Ordering & Graphing Irrational Numbers on a Number Line

Instructor: Kim Lackey

Kim has 20+ years of experience teaching a wide variety of math courses from fundamental math skills to AP Calculus.

This lesson will help you identify irrational numbers, order irrational numbers, and place the irrational numbers in the correct order on a number line.

What is an Irrational Number?

Do you remember the Sesame Street game, One of These Things is Not Like the Other? Usually there were four objects and one out of the four didn't belong in some way. Well, that is the case with an irrational number. In short, it is a decimal number that doesn't belong with most other decimals. An irrational number has decimal places that go on forever and don't repeat in a pattern.

Examples include:

3.1428571428571... =


1.41421356237... =

square root 2

2.2360679775... =

square root of 5

How are the Irrational Numbers Graphed on a Number Line?

The first step in graphing any group of numbers on a number line is figuring out their order. It works just like alphabetizing but with numbers; start with the smallest number, which is like starting with the letter A. Using the examples that were given above, square root of 2 would come first, then square root of 5, then pi. This is what it would look like on a number line.

Irrationals on Number Lines

Notice how the red tick marks aren't on the whole number tick marks. All irrational numbers are decimals, so they fall in between the whole numbers. When the first number of the decimal portion of the irrational number is less than 5, the tick mark made on the number line should be closer to the whole number part of the irrational number. When the first number of the decimal portion of the irrational number is greater than 5, the tick mark made on the number line should be farther away from the whole number part of the irrational number. What if the first number of the decimal portion of the irrational number is exactly 5? Then the tick mark will be exactly halfway between the the whole number part of the irrational number and the next whole number on the number line.

Let's take a look at a couple more examples to graph:




Use your calculator to find the decimal version of the Irrational Numbers.




Now we can put these in order.


Then last is


We have the order of these numbers, let's think about where to place their tick marks on the number line. For


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