√139 at a glance
- Exact value
- √139
- Decimal (10 places)
- 11.7898261226
- Rounded
- 11.8 · 11.79 · 11.790
- Perfect square?
- No — between 11² and 12²
- Rational?
- Irrational
- Both square roots
- ±11.789826
- Prime factorization
- 139
- Cube root
- 5.180101
How to simplify √139
139 is a prime number, so its only factors are 1 and 139. There is no perfect-square factor to pull out, which means √139 is already in its simplest radical form.
The square root of any prime is irrational. If √139 were a fraction a/b in lowest terms, then a² = 139b², so 139 would divide a — and then 139 would divide b too, contradicting “lowest terms.” That is why the decimal 11.7898261226 is only a rounded value.
Where √139 sits between perfect squares
121 = 11² and 144 = 12² are the nearest perfect squares, so √139 lies between 11 and 12. 139 is 18 above 121 and 5 below 144, so the root is closer to 12.
- Straight line between 121 and 144: 11.7826 (0.06% low)
- Tangent from 11, i.e. 11 + 18 ÷ 22: 11.8182 (0.24% high)
- Tangent from 12, i.e. 12 − 5 ÷ 24: 11.7917 (0.02% high)
For √139 the tangent at 12 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 139 is just 5 below 144.
Finding √139 with the Babylonian method
If a guess is too big, 139 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√139) in one step.
Start from the nearest whole number, 12 (12² = 144):
| Step | Guess x | 139 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 12.0000000000 | 11.5833333333 | 11.7916666667 | 2 |
| 2 | 11.7916666667 | 11.7879858657 | 11.7898262662 | 6 |
| 3 | 11.7898262662 | 11.7898259789 | 11.7898261226 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √139 = 11.7898261226 to every decimal shown.
√139 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √139 the pattern is [11; 1, 3, 1, 3, 7, 1, 1, 2, 11, 2, 1, 1, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √139 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 |
|---|---|---|
| 11/1 | 11.0000000000 | 7.9 × 10⁻¹ |
| 12/1 | 12.0000000000 | 2.1 × 10⁻¹ |
| 47/4 | 11.7500000000 | 4.0 × 10⁻² |
| 59/5 | 11.8000000000 | 1.0 × 10⁻² |
| 224/19 | 11.7894736842 | 3.5 × 10⁻⁴ |
| 1,627/138 | 11.7898550725 | 2.9 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 139y² = 1. Its smallest solution in positive whole numbers is x = 77,563,250, y = 6,578,829.
√139 in geometry and everyday measurements
- A square room or garden bed covering 139 square feet measures about 11.79 ft (11 ft 9 in) along each wall.
- 139 is not a sum of two whole-number squares — 139 is itself a prime that is one less than a multiple of 4, which rules that out — so √139 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 3 × 11 box, because 3² + 3² + 11² = 139.
Square roots near √139 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √136 | 2√34 | 11.6619 | No |
| √137 | √137 | 11.7047 | No |
| √138 | √138 | 11.7473 | No |
| √139 | √139 | 11.7898 | No |
| √140 | 2√35 | 11.8322 | No |
| √141 | √141 | 11.8743 | No |
| √142 | √142 | 11.9164 | No |
- The cube root of 139 is about 5.180101.
- Four times the radicand doubles the root: √556 = 2 × √139 ≈ 23.579652.
Frequently asked questions
What is the square root of 139?
The square root of 139 is √139, about 11.7898261226. The negative root, −11.789826, also squares to 139.
Is the square root of 139 rational or irrational?
Irrational. 139 is not a perfect square — it falls between 121 and 144 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √139 be simplified?
No. 139 is prime, so there is no perfect square to take out of the radical.
What is √139 rounded to two decimal places?
√139 ≈ 11.79 to two decimal places (11.8 to one, 11.790 to three). Check: 11.79² = 139.0041, close to 139.