surd

In mathematics, a surd refers to an irrational root, such as (\sqrt{3}), or more broadly, an irrational number. This concept was significant in ancient Greek mathematics, particularly for the Pythagoreans, who believed all quantities could be expressed by whole numbers or their ratios (rational numbers).12

The discovery of surds, or incommensurable quantities, challenged this belief. For example, the diagonal of a square with sides of length 1 unit is (\sqrt{2}), which cannot be expressed as a ratio of two integers. This led to an impasse in mathematical reasoning until mathematicians like Theaetetus and Eudoxus developed definitions of proportion that could account for these irrational numbers.2