Surds - Revision Notes

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In these revision notes for Surds, we cover the following key points:

  • What are surds?
  • What are transcendental numbers? Where do they differ from surds?
  • What number set do surds and transcendental numbers belong to?
  • What are the different types of surds?
  • What are the properties of surds?
  • What are the four basic operations with surds?

Surds Revision Notes

By definition, surds are all numbers representing unresolved roots of any real number. In other words, a surd is a root of a positive real quantity if its value cannot be exactly determined.

On the other hand, 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.

Hence, surds & transcendental numbers give the set of rational numbers.

Surds may belong to one (or more) of the following categories:

  1. Simple surds. This category includes surds that contain only one number inside the root.
  2. Pure surds. This category includes only surds obtained when taking the root of a prime number.
  3. Similar surds. This category includes surds that have common factors.
  4. Mixed surds. This category includes surds that contain numbers that can be expressed as products of rational and irrational numbers.
  5. Compound surds. This category includes the sum or difference of two or more individual surds.
  6. Binomial surds. This category includes two or more surds that when added or subtracted give a single surd.

Surds have four main properties, where the first two are identical to the properties of roots. The properties of surds are as follows:

Property 1

For any positive numbers a and b (or they may also be negative when n is odd), we have

a ∙ b = ab

Property 2

For any positive numbers a and b (or they may also be negative when n is odd), we have

a/b = a/b

Property 3

For any numbers a and b where b is positive, we have

a/√b = a√b/b

Property 4

For any numbers a, b and positive c (or it can also be negative when n is odd), we have

ac ± bc = (a ± b)c

We can complete basic operations with surds in the same way as with all the other types of numbers.

Addition and subtraction of surds

We can add or subtract only like surds. Some surds apparently look different, but after a few operations, we can convert them into like surds.

Multiplication and division of surds

Multiplication and division of surds is nothing more but the applications of the first two properties of surds in the reverse direction (from end to start). The only restriction is that surds must have the same index.

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  4. Powers and Roots Practice Questions: Surds. Test and improve your knowledge of Surds with example questins and answers
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  6. Continuing learning powers and roots - read our next math tutorial: Rationalising the Denominator

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