Step-By-Step Guide to Adding Irrational Numbers

Step-By-Step Guide to Adding Irrational Numbers
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What is an Irrational Number?

Before we go ahead to adding, first you have to understand what makes a number irrational. The definition of an irrational number is a

number that cannot be written as a ratio of two integers. So the number 1.25, for example, would be rational because it could be written as 5/4. The number 0.3333333 (with a repeating 3) could be written as 1/3. In fact, any terminating decimal (decimal that stops after a set number of digits) or repeating decimal (decimal in which one or several digits repeat over and over again, without terminating), is rational.

So what numbers are irrational? The main example of an irrational number is a number that contains a square root. Therefore, √2 is an irrational number, as is 2√57. (Obviously, √4 is rational, because it is equal to “2,” a rational number.) Other examples of irrational numbers are pi(∏) and e, neither of which can be represented by a ratio of two integers.

Adding Square Roots

In order to add square roots, you can add only “like terms.” This may sound familiar from pre-algebra, in which you had to find “like terms” in order to add coefficients together. In pre-algebra, you looked at 3x2 and x2 as like terms because they both contained x2. You then added the coefficients - or the digits before the like terms - together to get 4x2.

The same works with square roots. For example, you can add 3√3 and 2√3 to get 5√3, in the same way that you can add 3x and 2x to get 5x. You cannot, however, add 3√3 and 2√2, in the same way that you cannot add 3x and 2x2.

You may, however, need to simplify the square roots before you can see whether they contain like terms. For example, if you were given the problem “3√2 + √8,” you might think that the terms cannot be added together. In truth, you could simply rewrite √8 to be √4 X √2, which is 2√2. So the results would be 3√2 + 2√2, which is 5√2.

Adding Other Irrational Numbers

The same would be true with adding other irrational numbers, such as ∏ and e. For example, if you had the problem “2∏ + 6∏,” you could easily add them together to get 8∏. If you had the problem “2∏ + 8_e_,” however, you would not be able to add the two terms together. The same would be true if you had a problem that contained both a square root and another irrational number, such as “2∏ + 2√2.”

Adding irrational numbers is actually quite simple, once you get the hang of it. The key is to find any like terms, and then add the coefficients together. Try it with the following problem, to make sure you have it right.

2√18 + 3√2 + √32 + √2 = ?

(The answer would be 6√2 + 3√2 + 4√2 + 1√2 = 14√2.)


This post is part of the series: Math Study Guides

Confused in math class? These math study guides span various topics, from square roots to improper fractions.

  1. Adding Irrational Numbers: A Step-by-Step Guide
  2. Two Techniques for Adding Mixed Fractions - With Examples
  3. Steps for Dividing Mixed Fractions with Examples and Resources
  4. Learn How to Solve Square Root Math Problems: Examples and Resources