# Square Root of 5 — Value, How to Find It, and Examples [Algebra](/content/tag/algebra/index.html) 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 | n | \( n \sqrt{n} \) (exact) | \( n \sqrt{n} \) (4 d.p.) | | --- | --- | --- | | 1 | 1 | 1.0000 | | 2 | \( \sqrt{2} \) | 1.4142 | | 3 | \( \sqrt{3} \) | 1.7321 | | 4 | 2 | 2.0000 | | **5** | \( \sqrt{5} \) | 2.2361 | | 6 | \( \sqrt{6} \) | 2.4495 | | 7 | \( \sqrt{7} \) | 2.6458 | | 8 | \( 2\sqrt{2} \) | 2.8284 | | 9 | 3 | 3.0000 | | 10 | \( \sqrt{10} \) | 3.1623 | | 11 | \( \sqrt{11} \) | 3.3166 | | 12 | 4 | 4.0000 | | 13 | \( \sqrt{13} \) | 3.6056 | | 14 | \( \sqrt{14} \) | 3.7417 | | 15 | \( \sqrt{15} \) | 3.8729 | | 16 | 4 | 4.0000 | | 17 | \( \sqrt{17} \) | 4.1231 | | 18 | \( 3\sqrt{2} \) | 4.2426 | | 19 | \( \sqrt{19} \) | 4.3589 | | 20 | \( \sqrt{20} \) | 4.4721 | \( \sqrt{5} \) 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 \( 2.24 \).