Real Numbers

Background

We've talked about rational numbers, which are numbers that we can make using fractions of integers (the denominator being non-zero), which also means they can be made by terminating or repeating decimals. We've also talked about irrational numbers, which are numbers that can't be made using fractions of integers, which also means they are made by decimals that neither terminate or repeat. The two types of numbers together constitute the Real Numbers.

Question

What are some examples of Real Numbers?

Answer

All the numbers on the number line are real numbers, such as 

Analysis

Let's first look at the number line between the integers 0 and 1:



Note that between 0 and 1, there are an infinite number of numbers and of those, there are two kinds of numbers: rational and irrational.

For instance, there is 0.3, which is a terminating decimal (the red dot):



There is also the repeating decimal  (the blue dot):



Between those two points, there are irrational numbers such as 0.32332233322233332222.... (purple dot):



No matter what the decimal is, we can find it along the number line. If we graph a point for each rational and irrational, we end up with a solid line:



and, in fact, we can then look at all the rational and irrational numbers between all of the integers - and so we end up with a solid line that runs forever to the left (bigger and bigger negative numbers) and also to the right (bigger and bigger positive numbers)(I'm showing the numbers from -100 to 100 in the graph below, but the graph continues on forever in both directions):



Vocabulary used:

For more information check out these links (comment to add your favourite link):

Where might you have come from?

Fact-orials Index

Numbers:

• Rational Numbers
• Irrational Numbers

Graphing:

• Number Line

Where might we go?

Numbers:

• Scientific Notation
• Variables 

Associated Operations:

• Estimation and Rounding