Surds are used commonly in math, they just are not referred to as surds. A surd is any positive number that is in square root form. Once you simplify the surd it must form a positive irrational number. If a natural number is formed, it is not considered to be a surd.

Infinite surds are just surds forming a sequence that goes on forever. The exact value of an infinite surd is expressed in the square root form. When the infinite surds in those sequences are simplified, a never ending sequence of irrational numbers are revealed.

The following is an the first example of an infinite surd:

You may be wondering why this can be classified as a surd when is not a surd. simplified forms natural number and cannot be classified as a surd. As you can see in the first 10 terms of the infinite surd, they are all irrational numbers.

a1: = 1.41421 ...

a2: = 1.55377 ...

a3: = 1.59805 ...

a4: = 1.61184 ...

a5: = 1.61612 ...

a6: = 1.61744 ...

a7: = 1.61785 ...

a8: = 1.61797 ...

a9: = 1.61801 ...

a10: = 1.61802 ...

From the first ten terms of the sequence you can see that the next sequence is

the previous term. Turning that into a formula for an+1 in terms of an makes:

From the plotted points of the infinite surd in graph 1, you can see that the more n increases, the closer an gets to the value of about 1.6181 but never touches it. This shows that the rise of the slope is getting increasingly smaller, so as n gets very large, an- an+1 gets closer to 0. Since the sequence goes on forever, it cannot be determined if an- an+1=0.