Quare root 123Seattle
Executive summary
The square root of 123 is the number which, when multiplied by itself, gives 123; in radical form it is written √123 [1]. It is not a perfect square and evaluates to approximately 11.0905365 (positive principal root) — often rounded to 11.09 or 11.091 in common references — and also has a negative counterpart −11.0905365 when both algebraic solutions are listed [2] [3] [4].
1. What the expression means and why it matters
The notation √123 denotes the principal (non‑negative) square root of 123, meaning the solution x ≥ 0 to x² = 123, and by definition every positive real number has two square‑root solutions ±√n though √n usually refers to the positive one [2] [1]. That concept is the inverse of squaring: if 11.0905365 × 11.0905365 ≈ 123 then 11.0905365 is the principal root, a fact repeated across multiple educational calculators and lessons [2] [5] [1].
2. Numerical value and commonly used approximations
Calculators and math resources list the decimal approximation of √123 as about 11.0905365; educational pages and quick calculators commonly round that to 11.09 or 11.091 depending on the required precision [2] [3] [6]. For three decimal places most sources recommend 11.091, while shorter forms use 11.09; these rounded values are consistent with the more precise value given by computational tools [1] [3] [6].
3. Is √123 rational or irrational, and can it be simplified?
Because 123 is not a perfect square (121 and 144 bracket it), its square root does not simplify to an integer and is an irrational, non‑terminating decimal; multiple tutoring sites and explanations state that √123 cannot be expressed as a simple rational fraction and therefore is irrational [1] [6]. Prime factorization of 123 = 3 × 41 produces no repeated factor pairs to extract, so the radical stays √(3×41) and cannot be reduced to a simpler radical with integer factors [2] [7].
4. Exact form, algebraic context and sign convention
In exact algebraic notation the number is written ¹ᐟ² or √123, and when solving x² = 123 algebraically one usually records both solutions x = ±√123 though the principal root is positive [2] [1]. Educational writeups emphasize that the ± notation indicates both algebraic roots but that the radical symbol by convention denotes the non‑negative value unless context indicates otherwise [2] [4].
5. Methods to compute or estimate √123
Common classroom methods to estimate √123 include locating the nearest perfect squares (√121 = 11 and √144 = 12) and interpolating or averaging, using long‑division style algorithms, or employing calculators and online root tools for high precision; tutorials and worked examples describe averaging between bounds and the long‑division method as practical ways to get 11.09… [1] [3] [8]. Online calculators and algebra tools return the same numerical approximations and are useful when more decimal places are required [9] [5].
6. What the provided reporting does not and implicit editorial signals
The assembled sources are educational pages and calculators that consistently present the same numeric approximation and the same algebraic facts, so there is no substantive controversy in the reporting; however, the sources vary in how they round and in pedagogical focus (some emphasize estimation, others give a more precise decimal), which can produce slightly different rounded outputs like 11.09 versus 11.091 without changing the mathematical truth [3] [6]. There is no information among the provided sources linking the stray word “Seattle” in the user query to any mathematical meaning, and the reporting does not cover any geographical or contextual use of that term, so no claim about Seattle can be made based on these sources (no source provided).