Square root 123 World square root 123 World 속보
Executive summary
The square root of 123, written √123, is an irrational number with principal (positive) value approximately 11.0905365; the equation x² = 123 has two algebraic solutions ±11.0905365, but the convention √123 denotes the positive root [1]. Because 123 is not a perfect square and has no square factors beyond 1, √123 cannot be reduced to a simpler integer or rational radical [2] [3].
1. What the number is and why it’s irrational
By definition the square root of 123 is the number that, when multiplied by itself, gives 123; solving x² = 123 yields the two algebraic solutions ±11.0905365, and the principal square root is the positive value 11.0905365 [1]. Multiple educational sources emphasize that 123 is not a perfect square and therefore its square root is a non‑terminating, non‑repeating decimal — the textbook definition of an irrational number [2] [4].
2. Exact form, simplest radical, and prime factors
In radical notation the exact form is simply √123; factorizing 123 into primes gives 3 × 41, neither of which is a perfect square, so there is no simplification into a product of an integer and a smaller radical — √123 is already in simplest radical form [1] [3] [2].
3. Decimal approximations and rounding conventions
Decimal approximations vary by source depending on rounding: common published approximations include 11.09 (rounded to two decimal places), 11.0905 (rounded to four decimal places), and more precisely 11.0905365 in some educational writeups [5] [6] [1]. For most practical work, rounding to three or four decimal places (11.091 or 11.0905) is sufficient, but the full non‑terminating expansion cannot be represented exactly in decimal form [6] [7].
4. How to compute it: methods and tutorials
Standard high‑school techniques shown across tutorials include estimating between the nearest perfect squares (√121 = 11 and √144 = 12) and refining with long‑division style square‑root algorithms, or using iterative numerical methods such as the Babylonian/Newton method to converge to the decimal value; calculators and spreadsheet functions (SQRT) give rapid numeric answers [8] [9] [4] [2].
5. Common mistakes and reporting inconsistencies
Instructional sites sometimes truncate differently — one source lists 11.091 while another gives 11.0905365 — which can confuse non‑specialists about the “correct” form; these are rounding differences rather than contradictory values [7] [1]. Some pages also conflate the ± symbol with the principal square root; mathematically x² = 123 has ± solutions but the radical symbol √ denotes the positive root, a distinction emphasized in multiple sources [1] [4].
6. Practical implications and verification
For engineering, physics, or calculation tasks where precision matters, use a numerical method or calculator and report the number to the required significant figures (examples: 11.0905 to four decimals, or 11.0905365 when more precision is needed) and note that exact representation remains √123 [6] [1]. Educational resources consistently show how to verify by squaring the computed approximation to check it approaches 123 as accuracy increases [9] [2].
7. Reporting limitations and source perspective
The available reporting is dominated by educational and calculator sites that agree on the mathematical facts but differ in presentation and rounding; there are no primary peer‑reviewed mathematical disputes here, and the citations used are tutorial and reference pages rather than formal proofs, which is sufficient for this elementary arithmetic question [8] [1] [2].