot be written as a fraction of any numbers belonging to Z: sqrt(2), sqrt(3), π, e, ...
This does not include numbers like 0.333333.... since it can be written as 1/3.
Below is a diagram showing the sets included in ℝ.
To get a better idea of what a real number IS, it may be useful to determine what a real number IS NOT.
To be blunt and real number is NOT and imaginary number. But what is an imaginary number?
Imaginary numbers are those written in the form:
a + bi
Where i is the imaginary unit, or sqrt(-1).
This number does exist, and is by no means 'imaginary' it is just called imaginary because we cannot represent it in our universe. It has a value, but does not have a representation. Can you show you sqrt(-1) bananas? Can you even approximate sqrt(-1) bananas?
This idea of being able to be 'represented' in our universe helps to explain what a real number is.
Some real numbers may be impossible to find their exact value, but an approximation can be made and we can perceive values such as pi as being half the distance around the unit circle. Or when we see e, the golden ratio all over nature.
What is NOT a real number?
http://www.mathsisfun.com/numbers/imaginary-numbers.html
http://en.wikipedia.org/wiki/Imaginary_number
http://mathworld.wolfram.com/ImaginaryNumber.html
//list and compare the differences.
