If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Algebra 1

### Course: Algebra 1>Unit 15

Lesson 2: Sums and products of rational and irrational numbers

# Sums and products of irrational numbers

The sum of two irrational numbers can be rational and it can be irrational. It depends on which irrational numbers we're talking about exactly. The same goes for products for two irrational numbers. This video covers this fact with various examples.

## Want to join the conversation?

• at 4.50 Sal says that pi squared is irrational .how do we know that?
• Is it always the case that when we multiply two irrational numbers, the product will be EITHER an irrational number OR an integer? I am asking because I have yet to see an example where the product of two irrational numbers yields a rational non-integer.
• This is not always the case. A counter example would be √1.125 * √2 = 1.5
• how is pi+1-pi= 1 because those numbers are both irrationals. Also if pi+pi=2 pi
then what difference does it make to pi being subtracted by 1, pi has an infinite amount of numbers so you cant subtract it by 1. Even if you can wouldn't the number replace itself or something because the numbers pi have are INFINITE!!. I also want to confirm this method to so that i don't have anything else to say,let me assume that a very dumb person; ahmm (myself), assumed that if pi + pi = 2 pi but there is a minus 1 there so i can subtract 2 by 1 it would still make it pi? please dont make fun of my stupidity im only 11 so if you can pleas answer my question i will be very grateful
• pi + 1 - pi addition is commutable, so you can move things around as long as you keep the sign, so pi - pi + 1 is the same, and anything minus itself (even irrational numbers) is always 0, so all that is left is 1 = 1
If pi is 3.14159..., so if you subtract 1, you would have 2.14159..., but as you expected it does not mean much.
• at , could someone please explain why 1/π is irrational, because isn't it just the ratio of two numbers?
• Rational numbers must be the ratio of 2 integers. Pi is not an integer. It is an irrational number. 1/Pi is also an irrational number.
• can you please teach me the sum of rational numbers
• No.
But fine:
By definition, a rational number can be expressed as a fraction with integer values in the numerator and denominator (denominator not zero). So, adding two rationals is the same as adding two such fractions, which will result in another fraction of this same form since integers are closed under addition and multiplication. Thus, adding two rational numbers produces another rational number.

statementbox "The sum of two irrational numbers is SOMETIMES irrational."

Product of a two rational numbers is rational
• At , he says pi-pi=c when he plugged it in for a and b, but wouldn't it be a + a then because he is using the same number for different variables?
• Yes your right the variables are not a and b as a and b are differet variables, it should have been a+a or 2a.
(1 vote)
• Irrational minus irrational is equal to
(1 vote)
• At in the video, Sal tells you that the result can be rational or irrational. It depends upon what irrational numbers are being subtracted.
• How do you prove that the product of two irrational numbers can be rational?
(1 vote)
• Not always. We cannot prove this but it can be seen through examples. For example, √2*√2=2, which is rational but √2*√3=√6, which is irrational.