How does the square root of 123 compare to nearby integers' square roots (like 11 and 12)?
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Executive summary
The square root of 123 is a non‑integer irrational number approximately 11.0905365, which places it squarely between the square roots of the nearby perfect squares 121 (√121 = 11) and 144 (√144 = 12) and much closer to 11 than to 12 [1] [2] [3]. Standard estimation and iterative methods — using bounding perfect squares and Newton’s (average) method — explain both why √123 lies in that interval and how to refine its decimal approximation [2] [4].
1. What the number is and where it sits
By definition, the square root of 123 is the positive number that, when multiplied by itself, equals 123; numerical tools and pedagogical sources give √123 ≈ 11.0905365 (often rounded to 11.091) and label it irrational because its decimal expansion does not terminate or repeat [1] [3] [2]. Because 121 = 11^2 and 144 = 12^2, the inequality √121 < √123 < √144 shows immediately that √123 is greater than 11 and less than 12 — a basic bounding argument used in textbooks and calculators [2] [5].
2. How much closer it is to 11 than to 12
Simple subtraction quantifies proximity: √123 − 11 ≈ 0.0905 while 12 − √123 ≈ 0.9095, so √123 is roughly ten times closer to 11 than to 12; this consequence follows directly from the decimal approximation cited in multiple references [1] [2]. That asymmetry is expected because 123 is only two units above 121 but twenty‑one units below 144, so the square‑root function’s local slope around 11 keeps small changes in the radicand mapped to proportionally small changes in the root (sources that explain bounding and approximation illustrate this relationship) [2] [4].
3. Why the value is irrational and how that matters
Authors and math sites state √123 is irrational — it cannot be written as a ratio of two integers — because 123 is not a perfect square and its square root yields a nonterminating, nonrepeating decimal [3] [1]. For practical work this means exact arithmetic uses the radical symbol √123 or algebraic manipulation, while numerical work uses approximations to the desired precision via calculators or iterative methods [5] [4].
4. How to estimate and refine √123 in practice
A common classroom approach uses the nearest perfect squares as bounds (√121 = 11 and √144 = 12) and then refines with methods such as averaging (Newton’s method) or long division to produce more accurate decimals; several educational sources walk through taking an initial guess , dividing 123 by that guess, averaging the guess with the quotient, and repeating to converge on ≈11.0905 [2] [4] [6]. Calculator tools provide immediate high‑precision values and step‑by‑step demonstrations of these algorithms for users who need more digits or want to see the convergence [7] [4].
5. Where popular explanations converge and where they differ
Most referenced pages agree on the numeric neighborhood (between 11 and 12) and on approximate decimal values like 11.0905–11.091, and they uniformly describe √123 as non‑integer and irrational [2] [3] [1]. Presentation differences arise in pedagogy: some emphasize prime‑factor or long‑division methods for hand calculation while others highlight Newton’s iterative averaging or simply use calculators to obtain many digits at once [6] [4] [7].
6. Bottom line for comparison to “nearby integers’ square roots”
Compared to the integer square roots 11 and 12, √123 is concretely between them and almost immediately recognizable as much closer to 11; numerical approximations place it at ≈11.0905365, meaning it exceeds 11 by about 0.09 and falls short of 12 by about 0.91 — a fact consistently reported across calculators and educational explanations [1] [2] [3].