Concept
What Is a Square Root?
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The Definition
The square root of a number n is a value x such that x² = n. In notation:
√n = x means x × x = n, where x ≥ 0
For example: √36 = 6 because 6 × 6 = 36. The symbol √ is called the radical sign or radical symbol, and the number under it is called the radicand.
Perfect Squares
Numbers whose square roots are whole integers are called perfect squares:
| Number | Square root | Check |
|---|---|---|
| 1 | 1 | 1 × 1 = 1 |
| 4 | 2 | 2 × 2 = 4 |
| 9 | 3 | 3 × 3 = 9 |
| 16 | 4 | 4 × 4 = 16 |
| 25 | 5 | 5 × 5 = 25 |
| 100 | 10 | 10 × 10 = 100 |
Non-Perfect Squares
Most numbers are not perfect squares. Their square roots are irrational — they go on forever without repeating. For example:
- √2 ≈ 1.41421356…
- √3 ≈ 1.73205080…
- √5 ≈ 2.23606797…
These cannot be written as exact fractions. They are computed using calculators or approximation methods.
The Radical Sign: Parts Explained
The full radical notation has three parts:
- Radical sign (√) — the tick-shaped symbol itself
- Radicand — the number or expression under the symbol
- Vinculum — the horizontal bar extending over the radicand (in typeset math)
- Index — a small number in the notch showing which root (2 for square root, 3 for cube root ∛, etc.); for square roots the index 2 is usually omitted
Is the Result Always Positive?
Yes. By convention, √ always returns the principal root — the non-negative value. So √9 = 3, not −3, even though (−3)² = 9 as well. To express both roots of an equation, mathematicians write ±√9 = ±3.
Square Roots in Real Life
- Geometry: The diagonal of a square with side s is √2 · s
- Physics: The speed of a falling object involves √(2gh)
- Statistics: Standard deviation uses the square root of variance
- Engineering: Signal processing, RMS calculations, impedance
Frequently Asked Questions
Is √9 equal to 3 or ±3?
By convention, √9 = 3 only (the principal root). To include both roots, write ±√9 = ±3. The symbol √ on its own always means the non-negative root.
What is the square root of 0?
√0 = 0. Zero is its own square root because 0 × 0 = 0. It is the only number with just one square root.
Can you take the square root of a negative number?
Not in real numbers. In complex numbers, √(−1) = i (the imaginary unit). So √(−4) = 2i, and so on. Real-number calculators will show an error for negative inputs.
What is the difference between square root and cube root?
The square root (√) finds a value x where x² = n. The cube root (∛) finds a value where x³ = n. For example: ∛27 = 3 because 3³ = 27. The cube root can be negative: ∛(−8) = −2.
How do I calculate a square root without a calculator?
For perfect squares, memorise the list above. For other numbers, use the Babylonian method: start with a guess g, then repeatedly compute g = (g + n/g) / 2 until it converges. This gives a close approximation within a few iterations.