Square Root Table: Quick Reference for Numbers 1–100

Comprehensive Square Root Table: Roots, Squares, and TipsA solid grasp of square roots and squares is a foundation of arithmetic and algebra. This comprehensive guide provides an easy-to-use square root table, explains the relationship between squares and square roots, shows quick mental methods for estimating roots, and offers practical tips for using roots in math problems, engineering, and everyday life.

What is a square and a square root?

• A square of a number x is the result of multiplying x by itself: x^2.
• A square root of a nonnegative number y is a number r such that r^2 = y. The principal square root is the nonnegative value, written as √y.
• Example: The square of 7 is 49, and √49 = 7.

Why square roots matter

Square roots appear in geometry (areas, Pythagorean theorem), physics (wave equations, kinematics), statistics (standard deviation), finance (volatility measures), and many applied problems. Knowing common square roots and how to approximate others quickly saves time and reduces calculation errors.

Square root table (1–200) — principal roots rounded to 4 decimal places

Below is a compact reference for principal square roots of integers from 1 through 200, rounded to four decimal places.

How to memorize and use the table efficiently

• Memorize perfect squares up to at least 15^2 (225). Those anchor points let you estimate nearby roots quickly.
• Group numbers by decade (1–10, 11–20, etc.) and practice estimating midpoints.
• Use the method of linear approximation: for n near a perfect square a^2, √n ≈ a + (n − a^2)/(2a). Example: for n = 50, nearest square 49 (a=7): √50 ≈ 7 + (1)/(14) = 7.0714 (actual 7.0711).
• For mental checks, square the estimated root to see how close it gets to the original number.

Quick methods to approximate square roots

1. Newton’s method (Babylonian method)

   • Start with guess x0. Iterate x_{k+1} = (x_k + n/x_k)/2. Converges quadratically.
   • Example: √10 start x0=3, x1=(3+⁄₃)/2=3.1667, x2≈3.1623.

2. Binary search

   • For positive n, search interval [0, max(1, n)] halving until desired precision.

3. Continued fractions

   • Useful for high-precision rational approximations; more advanced.

Common pitfalls and mistakes

• Confusing ± roots: equation x^2 = 9 has solutions x = 3 and x = −3, but the principal root √9 = 3.
• Rounding too early in multi-step problems can amplify error—keep 4–6 significant digits until the final step.
• Using approximate roots without checking bounds can produce incorrect inequality conclusions.

Applications and examples

• Geometry: side of square with area 200 is √200 ≈ 14.1421.
• Pythagorean theorem: for legs 5 and 12, hypotenuse = √(5^2+12^2)=√169=13.
• Physics: RMS speed uses square roots in averaging squared velocities.
• Finance: volatility scaling uses square root of time.

Printable and cheat-sheet suggestions

• Create a one-page printable with perfect squares, square roots for 1–100 (2 decimal places), and the linear approximation formula.
• For exams, fold the sheet so you can quickly scan decade groups; practice estimating without a calculator.

Reference: ready-to-use formulas

• Square: x^2
• Principal square root: √x (nonnegative)
• Linear approximation near a^2: √n ≈ a + (n − a^2)/(2a)
• Newton iteration: x_{k+1} = (x_k + n/x_k)/2

If you want, I can: provide a printable PDF, expand the table beyond 200, or add step-by-step examples converting decimals to roots.