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## Algebra 1

### Course: Algebra 1>Unit 15

Lesson 1: Irrational numbers

# Intro to rational & irrational numbers

Learn what rational and irrational numbers are and how to tell them apart. Created by Sal Khan.

## Want to join the conversation?

• Is Sal saying there are more irrational numbers than rational numbers? •   Wrath,
Actually, Sal was saying that there are an infinite number of irrational numbers. And there is at least one irrational number between any two rational numbers. So there are lots (an infinite number) of both.

And saying one thing that is infinite is more than another infinite thing is questionable because you can't add to infinite. Infinite goes on forever. It is a hard concept to completely comprehend.

For instance, there are an infinite number of decimals between 0 and 1.
And there are an infinite number of decimals between 0 and 2.
And there are numbers between 0 and 2 that are not between 0 and 1. But "goes on forever" so you can't really say there are more numbers between 0 and 2 than between 0 and 1. Both go on forever.

Infinite is a concept of "going on forever" and is not something that can be added to to or multiplied. And you can't say that one infinite is more than another infinite, even though logically you might think there are twice as many numbers between 0 and 2 that there are between 0 and 1. It is also not correct to say that the numbers between 0 and 1 and the numbers between 0 and 2 are the same. One infinite is not greater than, less than, or even equal to, another infinite. Infinite goes on forever.

(Edited)
There are some math concepts that do compare infinities. See the Peter Collingridge comment below. The infinity of irrational numbers is more than the infinity of positive integers. So I was partly incorrect. That is not to say the the infinity of irrational numbers is larger than the infinity of rational numbers.

The concept of infinity is very hard to grasp.
• I'm getting stuck on the irrational number part. An irrational number is any number that doesn't divide into a fraction? Anything that is pi? • Can some one explain what a rational number is i am still confused • A rational number is a number that can be express as the ratio of two integers. A number that cannot be expressed that way is irrational.
For example, one third in decimal form is 0.33333333333333 (the threes go on forever). However, one third can be express as 1 divided by 3, and since 1 and 3 are both integers, one third is a rational number.

Likewise, any integer can be expressed as the ratio of two integers, thus all integers are rational.

However, numbers like √2 are irrational because it is impossible to express √2 as a ratio of two integers.

The first irrational numbers students encounter are the square roots of numbers that are not perfect squares. The other irrational number elementary students encounter is π.
• Sal had a list of intriguing irrational numbers. What are they, and how can they be applied?

e?
square root of 2?
golden ratio?

Thanks. • You would probably not need to apply those numbers in Algebra 1 however they are quite useful. For example, "e" is the basis of calculus and appears in a lot of limits and functions. The square root of 2 is the hypotenuse of a right-angled triangle with both sides 1 and can be seen through the exact value of certain trigonometric functions. The golden ratio is a number that people claim is spread all throughout nature and can be seen through many series such as the Fibonacci numbers.
Hope this helps.
• Is a two digit, repeating decimal ( 4 example: 0.12121212...) a rational number
or an irrational number? • How is 0.3 = 1/3 • pie is an irrational number, which means it cannot be expressed in p/q form, where p and q are integers, but pie = 22/7 pls explain. • Pi does not equal 22/7. It is just an approximation for Pi. 22/7 is a repeating decimal. Pi is a never ending and never repeating decimal.
22/7 = 3.142857142857...
Pi = 3.1415926535897932384...
As soon as you get to the 3rd decimal digit, the numbers are different.

Hope this helps.
• How do you know what numbers are rational • A another trait of rational numbers is their decimal form. It will be either a terminating decimal (a decimal with a few decimal places, then stops), or a repeating decimal (a decimal with digits going on forever, but in a pattern so you know what comes next.
Fun Fact: if a number is not divisible by nine, then the repeating digit is what the remainder is, so if the remainder is 4, the decimal goes 44444444444444... forever.  