The Real Number system
- In math, numbers are classified into types in the Real Number system.
- Number systems can be subsets of other number systems.
- So, a number can have more than 1 type.
Clear as mud? ☺ Well, let’s learn more to make it clearer than that!
When we first learned to count, we started with 1, 2, 3, 4….and kept learning until we got to the millions and trillions, right? These counting numbers (1, 2, 3, 4, 5, 10, 100, 1000, 1,000,000…∞) are called natural numbers.
Did you notice something missing in natural numbers? Yes, the number 0! Good catch!Add that to natural numbers (0,1,2,3……∞) and you have whole numbers!
- So, as you can see, natural numbers are a subset of whole numbers.
- Also, see how the numbers 1,2, 3….are both natural numbers and whole numbers?
- 0 is only a whole number, and not a natural number.
Integers include all whole numbers (0) and also the negatives of the natural numbers: so (∞…,-4, -3, -2, -1, 0, 1, 2, 3,4, …∞).
So let’s look at the pattern again –
- All natural numbers are whole numbers and integers
- All whole numbers are integers
- Negative numbers are integers only
These numbers include all the above (natural, whole, integers) PLUS some types of fraction/decimal.
So, what makes a fraction/decimal rational?
A fraction x/y, where numerator x is an integer (…,-4, -3, -2, -1, 0, 1, 2, 3,4,…) and denominator y is a natural number (1, 2, 3,4) is rational.
In the same way, a decimal that does not keep repeating (.eg. ¼ = 0.25, ¾ = 0.75) is also rational. These are also called terminating decimals.
So rational numbers can look like this:
(∞…,-4, -3.5, -3, -2¾, -2, -1½, -1, 0, 0.88, 1, 1¼, 2, 2.38, 3, 3.91, 4, 4¼, …∞)
An irrational number is a real number that cannot be written as a simple fraction. In other words, it’s a decimal that never ends and has no repeating pattern.
- A decimal that keeps repeating is a good example of this.
- The most famous example of an irrational number is Π or pi.
- Π is the ratio of a circle’s circumference to its diameter.
- While it can be approximated to 3.14159, the actual value of Π only begins with 3.14159. The last known record calculation of Π is up to 2.7 TRILLION digits!
- Remember also, these are never-ending digits, with no repeating pattern.
- More important – Π = 3.14159 cannot be expressed as a fraction!
- 22/7 is an approximation that we use for calculations.
- So Π cannot be expressed as a fraction, decimal digits keep going forever and do not repeat in a pattern. This makes it an irrational number!
- Another good example is √2 or square root of 2.
- If you calculate its value, it approximates to 1.4142135623730950…
- This also cannot be expressed as a fraction, and so it’s irrational
- Other examples – √3, √5, √7, √11 and so on…