SURDS VS IRRATIONAL NUMBERS

Let us recall what is an irrational number and what is a surd.

Rational numbers can be expressed in the form where p and q are integers.
Rational numbers are non-terminating but have a recurring decimal part.

The numbers that can not be expressed in the form of are known as irrational numbers.
When written as decimals, irrational numbers neither terminate nor repeat.

When a positive rational number is not a power of 'n',
then the nth root of that number is irrational.
It is called as a surd.

Examples of irrational numbers that are not surds:

All non-terminating and non-repeating decimal numbers are irrational but not surds.
  • 2.3154261587.....
  • 3.1415926535... which is π(pi - ratio of circumference of a circle to its diameter)
  • 2.7182818284... which is e(Euler's number)
  • 1.6180339887... which is Φ(Golden ratio)
  • 0.30102999566... which is log 2(logarithm of 2 to the base 10)

We can not express them as

Therefore, all these are not surds.

Image

CONCLUSION :

All surds are irrational. But the converse is not true.
i.e, all irrational numbers are not surds.
This is represented by Venn diagrams in Set theory as: