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