Math Lesson 7.4.1 - What Are Surds?

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Welcome to our Math lesson on What Are Surds?, this is the first lesson of our suite of math lessons covering the topic of Surds, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

What Are Surds? What is the Difference between Surds and Transcendental and Irrational Numbers?

When the term "surds" was first introduced, mathematicians intended it as an alternative name for irrational numbers. So, basically, in that original sense, they considered surds and irrational numbers as representing the same thing.

However, now surds and irrational numbers mean different things. More precisely, surds represent only a subset of irrational numbers. This means there are many irrational numbers that are not surds but on the other hand, all surds are irrational numbers.

Let's explain in more detail the difference between surds and irrational numbers by introducing the subset of irrational numbers that do not belong to surds. All numbers included in this subset are called 'transcendental numbers' and we may encounter them in the form of constants used in science. For example, π, e (Euler's Number), trigonometric functions such as sin α, cos α, tan α and cot α if the angle is measured in radians, etc., are all transcendental numbers.

By definition, a transcendental number is an irrational number that is not the root of any number, meaning that it is not an algebraic number of any degree.

From here, by induction, we obtain the definition of surds. Thus, surds are all numbers representing unresolved roots of any real number.

For example, √2;√3; √8; ∛11; 29; √1/6; etc., are all surds because their value (although irrational) is obtained by calculating the roots of either integers or rational numbers (integers are rational numbers though). Indeed, when a calculator is used to find the value of the above numbers, it gives

√2 = 1.4142135623730950488016887242097…
√3 = 1.7320508075688772935274463415059…
√8 = 2.8284271247461900976033774484194…
∛11 = 2.2239800905693155211653633767222…
29 = 1.9610090574545480132063568500978…
1/6 = √0.16666666… = 0.40824829046386301636621401245098…

On the other hand, numbers like √25; √2.8224; ∛0.125; etc., are not surds because the result of these roots is a finite number. Indeed,

√25 = 5 because 52 = 25
√2.8224 = 1.68 because 1.682 = 2.8224


∛0.125 = 0.5 because 0.53 = 0.125

The following diagram shows the relationship between surds, transcendental numbers and irrational ones.

Math Tutorials: Surds Example

To summarize, a surd is a root of a positive real quantity if its value cannot be exactly determined.

Example 1

Which of the following numbers represents a surd?

  1. √121
  2. ∛125
  3. √179.56
  4. √21

Solution 1

Using a calculator to make the operations, we obtain

  1. √121 = 11
  2. ∛125 = 5
  3. √179.56 = 13.4
  4. √21 = 4.582575694955840006588047193728…

From all numbers, only √21 gives an infinite non-recurring part after the decimal point. All the other roots are finite, so they are not surds. Only √21 is a surd.

More Surds Lessons and Learning Resources

Powers and Roots Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
Lesson IDMath Lesson TitleLessonVideo
7.4.1What Are Surds?
7.4.2What are the different Types of Surds
7.4.3Properties of Surds
7.4.4Basic Operations with Surds

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