Square Root of 2 Formula Properties Proof and Solved Examples
The concept of square root of 2 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Knowing the value of √2 and learning how to estimate or prove its properties will help you solve questions in geometry, algebra, and competitive exams easily.
What Is Square Root of 2?
The square root of 2 is the unique positive number which, when multiplied by itself, equals 2. It is written as √2 in mathematics. You’ll find this concept applied in areas such as right triangle geometry, trigonometry, and the number system (especially irrational numbers). In formula terms: if \( x = \sqrt{2} \), then \( x \times x = 2 \).
Key Formula for Square Root of 2
Here’s the standard formula: \( \sqrt{2} = 1.41421356... \)
Decimal Value and Fraction Approximation
The decimal expansion of the square root of 2 never ends and never repeats. The value of √2 up to 8 decimal places is 1.41421356. There is no simple fraction for √2, but it is sometimes approximated as 99/70 for rough calculations. This decimal is non-terminating and non-repeating, which is a key sign that it is irrational.
| Symbol | Decimal Value | Fractional Approximation |
|---|---|---|
| √2 | 1.41421356… | 99/70 (approx) |
Is the Square Root of 2 Rational or Irrational?
Many students wonder if √2 is rational or irrational. The answer is: √2 is irrational. This means it cannot be written as a fraction p/q where p and q are integers (q ≠ 0). Here’s a classic step-by-step proof by contradiction:
1. Suppose √2 is rational. Then, write √2 = p/q where p and q have no common factors.
2. Square both sides: 2 = p2/q2, so p2 = 2q2.
3. This shows p2 is even, so p must be even (let p=2k).
4. Replace p: (2k)2 = 2q2 → 4k2 = 2q2 → 2k2 = q2.
5. So q2 is also even → q is even.
6. But if both p and q are even, they have a common factor 2. Contradicts assumption!
7. Thus, √2 cannot be rational. It is irrational.
How to Calculate the Square Root of 2
There are a few main methods to find or estimate the value of √2:
- Estimation: Since 12 = 1 and 22 = 4, √2 must lie between 1 and 2. Try 1.42 = 1.96 and 1.52 = 2.25 to see that √2 ≈ 1.41. Refine further for more decimal places.
- Division (Long Division) Method:
1. Group digits of “2” as “2.00 00 00…” for decimals.
2. Find the largest square ≤ 2 (which is 12=1).
3. Subtract and bring down pair “00”. Divide as per the long division algorithm.
4. Repeat for as many decimals as you need.
See the full division method here. - Calculator: Type “2” and press the square root (√) button to get a decimal answer instantly.
Where is Square Root of 2 Used?
The square root of 2 appears all across math and even daily life! Here are some key uses:
- Geometry: The length of the diagonal of a square with side 1 is √2 units (from Pythagoras’ theorem).
- Trigonometry: The value √2/2 occurs for sin(45°) and cos(45°).
- Paper sizes: International A4/A3 paper are based on a √2 aspect ratio.
- Engineering and Technology: Calculating root mean square (RMS) values.
- Science and Construction: Measuring diagonals for right-angled layouts.
Visual Representation on the Number Line
To draw √2 on a number line using geometry:
1. Draw a line segment of 1 unit.
2. At one end, construct a perpendicular line, also of 1 unit.
3. Connect their ends; the hypotenuse = √2 (using Pythagoras' theorem).
4. Use a compass to mark that distance on the number line—this point is at √2.
Comparison: Square Root of 2 vs Other Roots
| Square Root | Value | Type |
|---|---|---|
| √1 | 1 | Rational |
| √2 | 1.41421356… | Irrational |
| √3 | 1.73205080… | Irrational |
| √4 | 2 | Rational |
Frequent Errors and Misunderstandings
- Thinking √2 is exactly 1.414. In reality, the decimals do not end!
- Trying to write √2 as a simple fraction — it cannot be done.
- Confusing square root of 2 (√2) with square of 2 (22=4).
- Not recognizing that irrational numbers have non-terminating, non-repeating decimals.
Try These Yourself
- Find √2 up to four decimal places using the long division method.
- Prove, using a contradiction, that √2 is not rational.
- Construct a length of √2 using only a ruler and compass.
- Calculate the diagonal of a square with side 5 cm.
Relation to Other Concepts
The idea of square root of 2 connects closely with Pythagorean Theorem, Irrational Numbers, and Surds. Mastering its calculation and properties helps you understand geometry, trigonometry, and more advanced number theory topics.
Classroom Tip
A quick way to remember the square root of 2 is that it’s always slightly more than 1.4 and less than 1.5—the “diagonal of a square with sides 1” also helps as a visual cue. Vedantu’s teachers always encourage drawing and visualizing for better retention!
We explored square root of 2—from its definition, decimal value, irrationality proof, calculation tricks, real-life applications, and connections to other math ideas. Continue practicing with Vedantu to become confident and fast when solving root and surd problems in exams or classwork!
Related Learning Links
FAQs on Square Root of 2 Explained with Definition and Value
1. What is the value of square root of 2?
The value of the square root of 2 is approximately 1.41421356. It is a non-terminating, non-repeating decimal, which means it cannot be written exactly as a fraction. In radical form, it is written as √2, and its decimal value is often rounded to 1.414 or 1.41 for practical calculations.
2. Is √2 a rational or irrational number?
The number √2 is an irrational number because it cannot be expressed as a ratio of two integers. Its decimal expansion is non-terminating and non-repeating. This was one of the earliest known proofs of irrational numbers in mathematics, showing that no fraction like a/b can exactly equal √2.
3. How do you calculate the square root of 2?
You can calculate the square root of 2 using long division, approximation, or a calculator. One common manual method is the long division method:
- Pair digits of 2 as 2.00 00 00...
- Find the largest number whose square is ≤ 2 (which is 1).
- Subtract and bring down pairs of zeros.
- Continue the division process to get decimal places.
This gives an approximate value of 1.414213....
4. Why is the square root of 2 important in Maths?
The square root of 2 is important because it appears in geometry, algebra, and trigonometry. In a square with side length 1, the diagonal has length √2 using the Pythagorean theorem:
- Diagonal² = 1² + 1² = 2
- Diagonal = √2
It is also fundamental in proofs about irrational numbers and appears in coordinate geometry and trigonometric ratios like sin 45° and cos 45°.
5. What is the square of √2?
The square of √2 is 2. By definition, squaring a square root returns the original number:
- (√2)² = 2
This follows from the property that (√a)² = a for any non-negative real number a.
6. What is the cube of √2?
The cube of √2 is 2√2. This is because:
- (√2)³ = (√2) × (√2) × (√2)
- = 2 × √2
In exponential form, √2 = 21/2, so (21/2)³ = 23/2 = 2√2.
7. How do you prove that √2 is irrational?
You can prove √2 is irrational using proof by contradiction. The key steps are:
- Assume √2 = a/b, where a and b are integers with no common factors.
- Square both sides: 2 = a²/b².
- Rearrange: a² = 2b², so a² is even, meaning a is even.
- Substitute a = 2k → leads to b being even.
This means both a and b are even, contradicting the assumption that they have no common factor. Therefore, √2 is irrational.
8. What is √2 in fraction form?
The square root of 2 cannot be written exactly as a fraction because it is irrational. However, it can be approximated by fractions such as:
- 99/70 ≈ 1.4143
- 1414/1000 = 1.414
These are rational approximations but not exact values.
9. What is the value of √2 up to 3 decimal places?
The value of √2 up to 3 decimal places is 1.414. The full decimal starts as 1.41421356..., and rounding to three decimal places keeps the first three digits after the decimal point, giving 1.414.
10. Where is the square root of 2 used in real life?
The square root of 2 is used in geometry, engineering, and design, especially when working with squares and right triangles. Common applications include:
- Finding the diagonal of a square (diagonal = side × √2)
- Paper sizes like A4, which follow the √2 aspect ratio
- Trigonometry values such as sin 45° = √2/2
Its frequent appearance in right-angled triangles makes √2 an essential constant in mathematics.
