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The Exact Truth
The Exact Truth
When ancient Greek mathematicians discovered that the diagonal of a unit square has length √2, they found a number that couldn't be written as a fraction. This discovery shook mathematics! Today, we keep these 'irrational' numbers in their exact root form - called surds - because they're more precise than any decimal approximation. Engineers use surds for exact calculations in bridge designs, physicists for quantum mechanics, and even your phone's GPS calculations involve surds.
What is a surd?: An irrational root that cannot be simplified to a whole number
Examples: √2, √3, √5, ∛7
NOT surds: √4 = 2, √9 = 3 (these simplify to whole numbers)
Mathematics glossary
- What is a surd?
- An irrational root that cannot be simplified to a whole number Examples: √2, √3, √5, ∛7 NOT surds: √4 = 2, √9 = 3 (these simplify to whole numbers)
Earn the mark scheme marks
Common Mistakes to Avoid
❌ Wrong: √a + √b = √(a+b)
✅ Right: Cannot combine different surds
√9 + √16 = 3 + 4 = 7, NOT √25 = 5
❌ Wrong: √18 = 9√2
✅ Right: √18 = 3√2
√18 = √(9×2) = 3√2 (take square ROOT of 9)
❌ Wrong: 3√2 × 2√3 = 5√5
✅ Right: 3√2 × 2√3 = 6√6
Multiply coefficients AND surds: 3×2=6, √2×√3=√6
❌ Wrong: Leaving denominator with surd
✅ Right: Always rationalize
1/√2 should be written as √2/2
Now try it yourself
Quiz · Question 1 of 14
Which of these is the simplified form of √48?
Tap an answer to check it
This topic in real past papers
Every real exam question we've found on surds, with a full worked answer.