Part of the Stem Teachers Script Lesson

In this lesson you Introduction Hello Mathematicians, after this lesson, you will be able to state and apply the are going to definition of irrational numbers. learn…by 10-20 sec doing/using…

Connection You know that rational numbers are any number that can be expressed as a a/b where both a and b are and b is not equal to zero. Rational numbers (Define Terms/ include all integers, all and mixed numbers (made up of only integers) and Building on Prior You know that… Knowledge) all terminating or repeating .

30-60 sec

As you can see, rational numbers include nearly every number you can think of. But there are some that are not included, and these are our irrational numbers.

An is any number which cannot be expressed as a fraction a/b where a and b are both integers and b is not zero.

Demonstration I’m going to explain this idea 1-3 minc by showing you¦ As you think back to how the rationals included all the integers, any fraction made up of integers, and all terminating and repeating decimals, perhaps you can see that really the only type of number not already taken by the rationals is a that does not terminate and does not repeat. So irrational numbers are decimals that continue forever without terminating or repeating.

So if they never terminate and never repeat, how does one right them? Good question. Irrational numbers will always have to be represented by some type of symbol. These are the most common irrational numbers we will be dealing with:

... - One way to write irrational numbers is to write a decimal with a ... at the end. The ... indicates the decimal continues forever and never repeats. If the decimal did repeat, the repeating bar would have been used and the number would have been rational.

Square roots: When a number is a perfect , its will be an . But if a number is not a perfect square, its square root is an irrational number, a decimal that goes on forever without repeating. So the square root of 5 is an irrational number. In a previous lesson, we learned how to estimate the square root of 5, and it is just a little bigger than 2. But the result is a decimal that continues forever without repeating. The square root of any number that is not a perfect square will be an irrational number.

Pi: You have worked with the number when dealing the area and circumference of circles. Pi is an irrational number. It is a decimal that continues to the right forever and never repeats. Pi is typically estimated as the 3.14. But that is simply an estimate. The pi continues forever without repeating. Also, any term or fraction with pi in it will be irrational. Such as 7pi or pi/3.

There are other irrational numbers such as the number "e" and The , but you will visit those later in high school.

Let's try a quick application.

Which of the following are irrational numbers?

2pi, 3.14, 2.71830, 8.2 repeating, radical 7, radical 9, 4.212302...

Application Let’s see how this There are only three irrational numbers, did you figure them out? works in a 1-2 min problem¦

2pi is because any term that has pi in it is irrational.

3.14 is not, because it is a decimal that terminates. It is the estimation of pi, but it is a rational estimation of pi.

2.71830 is a long decimal, but it does end and is therefore rational.

8.2 repeating repeats and therefore is rational

Radical 7 is irrational because 7 is not a perfect square Radical 9 is rational because radical 9 is 3, which is an integer, which is rational

4.212302... is irrational because the ... at the end indicates the number continues forever without repeating.

How did you do?

Conclusion Now you know that an irrational number is a decimal that continues forever without So, now you know repeating and that irrational numbers must be represented by symbols. You can also how to…by… 10-20 sec determine whether a number is irrational.