1. Surds
An Irrational Number is a real number that cannot be written as a simple fraction.
Irrational means not Rational
Example: 1.5 is rational, because it can be written as the ratio 3/2
Example: 7 is rational, because it can be written as the ratio 7/1
Example (3 repeating) is also rational, because it can be written as the ratio 1/3

4. How did we get the word "Surd" ?
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!
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")

5. Surds Surds are square roots that can’t be written as a whole number.
is not a surd, as it can be written as 5
7 is a surd, as it can’t be written as a whole number.
There are rules that we can follow that will allow us to simplify a surd as much as possible.

6. 𝒂𝒃 = 𝒂 𝒃 Simplify 12 + 2 27 = 4 3 + 2 9 3 = 2 3 + 6 3 Example 1:
𝒂𝒃 = 𝒂 𝒃
This rule helps us simplify surds by splitting the surd into a surd and a non-surd The non-surd can then be simplified into a whole number.
Example 1:
Simplify   = = =

7. 𝒂 𝒃 = 𝒂 𝒃 Simplify 44 11 + 25 36 = 44 11 + 25 36 = 4 + 5 6 = 2 + 5 6
𝒂 𝒃 = 𝒂 𝒃
Example 2:
Simplify = = = =