What is the square root of 2?

1. What “square root of 2” means in plain terms

The square root of 2, written √2, is the number which, when multiplied by itself, equals 2; that is the defining equation x^2 = 2 ^{[1] [3]}. Geometrically it is the length of the diagonal of a square whose sides measure one unit, a direct consequence of the Pythagorean theorem ^[1]. Practical descriptions for learners repeat this same idea: √2 is “the number that must be raised to the 2nd power to produce 2” ^{[4] [5]}.

2. Exactness versus approximation: why you usually see 1.4142

√2 cannot be written exactly as a finite decimal or simple fraction, so every decimal form is an approximation; common rounded values include 1.414 or 1.4142 for everyday use ^{[6] [3]}. More sophisticated computations have produced trillions of digits, but everyday arithmetic truncates or rounds for convenience; instructional sources often show iterative methods (long division, Newton’s method) that converge to 1.41421356… ^{[6] [7]}.

3. A fundamental irrational number with a long history

Historical surveys emphasize that √2 was likely the first number proved irrational, a discovery that challenged the Pythagorean belief that all magnitudes are ratios of integers; classical accounts connect this to ancient Greek mathematics and later formalizations in Renaissance and modern mathematics ^{[1] [8]}. Quanta’s account traces how mathematical language and rigor evolved so that numbers like √2 could be treated as bona fide mathematical objects ^[8].

4. How people compute or approximate it in practice

Textbooks and online tools demonstrate many algorithms: manual long-division-style methods, Newton-Raphson iterations, continued fractions, and modern high-precision computer algorithms; these all produce progressively better approximations to √2 ^{[6] [7]}. Educational pages and calculators show the same result in different formats — exact symbolic form √2 or decimal approximations — and code snippets or web tools exist to get as many digits as needed ^{[9] [10]}.

5. Useful rational approximations and where they matter

Although irrational, √2 can be approximated by fractions for engineering or design: 99/70 is a historical/industrial approximation cited in paper sizing standards and elsewhere because it is close and easy to work with ^[2]. Sources note the ISO paper series uses ratios related to √2; such approximations are practical compromises rather than exact values ^[2].

6. Two perspectives: symbolic exactness and numerical practicality

Mathematicians treat √2 exactly as an algebraic, irrational constant and often prefer the symbolic form √2 in proofs and formulas ^{[1] [8]}. Engineers, designers and teachers frequently rely on decimal truncations (1.414, 1.4142) or simple fractions for calculation and explanation ^{[6] [2]}. Both perspectives are consistent: exact symbolic identity coexists with approximate numeric representations chosen for context.

7. What the available sources do not address

Available sources do not mention any competing claims that √2 equals a simple rational number; they uniformly present it as irrational ^{[1] [6]}. They do not provide a single canonical decimal expansion in the snippets shown beyond noting common approximations and that many digits have been computed ^[6].

8. Bottom line — how to answer the original question now

If you need a concise answer: √2 is the positive solution to x^2 = 2, symbolically √2; numerically use 1.41421356237… and round as needed (typical short form 1.4142) ^{[1] [6]}. For proofs, leave it as √2; for practical calculations, pick the decimal precision or rational approximation appropriate to your tolerance for error ^{[1] [2]}.