Surds · ⁨Radicales⁩

The number that broke the Greeks

• The diagonal of a unit square is $\sqrt{2} \approx 1.41421356\dots$ — the digits never repeat and never end.
• No fraction equals it exactly. That's why we call it a surd 根式: an irrational 无理数 root that we leave in its exact form.

What is a surd?

• A surd is a root that cannot be simplified to a rational number.
• $\sqrt{5}$, $\sqrt{7}$, $\sqrt[3]{10}$ are surds — they are irrational.
• $\sqrt{4} = 2$ and · ⁨ y ⁩$\sqrt[3]{27} = 3$ are not · ⁨no⁩ surds — they simplify to integers.

Surds come from diagonals: a $1 \times 1$ square has diagonal $\sqrt{2}$; a $2 \times 2$ square has diagonal $\sqrt{8} = 2\sqrt{2}$.

Surd rules

• $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ — combine under one root.
• $\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}$ — divide under one root.
• Simplify by pulling out the largest perfect square factor:

Rationalising 有理化 the denominator

• A fraction with a surd in the denominator is not in simplest form.
• Multiply top and bottom by the surd to remove it from the bottom:

Why keep surds exact?

• $\sqrt{2} \approx 1.414$ is an approximation; $\sqrt{2}$ is exact.
• In engineering and physics, rounding too early causes errors to accumulate — keeping surds preserves accuracy through long calculations.

Like surds and two-term denominators

• Simplify each root first: $\sqrt{200}-\sqrt{32}=10\sqrt2-4\sqrt2=6\sqrt2$. Combine only matching surd parts.
• For · ⁨A favor ⁩$1/(\sqrt3-1)$ multiply by the conjugate $\sqrt3+1$: the denominator is $(\sqrt3-1)(\sqrt3+1)=3-1=2$. Thus the result is $(\sqrt3+1)/2$; the multiplying fraction equals 1.

You've got it