Concept
Is the Square Root Symbol Always Positive?
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√9 = 3 (not ±3)
The square root symbol always returns a non-negative value by definition.
The Principal Root Convention
Every positive number has two square roots — one positive and one negative. For 9, those roots are +3 and −3, because both 3² and (−3)² equal 9.
But the square root symbol √ is defined to return only the principal root — the non-negative one. This convention was established to make √ a proper mathematical function: a function must give exactly one output for each input, so √9 must equal one number, not two.
The rule: √x always equals the non-negative square root of x, for all x ≥ 0.
Why Not Both Roots?
Functions require a unique output. If √9 could equal both 3 and −3, √ would not be a function — it would be a relation. By fixing √ to the principal root, mathematicians can write rules like:
- √(x²) = |x| for all real x
- √(ab) = √a · √b for a, b ≥ 0
- d/dx [√x] = 1/(2√x)
These identities only hold consistently when √ means the non-negative root.
When Both Roots Matter: ±√
In algebra, when solving equations like x² = 9, both solutions matter. Mathematicians make this explicit by writing ±√9:
| Expression | Value | Meaning |
|---|---|---|
| √9 | 3 | Principal (positive) root only |
| −√9 | −3 | Negative root only |
| ±√9 | 3 or −3 | Both roots (used in equations) |
What About √0 and √ of Negative Numbers?
√0 = 0 — zero has only one square root: itself.
√(−1) is not a real number. Square roots of negative numbers require the imaginary unit i, where i² = −1. In complex numbers, √(−9) = 3i, still taking the principal value by convention.
Common Mistakes
- Mistake: Writing √9 = ±3
Correct: √9 = 3. Use ±√9 when both roots are intended. - Mistake: Assuming √(x²) = x
Correct: √(x²) = |x|, because √ cannot return a negative value. - Mistake: Writing √(−4) = −2
Correct: √(−4) is not real; it equals 2i in complex arithmetic.
Frequently Asked Questions
Is √9 equal to 3 or −3?
By convention, √9 = 3. The square root symbol always denotes the principal (non-negative) root. To express both roots write ±√9 = ±3.
Why does √(x²) equal |x| and not x?
Because √ always returns a non-negative result. If x = −3, then x² = 9 and √9 = 3 = |−3|. Writing √(x²) = x would incorrectly give √9 = −3 when x is negative.
Can a square root ever be negative?
The value of the square root symbol √ is never negative for real inputs. However, the equation x² = 9 has two solutions: x = 3 and x = −3. The negative solution is expressed as x = −√9, not as √9 itself.
What is the square root of a negative number?
In real numbers, the square root of a negative number is undefined. In complex numbers, √(−1) = i (the imaginary unit), and √(−n) = √n · i for any positive n.