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Surds

If you 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 (it equals 2), so it is not a surd!

Have a look at some more examples:

Number Simplifed As a Decimal Surd or
not?
√2 √2 1.4142135(etc) Surd
√3 √3 1.7320508(etc) Surd
√4 2 2 Not a surd
√(1/4) 1/2 0.5 Not a surd
3√(11) 3√(11) 2.2239800(etc) Surd
3√(27) 3 3 Not a surd
5√(3) 5√(3) 1.2457309(etc) Surd

As you can see, the surds have a decimal which goes on forever without repeating, and that makes them Irrational Numbers.

In fact "Surd" used to be another name for "Irrational", but it is now used for a root that is irrational.

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")

 

Conclusion

If it is a root and irrational, it is a surd.

But not all roots are surds.