Surds (करणी) In Hindi
करणी, जिन्हें अंग्रेजी में “Surds” कहा जाता है, वे संख्याएँ होती हैं जिन्हें दो पूर्णांकों के भिन्न के रूप में नहीं व्यक्त किया जा सकता है। दूसरे शब्दों में, करणी अपरिमेय संख्याएँ होती हैं जिन्हें मूल चिह्न (√) के साथ व्यक्त किया जाता है, आमतौर पर वर्गमूल के रूप में।
उदाहरण:
√2 (दो का वर्गमूल)
√3 (तीन का वर्गमूल)
√5 (पांच का वर्गमूल)
√7 (सात का वर्गमूल)
Surds In English
Surds are the square roots (√) of numbers that cannot be simplified into a whole or rational number. It cannot be accurately represented in a fraction. In other words, a surd is a root of the whole number that has an irrational value.
Types of Surds
1. Simple Surds – A surd that has only one term is called a simple surd. Example: √2, √5, …
2. Pure Surds: These are surds that cannot be further simplified, meaning they have no rational factors other than 1 under the radical. Examples include √2, √3, and √5.
3. Mixed Surds: These surds contain both a rational number and an irrational number under the radical symbol. They can be simplified by factoring out the rational number. Examples include 3√2, 5√7, and -2√11.
4. Similar Surds – The surds having the same common surds factor.
5. Compound Surds: These are expressions that involve the addition or subtraction of two or more surds. Examples include √2 + √3, √5 – √7, and 2√6 + 3√10.
6. Binomial Surds: These are a specific type of compound surd where the expression consists of the sum or difference of only two surds. Examples include √2 + √3 and √5 – √2.
Six Rules for Surds
1. Multiplication: You can multiply two surds by multiplying the numbers under the radical symbols and simplifying if possible: √a × √b = √(a × b).
Example: √2 × √3 = √(2 × 3) = √6.
2. Division: You can divide two surds by dividing the numbers under the radical symbols and simplifying if possible: √a / √b = √(a / b).
3. Exponentiation: You can raise a surd to a power by raising both the number under the radical and the radical itself to the power: \( (√a)^n \) = \( (√a)^n \).
Example: \( (√2)^3 \) = \( (√2^3) \) = √8.
4. Simplifying: You can sometimes simplify surds by factoring out any perfect squares from under the radical symbol.
Example: √18 = √(9 × 2) = 3√2.
5. Addition and Subtraction: Generally, you cannot directly add or subtract surds unless they are already “like terms” (have the same radical). For example, √2 + √3 cannot be simplified further.
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