Surds
| English | Chinese | Pinyin |
|---|---|---|
| surd | 根式 | gēn shì |
| irrational | 无理数 | wú lǐ shù |
| rationalising | 有理化 | yǒu lǐ huà |
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.
Surd simplification route
Break a surd into square factors and simplify it.
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 $\sqrt[3]{27} = 3$ are not 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:
Roots don't add. $\sqrt{a} + \sqrt{b} \neq \sqrt{a+b}$. For example, $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$, but $\sqrt{25} = 5 \neq 7$.
Simplify √20 into the form k√5. What is k?
√20 = √(4×5) = √4 × √5 = 2√5, so k = 2.
Which surd rule is correct?
√a × √b = √(ab); roots do not add like √a + √b = √(a+b).
Simplify √72 into the form k√2. What is k?
√72 = √(36×2) = 6√2, so k = 6.
√9 + √16 = √25.
√9 + √16 = 3 + 4 = 7, but √25 = 5. Roots do not add: √a + √b ≠ √(a+b).
Complete the rule: √a × √b = √(____).
√a × √b = √(ab). You can combine two square roots into one by multiplying under the root.
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:
$\dfrac{6}{\sqrt{3}} = \dfrac{6\sqrt{3}}{3} = 2\sqrt{3}$. Multiply by $\dfrac{\sqrt{3}}{\sqrt{3}} = 1$, then simplify.
Rationalise 10/√5 into the form k√5. What is k?
10/√5 × √5/√5 = 10√5/5 = 2√5, so k = 2.
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.
You've got it
- a surd is an irrational root; keep it exact (don't approximate)
- simplify by the largest square factor: $\sqrt{20} = 2\sqrt{5}$, $\sqrt{72} = 6\sqrt{2}$
- rationalise the denominator by multiplying top and bottom by the surd
- $\sqrt{a} + \sqrt{b} \neq \sqrt{a+b}$ — roots don't add