What is a Square Root Calculator?
This calculator computes the square root of a number and simplifies radical expressions.
How it's calculated
√a = b ⇔ b² = a (b ≥ 0)
Step-by-step example
Simplifying √72
- Step 1 Break 72 into a perfect square times a remainder: 72 = 36×2
- Step 2 Since √36 = 6, √72 = √36×√2 = 6√2
- Step 3 Decimal approximation: 6×1.414 ≈ 8.485
Interpretation: Pulling out the largest possible perfect square gives the simplest form.
The irrational number that reportedly got someone killed
√2 cannot be expressed as a fraction — its decimal expansion (1.41421356…) goes on forever without repeating. The Pythagorean school believed everything could be expressed as a ratio of whole numbers, but the diagonal of a simple unit square works out to exactly √2 — a result from their own theorem that shattered their own worldview. Legend has it that Hippasus, who discovered or revealed this fact, was thrown into the sea for it. Whether the story is literally true or not, it captures how genuinely disruptive the discovery was.
Good to know
- Standard paper's aspect ratio is 1:√2 — the only ratio that stays exactly the same when folded in half, which is why paper sizes like A4/A3/A5 scale cleanly into each other.
- The √ symbol comes from a stylized letter 'r', for the Latin radix (root) — the same root behind calling it a 'square root.'
- √2 ≈ 1.414 and √3 ≈ 1.732 are commonly memorized as quick mental references, especially useful since √3 shows up constantly in three-phase electrical calculations.
- The square root of a negative number introduces the imaginary unit i — once dismissed as a mathematical curiosity, now essential to electrical engineering, quantum mechanics, and signal processing.
- Before calculators, students learned a manual long-division-style method for computing square roots by hand.
Frequently asked questions
Q. How do I simplify √8?
Since 8 = 4×2, √8 = 2√2 — pull the perfect square factor out.
Q. Is there a square root of a negative number?
Not in real numbers, but in complex numbers it's expressed using the imaginary unit i.
Q. Can I compute a cube root too?
Yes. An nth root is computed as a^(1/n).