Main content

### Course: Algebra 1 > Unit 15

Lesson 3: Proofs concerning irrational numbers# Irrational numbers: FAQ

Frequently asked questions about irrational numbers

## What is an irrational number?

An irrational number is a real number that cannot be written as a ratio of two integers. In other words, it can't be written as a fraction where the numerator and denominator are both integers. Irrational numbers often show up as non-terminating, non-repeating decimals.

Learn more with our Intro to rational & irrational numbers video.

## Where do irrational numbers come up in the real world?

Irrational numbers show up all over the place! For example, the number $\pi $ is irrational and it's key for working with circles. The square root of $2$ , another irrational number, is important for understanding right triangles.

## How can we tell if a number is rational or irrational?

If we can write the number as a fraction of two integers, then it's rational. Otherwise, it's irrational.

Practice with our Classify numbers: rational & irrational exercise.

## Are there any rules for adding or multiplying rational and irrational numbers?

Yes! When we add or multiply two rational numbers, we'll always get a rational number as the result. But when we add or multiply a rational number with an irrational number, we'll end up with an irrational number.

Learn more with our Proof: sum & product of two rationals is rational
video.

Learn more with our Proof: product of rational & irrational is irrational video.

Learn more with our Proof: sum of rational & irrational is irrational video.

## What do we know about the sum and product of two irrational numbers?

There are a few things to keep in mind. For one, the sum of two irrational numbers is not always irrational. For example, $\sqrt{2}+\sqrt{18}=4\sqrt{2}$ , which is another irrational number. However, $\sqrt{2}+(-\sqrt{2})=0$ , which is rational.

Likewise, the product of two irrational numbers is not always irrational. For example, $\sqrt{2}\times \sqrt{2}=2$ , which is rational.

Learn more with our Sums and products of irrational numbers video.

## Want to join the conversation?

- what do you get if you sqrt a negative number?(11 votes)
- You can't take the square root of a negative number. It will result in an imaginary number (which sounds made up, but is 100% real).

The imaginary number is denoted by i, and i² = -1.

(So the √-4 is 2i)

You can look at an intro to it here: https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:imaginary/v/introduction-to-i-and-imaginary-numbers(17 votes)

- what ifit was a negitive(8 votes)
- Regardless of the sign, if it can be represented by a fraction then it's rational(10 votes)

- The sum of two irrational numbers is SOMETIMES irrational.(5 votes)
- Most of the time it is irrational unless they are additive inverses such as √3 + )-√3) = 0.(13 votes)

- So, if I'm correct, then when you multiply an irrational number by a rational number, you will never get an exact number because you can never find the value of the irrational number. If so, then when do we ever get irrational numbers and when do we actually need to use these in real life?(9 votes)
- When you use irrational numbers, such as pi, you usually round to a certain decimal place. _e_ is an interesting example of a common irrational number. It is found in spirals, like the head of a sunflower.(5 votes)

- someone already asked my question, this is very helpful! thanks!(11 votes)
- irrational is a number that goes on forever like 3.14 rational is a number that can be turned into a fraction(7 votes)
- pi = 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482(6 votes)
- What do you mean when you say "write the number as a fraction of two integers?" Can you define integers and explain how to define whether or not a number is rational or irrational?(6 votes)
- An integer is a number with only 0 after the decimal point, such as 3 (3.0) or -645 (-645.0). If we can write a number as a fraction that has both an integer numerator and an integer denominator, then it is rational. If we can't, then it is irrational. Repeating decimals are rational, because we can write them as fractions. For example, if we take 0.7--->, we can write it as 7/9.(0 votes)

- does any body got cookies?](5 votes)
- when an irrational number is multiplied another irrational number, doesn't it become a rational?(2 votes)
- you can't determine whether the resulting number is rational or irrational. the square root of two times the square root of two is two, but pi times pi is still an irrational number.(5 votes)