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Interval Notation Calculator

Created by Anna Szczepanek, PhD
Reviewed by Rijk de Wet
Last updated: Jan 18, 2024


This interval notation calculator will help you with your math assignments by explaining how to write in interval notation — and the other way around, since our interval notation converter works in both ways, from inequalities to intervals and from intervals to inequalities!

If you're not yet very familiar with this topic or need a refresher, scroll down to read what interval notation is and how to use the various interval notation symbols. As a bonus, we will discuss the tricky problem of writing the set of all real numbers in interval notation!

What is interval notation?

Interval notation is a way of describing certain subsets of the real line. It concerns subsets that contain all numbers between some two bounds: the interval [a, b] corresponds to the set of all real numbers between a and b, including a and b, i.e., a ≤ x ≤ b. To exclude both a and b, we write (a, b), which is equivalent to a < x < b.

🔎 It's crucial that you understand intervals well, as they will keep reappearing as you learn more math and science. For instance, the so-called confidence intervals are one of the main tools of modern statistics!

How do I write in interval notation?

To write a set in the interval notation:

  1. Write the left-most number of the set (formally: its infimum).

  2. If the infimum belongs to the set, precede it by a square bracket [. Otherwise, precede it with a parenthesis (.

  3. Put a comma after the number.

  4. Write the right-most number of the set (formally: its supremum).

  5. Repeat Step 2 for the supremum, adding ] or ). That's it!

  6. Don't hesitate to check your result with an online interval notation calculator.

As you can see, it's not that hard to learn how to write in interval notation. However, remembering all the interval notation symbols and their order requires a bit of practice. Omni's interval notation converter is here to provide you with as many examples as you wish!

🙋 Want to learn about intervals and inequalities on the real line with the help of colorful pictures? Check out Omni's graphing inequalities on a number line calculator and absolute value inequalities calculator!

How to use this interval notation calculator?

To most efficiently use our interval notation converter:

  1. Choose the conversion direction: do you want to go from inequality to interval notation or the other way round?
  2. Input your problem: the endpoints of an interval or the inequality you want to convert. As you see, our interval notation calculator can deal with compound inequalities as well!
  3. The solution appears immediately. Our tool will simplify it for you if possible. Enjoy!

FAQ

How do I write all real numbers in interval notation?

The interval notation for the set of all real numbers reads (-∞,∞). Recall that is the symbol for infinity: all real numbers are those that lie between minus infinity and plus infinity.

When should I use brackets in interval notation?

Use square brackets in interval notation if you want to say that the endpoint of the interval belongs to the set in question. If both endpoints belong to the set, put square brackets around your interval: [a, b]. If only one of them belongs to the set, put one square bracket and one parenthesis: [a,b) or (a,b].

What is interval notation from 1 to 2?

There are four possible forms of the interval notation from 1 to 2, depending on whether these endpoints are included in the set:

  • If they are both included, write [1,2];
  • If neither are included, write (1,2);
  • If 1 is included but 2 is not, write [1,2);
  • If 1 is not included but 2 is, write (1,2].
Anna Szczepanek, PhD
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