How to Find the Value of Root 5 Manually?
The concept of Value of Root 5 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding the square root of 5 helps students solve equations, geometry problems, and appreciate the properties of irrational numbers.
What Is Value of Root 5?
The value of root 5 refers to that positive real number which, when multiplied by itself, equals 5. In mathematical terms, it is written as √5 or 51/2. This number is not a perfect square, so its square root cannot be expressed as a finite fraction or simple decimal. You’ll find this concept applied in areas such as quadratic equations, geometry (like finding diagonal lengths), and number systems.
Key Formula for Value of Root 5
Here’s the standard formula: \( \sqrt{5} = 2.236067977\ldots \)
The decimal expansion of √5 is non-terminating and non-repeating, confirming that it is an irrational number.
Decimal Value of Root 5
| Form | Representation |
|---|---|
| Radical Form | √5 |
| Exponential Form | 51/2 or 50.5 |
| Decimal (3 places) | 2.236 |
| Decimal (5 places) | 2.23607 |
Is Root 5 Rational or Irrational?
The value of root 5 is irrational. This is because there is no fraction a/b (where a and b are integers, b ≠ 0, and numerator and denominator have no common divisor) whose square is exactly 5. The decimal expansion of √5 goes on forever without repeating, which is the hallmark of irrational numbers.
For example, √4 = 2 (rational), but √5 = 2.236… (irrational).
How to Find Value of Root 5: Step-by-Step Methods
1. Long Division Method
The long division method is a manual way to find the square root of 5 up to as many decimals as needed:
1. Set up 5 as 5.00000000 and group digits after the decimal into pairs.2. Find the largest square smaller than 5, which is 4 (2×2=4). Place 2 as the first digit of the root.
3. Subtract 4 from 5 (remainder 1), bring down the next pair (00), and double the 2 to get 4 (the divisor base).
4. Find a digit (let's call it X) so that 4X × X is less than or equal to 100. X=2 fits (42×2=84).
5. Subtract 84 from 100 (get 16), bring down the next pair, update divisor to 44, and repeat steps for more decimals.
6. Continue for desired accuracy. You will arrive at 2.236 (to 3 decimals), 2.23607 (to 5 decimals), etc.
2. Average (Babylonian) Method—Quick Trick
You can estimate √5 using the average of nearby roots:
1. √4 = 2, √9 = 3. So, √5 is between 2 and 3.2. Try initial guess x1=2.2,
x2 = (x1 + 5/x1)/2 = (2.2 + 2.27)/2 ≈ 2.235,
Repeat to converge to 2.236.
Cross-Disciplinary Usage
Value of root 5 is not only useful in Maths, but also plays an important role in Physics (like solving equations with radicals), Computer Science (algorithms), architecture, and logical reasoning. Students preparing for JEE or NEET will see its relevance in various questions.
Applications of Value of Root 5
- Calculating diagonals in rectangles and parallelograms (using Pythagoras’ theorem).
- Solving quadratic equations like x² - 5 = 0.
- Geometry problems involving non-perfect squares.
- Standard deviation calculations when variance is 5.
- Used in trigonometry, statistics, and construction.
Comparison with Other Square Roots
| Root | Value (3 decimals) | Type |
|---|---|---|
| √2 | 1.414 | Irrational |
| √3 | 1.732 | Irrational |
| √4 | 2.000 | Rational |
| √5 | 2.236 | Irrational |
| √6 | 2.449 | Irrational |
Practice Problems: Find and Use Value of Root 5
Example 1: Solve x² = 5.
1. Take square root both sides: x = ±√5
2. x = ±2.236 (to 3 decimal places)
Example 2: What is 3√5 rounded to two decimals?
1. Multiply 3 × 2.236 = 6.708
Example 3: Calculate the diagonal of a rectangle with sides 1 and 2 units.
1. Diagonal = √(1² + 2²) = √5 = 2.236 units
Now You Try: What is the value of 5√5 to two decimals?
1. 5 × 2.236 = 11.18
Related Square Roots Table (1–10)
| Number | Square Root (3 decimals) |
|---|---|
| 1 | 1.000 |
| 2 | 1.414 |
| 3 | 1.732 |
| 4 | 2.000 |
| 5 | 2.236 |
| 6 | 2.449 |
| 7 | 2.646 |
| 8 | 2.828 |
| 9 | 3.000 |
| 10 | 3.162 |
Frequent Errors and Misunderstandings
- Writing √5 as 2.23 instead of 2.236 (rounding too early).
- Assuming √5 is rational just because its decimal looks “short”.
- Mixing up perfect and non-perfect squares when comparing roots.
Relation to Other Concepts
The idea of Value of Root 5 is closely tied to rational and irrational numbers, square roots, quadratic equations, and Pythagoras’ theorem. Mastering these concepts lays a strong foundation for advanced algebra and geometry.
Classroom Tip
A quick way to remember the value of root 5 is: “It’s a bit above 2 and below 3, and matches 2.236 to three decimals.” Vedantu’s teachers often teach square root estimations with number line visuals and everyday analogies for fast exam recall.
We explored Value of Root 5—from definition, methods, solved examples, practical tips, and connections to other root values. Keep practicing with Vedantu’s worksheets and square root calculators to master roots and strengthen your Maths fundamentals!
Square Root Finder | Square Root Table (1–20) | Value of Root 3 | Irrational Numbers
FAQs on Value of Root 5 in Maths: Decimal, Steps & Applications
1. What is the value of root 5?
The value of root 5 (√5) is approximately 2.236. This is an irrational number, meaning its decimal representation goes on forever without repeating. For most calculations, using 2.236 is sufficient.
2. Is root 5 rational or irrational?
Root 5 (√5) is an irrational number. This means it cannot be expressed as a simple fraction of two integers. Its decimal expansion is non-terminating and non-repeating.
3. How do you find the value of root 5 manually?
One common method is the long division method for finding square roots. There are also approximation techniques. Calculators provide the most efficient way to find √5.
4. What is the decimal value of root 5 up to 3 decimal places?
The decimal value of √5 up to 3 decimal places is 2.236.
5. Where do we use the value of root 5 in Maths?
The value of √5 appears in various mathematical contexts, including:
- Geometry: Calculations involving diagonals of rectangles and other geometric problems.
- Algebra: Solving quadratic equations and simplifying expressions involving surds.
- Trigonometry: Certain trigonometric calculations and identities.
6. How is root 5 used in quadratic equations?
Root 5 can be a solution to quadratic equations. For example, the equation x² - 5 = 0 has solutions x = √5 and x = -√5. Understanding irrational roots is crucial for solving these equations.
7. Can we express root 5 as a fraction or surd?
While we cannot express √5 as a simple fraction, it is already in its simplest surd form (√5). Surds represent irrational numbers using radicals.
8. How does root 5 compare to root 4 and root 6 in real-world measurements?
√4 = 2 and √6 ≈ 2.449. This shows √5 lies between these two values. In real-world scenarios, this difference may represent slight variations in length or area calculations.
9. Are there shortcuts to estimate other irrational roots like root 5?
Yes, there are various approximation methods. One involves identifying perfect squares close to the number (e.g., 4 and 9 for √5) and using their square roots to create an estimate. Calculators offer precise results.
10. What is the value of √5 up to 5 decimal places?
The value of √5 to 5 decimal places is approximately 2.23607.
11. Why is the decimal expansion of √5 non-repeating and non-terminating?
Because √5 is an irrational number. Irrational numbers, by definition, have decimal expansions that neither terminate (end) nor repeat a pattern of digits. This is a fundamental property of irrational numbers.
