Real NumbersDefinition and Density

Venn Diagram

And the Natural Numbers are located….?

Venn Diagram

Rational Numbers

Irrational Numbers

Irrational Numbers

Density of the Real Numbers• The points on the real number line are dense, meaning that the real line has no gaps.

• Between any two distinct points on the real number line is an infinite number of other points.  

Density of the Real Numbers• The rational numbers are dense since between any two rational numbers there is always another rational number. You can always add them and divide by 2. For example 1/2 and 1/3. You can add them and divide by 2. 3/6+ 2/6=5/6 and half of that is 5/12 (5/12 is certainly between 4/12 and 6/12)

• The whole numbers are not dense. Is there a whole number between 1 and 2?

• Irrational numbers are dense as well, as you can do the same thing you did with the rational numbers. Just add them and divide by 2, finding another number that is halfway between the two numbers.

Density of the Real Numbers

Of course there are many other numbers between each rational and each irrational number. The idea of adding and dividing by two just ensures the existence of at least one such number. Now if the density property applies to rational numbers and irrational numbers, it must apply to real numbers since they can be viewed as the intersection of these two sets.

Density of the Real Numbers

Density of the Real Numbers