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## 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?

• 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?
• anything that doesn't have a pattern, goes on forever, and has a decimal is basically an irrational number
• 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.
• Is Sal saying there are more irrational numbers than rational numbers?
• Although Sal didn't mention it, there are more irrational numbers than rational numbers. It's a tricky concept since there are an infinite number of both. However, in the nineteenth century, Gregory Cantor showed how it was possible to think about different cardinalities, sizes if you like, of infinite sets.

It turns out the set of all even numbers, the set of all integers and the set of all rational numbers are all the same "type" of infinity (called Aleph nought or null). While the set of all irrational numbers is larger (infinitely larger); it is uncountably large. That is, you can't map it onto the set of counting numbers.

See http://en.wikipedia.org/wiki/Aleph_number for more details.
• 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?
• 0.1212... is definitely a rational number. The fraction is 4/33
• How is 0.3 = 1/3
• Its not really. Just 0.3 equals 3/10. However, if you put a macron (the equivalent of a underscore that goes over characters) over the 3, you would show that it repeats forever.
(1 vote)
• 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.
• At about , Sal said that a rational number plus an irrational number equals an irrational number. what about an irrational number plus an irrational number?
• It depends.
Something like Sqrt(2) + Sqrt(3) is still irrational, but obviously something like Pi + (-Pi) = 0 is rational.