Square Root of 5
Calculate √5 instantly — exact value, step-by-step explanation, and properties.
How to Calculate √5
The square root of 5 is approximately 2.2360679775. Since 5 is not a perfect square, its square root is an irrational number. We know that √4 = 2 and √9 = 3, so √5 falls between 2 and 3. To estimate manually: start with 2.2, then refine using the Babylonian method — guess a value, divide 5 by it, average the result with your guess, and repeat.
The Babylonian Method (Step by Step)
- Start with an initial guess. Since √4 = 2 and √9 = 3, a good starting guess is 2.5.
- Divide 5 by your guess: 5 ÷ 2.5 = 2.0000
- Average the result with your guess: (2.5 + 2.0000) ÷ 2 = 2.250000
- Repeat with the new estimate until you reach the desired precision.
√5 ≈ 2.2360679775 Properties of √5
Perfect Square?
No
5 is not a perfect square. Its root is irrational.
Decimal Value
2.2360679775
Rounded to 10 decimal places.
Square Root Symbol
√5
Also written as 5^(1/2) or 5^0.5 in programming.
Frequently Asked Questions
What is the square root of 5?
The square root of 5 is approximately 2.2360679775. This is an irrational number.
Is 5 a perfect square?
No, 5 is not a perfect square. Its square root (2.2360679775...) is an irrational number that cannot be expressed as a simple fraction.
How do you calculate √5 by hand?
To calculate √5 by hand, use the Babylonian method: (1) Make an initial guess — since √4 = 2 and √9 = 3, start with 2.5. (2) Divide 5 by your guess. (3) Average the result with your guess. (4) Repeat steps 2-3 until you reach desired precision.
What is √5 in simplified radical form?
√5 is already in its simplest radical form. The number 5 has no perfect square factors other than 1.