Squares and roots
A square multiplies a number by itself. A square root reverses that relationship. Explore the reasoning, then represent suitable values on the frame.
From a square to a root
Because 7 × 7 = 49, √49 = 7. The symbol √ gives the nonnegative principal root. The equation x² = 49 has two solutions, 7 and −7. Negative numbers have no real square root; cube roots can be negative.
A written method for √20
- Bracket: 4² = 16 and 5² = 25, so 4 < √20 < 5.
- Refine: 4.4² = 19.36 and 4.5² = 20.25.
- Refine again: 4.47² = 19.9809 and 4.48² = 20.0704.
Thus 4.47 < √20 < 4.48. The website frame can display these trial values but cannot display their four-decimal squares exactly.
What this extension means
Squares and cubes use multiplication; roots use an inverse operation with trial, bounding or a specified algorithm. No new bead type is mathematically necessary to represent a result within range, but larger results need more whole-number rods, and finer approximations need more fractional rods.
This explorer supplies written reasoning and a numerical calculation. It does not establish a tested physical bead procedure for squaring or root extraction. The current five-place decimal explorer has range −999.99 to 999.99 and hundredths precision. Irrational roots cannot be represented exactly by any finite decimal frame.