Rational and Irrational Numbers

In this lesson we discuss the different kinds of number systems with a special emphasis on rational and irrational numbers. The slides go through what distinguishes a rational number from an irrational number, some common forms of each, and the best way to recognize and deal with these two number groups.

So without further ado, read through the slides below to get a feel for the different number systems and how to distinguish between rational and irrational numbers!

Phew, that’s a lot to take in. Once you’ve gone over this and found some practice problems to cement the idea in your head what integers are, you may move on to the next lesson below!

Decimals and Fractions

This lesson is a video generated from the fully animated PowerPoint I created that is available in my Teachers Pay Teachers store page. You may also head over to my YouTube channel to see the slideshows in video form there!

  • Number systems
  • Rational and irrational numbers
  • Objectives
  • By the end of this lesson you should feel comfortable:
  • Distinguishing between different types of numbers
  • Utilizing the definition of rational and irrational numbers
  • Types of numbers
  • Have you ever noticed the different kinds of numbers you see.
  • For example, what makes these two numbers different?
  • Aside from the obvious fact that they are different numbers, do you see the decimal point on the right?
  • How about these two numbers?
  • The number on the right is negative, which makes it an integer if you remember the last unit!
  • In fact, integers are their own group of numbers!
  • Number systems
  • Let’s begin with a diagram:
  • Every number you know is in a group called the real numbers.
  • Inside that you’ve got two categories, rational numbers and irrational numbers.
  • In the rational numbers you have the group we already learned about called the integers.
  • And finally, the positive integers and zero are called the whole numbers.
  • Number Systems
  • Now that we have seen the different number systems, we should explore what each of them mean.
  • Where better to start than the beginning? The number line!
  • Our first quantity is always zero. Moving to the right we see all of our whole numbers, or numbers without parts.
  • If you add in the negative numbers, then you have the next group up which we have already discussed, the integers.
  • Notice here that the group of integers is bigger than the whole numbers, so every whole number is also an integer!
  • Number systems
  • Alas, the world is not always perfect. There are times where you encounter parts of a whole number.
  • For example in measurement you may halve ½ an inch or you could owe someone $5.75 for a meal.
  • Both decimals and fractions belong to a group called the rational numbers.
  • Rather than simply following along the number line, these numbers take up every space between each whole number as well.
  • Infinitely many numbers can be between 0 and 1:
  • Number Systems
  • Our chart showed another group other than just the rational numbers, though.
  • This last definition is easy to remember: Irrational Numbers are all the numbers that are not rational.
  • But what is there other than a fraction or decimal…?
  • Let’s take a look at √7. Go ahead and type that in your calculator.
  • Decimals that go on forever and don’t repeat even once are called nonterminating, non-repeating decimals.
  • These will make up our irrational numbers.
  • Rational vs Irrational
  • Most of the time it should be easy to pick out a rational number.
  • Rational has the word ratio in it, usually meaning fraction, or at least a fairly short decimal number.
  • But what about this?
  • The ellipses at the end show it is nonterminating, but it repeats!
  • This means that it can be written as a fraction, in fact:
  • We also have irrational numbers that are famous constants such as pi, π, and Euler’s number, e, which are never-ending, never-repeating.
  • Examples
  • Let’s wrap things up with a chart to show some of each kind of number:
  • conclusion
  • Whole Numbers start at 0 and count up by 1.
  • Integers are positive and negative whole numbers along with 0.
  • Rational Numbers include whole numbers and integers as well as all partial numbers in between.
  • Irrational Numbers are nonterminating, non-repeating decimals.
  • Real Numbers contain all of these numbers.
  • As groups get larger, the group before is contained inside the next.
  • For example, every integer is a rational number, but not every rational number is an integer.