What is the Square Root of 5? (Value & Calculation)

The number 5 is a prime number, meaning it is not a perfect square. Because of this, its square root (√5) is mathematically proven to be an Irrational Number. This means that the decimal value (2.23606797749...) will continue endlessly into infinity without ever stopping, and the numbers will never fall into a repeating pattern. You absolutely cannot write √5 as a clean, simple fraction (p/q).

Since you cannot use a calculator in a board exam, you must use the classic Long Division method to find the value of √5 up to 3 decimal places:

1. Step 1: Write 5 as a decimal with pairs of zeros: 5. 00 00 00
2. Step 2: Find the largest square number smaller than 5. That is 4 (which is 2×2). Write 2 at the top. Subtract 4 from 5 to get a remainder of 1.
3. Step 3: Bring down the first pair of zeros (making it 100). Place a decimal point at the top.
4. Step 4: Double the number at the top (2 becomes 4). Now, find a number 'x' such that 4x multiplied by 'x' is just below 100. (42 × 2 = 84). Write 2 at the top.
5. Step 5: Subtract 84 from 100 to get a remainder of 16. If you continue this highly mechanical process for two more pairs of zeros, you will perfectly derive the famous answer: 2.236.

In advanced mathematics and art, √5 is incredibly famous because it is the core biological foundation of the Golden Ratio (1.618). The strict algebraic formula for the Golden Ratio is exactly: (1 + √5) / 2. This ratio defines the perfect, beautiful spirals found in galaxy stars, ocean shells, and the Mona Lisa painting.