√101 at a glance
- Exact value
- √101
- Decimal (10 places)
- 10.0498756211
- Rounded
- 10.0 · 10.05 · 10.050
- Perfect square?
- No — between 10² and 11²
- Rational?
- Irrational
- Both square roots
- ±10.049876
- Prime factorization
- 101
- Cube root
- 4.657010
How to simplify √101
101 is a prime number, so its only factors are 1 and 101. There is no perfect-square factor to pull out, which means √101 is already in its simplest radical form.
The square root of any prime is irrational. If √101 were a fraction a/b in lowest terms, then a² = 101b², so 101 would divide a — and then 101 would divide b too, contradicting “lowest terms.” That is why the decimal 10.0498756211 is only a rounded value.
Where √101 sits between perfect squares
100 = 10² and 121 = 11² are the nearest perfect squares, so √101 lies between 10 and 11. 101 is 1 above 100 and 20 below 121, so the root is closer to 10.
- Straight line between 100 and 121: 10.0476 (0.02% low)
- Tangent from 10, i.e. 10 + 1 ÷ 20: 10.0500 (0% high)
- Tangent from 11, i.e. 11 − 20 ÷ 22: 10.0909 (0.41% high)
For √101 the tangent at 10 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 101 is just 1 above 100.
Finding √101 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 101: following the tangent line down to zero simplifies to averaging x with 101 ÷ x.
Start from the nearest whole number, 10 (10² = 100):
| Step | Guess x | 101 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 10.0000000000 | 10.1000000000 | 10.0500000000 | 3 |
| 2 | 10.0500000000 | 10.0497512438 | 10.0498756219 | 9 |
| 3 | 10.0498756219 | 10.0498756204 | 10.0498756211 | all 10 shown |
The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √101 = 10.0498756211 to every decimal shown.
√101 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √101 the pattern is [10; 20] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 101 is one more than a perfect square (10² + 1). A pattern that never ends is one more proof that √101 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 10/1 | 10.0000000000 | 5.0 × 10⁻² |
| 201/20 | 10.0500000000 | 1.2 × 10⁻⁴ |
| 4,030/401 | 10.0498753117 | 3.1 × 10⁻⁷ |
| 80,801/8,040 | 10.0498756219 | 7.7 × 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 101y² = 1. Its smallest solution in positive whole numbers is x = 201, y = 20. Because the period is odd, the equation with −1 on the right also has a solution: 10² − 101 × 1² = −1.
√101 in geometry and everyday measurements
- A square room or garden bed covering 101 square feet measures about 10.05 ft (10 ft 1 in) along each wall.
- 101 = 1² + 10², so by the Pythagorean theorem √101 is the diagonal of a 1 × 10 rectangle — and the distance between the points (0, 0) and (1, 10) on a grid.
Square roots near √101 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √98 | 7√2 | 9.8995 | No |
| √99 | 3√11 | 9.9499 | No |
| √100 | 10 | 10.0000 | Yes |
| √101 | √101 | 10.0499 | No |
| √102 | √102 | 10.0995 | No |
| √103 | √103 | 10.1489 | No |
| √104 | 2√26 | 10.1980 | No |
- The cube root of 101 is about 4.657010.
- Four times the radicand doubles the root: √404 = 2 × √101 ≈ 20.099751.
Frequently asked questions
What is the square root of 101?
The square root of 101 is √101, about 10.0498756211. The negative root, −10.049876, also squares to 101.
Is the square root of 101 rational or irrational?
Irrational. 101 is not a perfect square — it falls between 100 and 121 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √101 be simplified?
No. 101 is prime, so there is no perfect square to take out of the radical.
What is √101 rounded to two decimal places?
√101 ≈ 10.05 to two decimal places (10.0 to one, 10.050 to three). Check: 10.05² = 101.0025, close to 101.