Square Root of 5 — Value, How to Find It, and Examples

Algebra

TL;DR

The square root of 5 is about 2.236, and because 5 is prime, ( \sqrt{5} ) is irrational and already in simplest radical form. This article gives the exact and decimal value, two by-hand methods, the link between ( \sqrt{5} ) and the golden ratio, and the slips students make most.

The Answer At A Glance

Quick Answer:

Result: ( \sqrt{5} \approx 2.236068 )

Notation: Decimal approximation; exact form is ( \sqrt{5} ).

Method shown: Long division by hand, cross-checked with estimation between perfect squares.

Approximate value: 2.2361 (4 d.p.)

Exact form: ( \sqrt{5} ) — cannot be simplified, since 5 is prime.

Quick Reference Table — Square Roots From 1 to 20

( \sqrt{5} ) sits between 4 (=2) and 9 (=3), close to 2.24 because 5 is only just past 4.

Where ( \sqrt{5} ) Appears

( \sqrt{5} ) is the diagonal of a 1×2 rectangle — Pythagoras gives ( \sqrt{1^2 + 2^2} = \sqrt{5} ). Its most famous role is inside the golden ratio: ( \phi = \frac{1 + \sqrt{5}}{2} \approx 1.618 ), the number that the ratios of consecutive Fibonacci numbers converge toward. ( \sqrt{5} ) quietly sits behind golden rectangles, pentagons, and the spiral patterns the golden ratio describes.

What "square root of 5" Means

The square root of a non-negative number ( n ) is the value ( x ) such that ( x^2 = n ). For ( \sqrt{5} ), it is the positive ( x ) with ( x^2 = 5 ).

Is The Square Root of 5 Rational or Irrational?

( \sqrt{5} ) is irrational. A whole number is a perfect square only when every prime in its factorization appears to an even power; 5 is prime, so it carries the single factor 5 — an odd exponent — and is not a perfect square.

That means ( \sqrt{5} ) cannot be written as a fraction ( p/q ), and its decimal 2.2360680… never terminates and never repeats. As of recent computations, the digits have been calculated to trillions of places, with no pattern emerging — exactly what irrationality predicts.

How To Find ( \sqrt{5} ) — Two Methods

Method 1 — Long division (digit by digit)

Write 5 as 5.000000 and pair the digits after the decimal point.

Step 1. The largest integer whose square is at most 5 is 2 (( 2^2 = 4 )). Subtract: 5−4=1. Bring down 000000 to get 100.

Step 2. Double the quotient 2 to get 4. Find ( d ) with (40+d)⋅d≤100. Here, d=2 gives 42⋅2=84. Subtract: 100−84=16. Bring down 000000 to get 1600.

Step 3. Double 2 to get 44. Find ( d ) with (440+d)⋅d≤1600. Here, d=3 gives 443⋅3=1329. Subtract: 1600−1329=271.

Continuing produces 2.2360…

Final answer: ( \sqrt{5} \approx 2.2361 ).

Method 2 — Estimation between perfect squares

Since ( 2^2 = 4 ) and ( 3^2 = 9 ), start at 2: ( 2.2^2 = 4.84 ) is a little low. Try ( 2.23^2 = 4.9729 ). Try ( 2.236^2 = 4.999696 ). Each guess closes in on 5, settling at 2.236 for everyday work — the same logic as the average (Babylonian) method.

What are the most common mistakes with ( \sqrt{5} )?

Mistake 1: Trying to simplify a prime radicand

Where it slips in: A student applies the "pull out a square factor" rule before checking whether 5 has one.

Don't do this: Writing ( \sqrt{5} = \sqrt{4} + \sqrt{1} ).

The correct way: 5 is prime — no square factor — so ( \sqrt{5} ) is already simplest.

Mistake 2: Splitting the root over addition

Where it slips in: When ( \sqrt{5} ) shows up as ( \sqrt{1} + \sqrt{4} ).

Don't do this: 1+4=3.

The correct way: 1+4=5 ≈ 2.236.

Mistake 3: Misplacing ( \sqrt{5} ) in the golden-ratio formula

Where it slips in: Recalling ( \phi ) but forgetting the division by 2.

Don't do this: ( \phi = 1 + \sqrt{5} = 3.236 ).

The correct way: ( \phi = \frac{1 + \sqrt{5}}{2} \approx 1.618 ).

Examples of Square Root of 5

Example 1

Simplify ( \sqrt{20} ) using ( \sqrt{5} ).

( \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \approx 4.4721 ).

Example 2 (Wrong path first)

Find the diagonal of a 1×2 rectangle.

Wrong attempt. A student adds the sides: diagonal =1+2=3.

Correct. Use Pythagoras: diagonal = ( \sqrt{1^2 + 2^2} = \sqrt{5} \approx 2.236 ).

Example 3

Compute the golden ratio from ( \sqrt{5} ).

( \phi = \frac{1 + \sqrt{5}}{2} = \frac{3.236}{2} \approx 1.618 ).

Example 4

Rationalize ( \frac{2}{\sqrt{5}} ).

Multiply top and bottom by ( \sqrt{5} ): ( \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5} \approx 0.8944 ).

Example 5

Evaluate ( \sqrt{45} + \sqrt{5} ).

( \sqrt{45} + \sqrt{5} = 3\sqrt{5} + \sqrt{5} = 4\sqrt{5} \approx 8.944 ).

Conclusion

• The square root of 5 is approximately 2.236 — irrational, non-terminating, non-repeating.

• 5 is prime, so ( \sqrt{5} ) cannot be simplified.

• Long division and estimation both reach the value by hand.

• ( \sqrt{5} ) is the diagonal of a 1×2 rectangle and the engine behind the golden ratio.

• Square roots do not distribute over addition: ( a + b \neq \sqrt{a + b} ).

A practical next step

1. Find ( \sqrt{6} ) to three decimal places by long division and check by squaring.

2. Show that ( \sqrt{20} ) simplifies to ( 2\sqrt{5} ) while ( \sqrt{5} ) does not simplify.

3. Use ( \phi = \frac{1 + \sqrt{5}}{2} ) to verify that ( \phi^2 = \phi + 1 ).

Frequently Asked Questions

What is the value of the square root of 5?

Approximately ( 2.236 ) or ( 2.2360680… ).

Is the square root of 5 rational or irrational?

Irrational. 5 is prime, so ( \sqrt{5} ) cannot be written as a fraction.

Can the square root of 5 be simplified?

No, 5 has no square factor, so ( \sqrt{5} ) is already in simplest radical form.

Why does ( \sqrt{5} ) appear in the golden ratio?

The golden ratio solves ( x^2 - x - 1 = 0 ), and the quadratic formula puts ( \sqrt{5} ) in the answer: ( \phi = \frac{1 + \sqrt{5}}{2} ).

What is the square root of 5 to two decimal places?

Approximately ( 2.24 ).