Square root 123
Executive summary
The principal square root of 123 is an irrational number approximately 11.0905–11.091 depending on rounding; several reputable instructional sites give three‑decimal approximations of 11.090 or 11.091 (examples: Cuemath reports 11.090 and CK‑12/Thinkster report 11.091) [1] [2] [3]. Sources agree 123 is not a perfect square and its square root is most usefully treated as √123 or 123^(1/2) when an exact symbolic form is required [4] [5].
1. What "square root of 123" means — quick definitional context
By standard definition the square root of 123 is the number which, when multiplied by itself, equals 123; in most contexts the phrase refers to the principal (non‑negative) root, written √123 or 123^(1/2) [4] [5]. Sources emphasize that because 123 is not a perfect square you cannot express √123 as an integer; instead you give it in radical form or as a decimal approximation [5] [6].
2. Numeric approximations and small rounding differences
Multiple educational calculators and tutoring sites list the decimal approximation in the 11.090–11.091 range. Cuemath states √123 ≈ 11.090 to three decimal places [1]. CK‑12 and Thinkster/HelloThinkster list 11.091 [2] [3]. These tiny differences reflect rounding conventions and the number of digits shown by each source rather than substantive disagreement about value [1] [2] [3].
3. Why the root is irrational — what the sources say
Instructional pages explain that √123 cannot be written as a ratio of two integers because 123 is not a perfect square; those pages therefore treat √123 as an irrational, non‑terminating decimal and use numerical methods (long division, Babylonian/Hero’s method, or averaging) to compute approximations [3] [7] [6]. Sources explicitly call out that for non‑perfect squares the long division or iterative methods are appropriate to obtain more accurate decimals [3] [7].
4. Exact representation and prime factors
For exact symbolic work, sources show √123 left in radical form or expressed with fractional exponent notation 123^(1/2). They note the prime factorization of 123 is 3 × 41, which does not produce square pairs, so no simplification into an integer times a square root is possible beyond √(3×41) ≡ √123 [5] [4].
5. How calculators and methods produce the decimals
Sites offering step‑by‑step calculators explain methods used: Hero’s/Babylonian iteration, long division, or built‑in numeric evaluators produce converging approximations; CoolConversion and similar pages show the Babylonian method stepwise for numbers like 123 [7] [8]. Math tools (Mathway, DollarTimes, VisualFractions) present the same outcome in decimal form and often include rounding notes [8] [9] [6].
6. Sources, rounding and minor discrepancies — reading between the lines
The small discrepancy between 11.090 and 11.091 arises from different rounding cutoffs and the number of digits published: one site gives three decimals truncated or rounded to 11.090 [1], others show 11.091 [2] [3]. There is no substantive dispute in the sources about the mathematical value; they consistently treat √123 as an irrational number best expressed exactly as √123 or approximately as ≈11.0905… [5] [4] [3].
7. Practical takeaways and when to use each form
Keep √123 as √123 or 123^(1/2) in symbolic algebra and proofs [4]. Use the rounded decimal 11.090 or 11.091 for calculations where three‑decimal precision suffices [1] [2]. For higher‑precision needs, use a calculator or iterative method (Babylonian/long division) as described by CoolConversion or VisualFractions [7] [6].
Limitations and transparency: available sources do not mention an exact rational representation of √123 (not found in current reporting) and do not produce a single universally displayed rounded value — the tiny rounding differences reflect presentation choices rather than conflicting mathematics [1] [2] [3].