When we can't simplify a number to remove a square root (or cube root etc) then it is a surd.

Example: 2 (square root of 2) can't be simplified further so it is a surd

Example: 4 (square root of 4) can be simplified (to 2), so it is not a surd!

Have a look at these examples (including cube roots and a 5th root):

Number Simplified As a Decimal Surd or
2 2 1.4142135...(etc) Surd
3 3 1.7320508...(etc) Surd
4 2 2 Not a surd
¼ ½ 0.5 Not a surd
311 311 2.2239800...(etc) Surd
327 3 3 Not a surd
53 53 1.2457309...(etc) Surd

The surds have a decimal which goes on forever without repeating, and are Irrational Numbers.

How did we get the word "Surd" ?

Well around 820 AD al-Khwarizmi (the Persian guy who we get the name "Algorithm" from) called irrational numbers "'inaudible" ... this was later translated to the Latin surdus ("deaf" or "mute")

And now surd means a root that is irrational.


  • When it is a root and irrational, it is a surd.
  • But not all roots are surds.


2043,320, 2044, 3162, 3163, 3865, 3866, 3867, 3868, 3869