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05 - Irrational Numbers And Proof Of Root 2 Is Irrational - Class 10 - Maths Video Lecture
FAQs on 05 - Irrational Numbers And Proof Of Root 2 Is Irrational - Class 10 - Maths Video Lecture
| 1. What are irrational numbers? |  |
Ans. Irrational numbers are real numbers that cannot be expressed as a ratio or fraction of two integers. They are non-repeating and non-terminating decimals. Examples of irrational numbers include √2, π, and e.
| 2. How can we prove that √2 is irrational? | |
Ans. The proof of √2 being irrational is a classic mathematical proof. It involves assuming that √2 is rational and then arriving at a contradiction. The proof shows that if √2 is rational, then it can be expressed as a fraction of two integers. However, by simplifying this assumed fraction, we find that the numerator and denominator have a common factor of 2, which contradicts the assumption that the fraction is in its simplest form. Therefore, √2 is irrational.
| 3. Are all square roots irrational numbers? | |
Ans. No, not all square roots are irrational numbers. Square roots of perfect squares (numbers that are the square of an integer) are rational numbers. For example, the square root of 25 is 5, which is a rational number. However, square roots of non-perfect squares, such as √2 and √7, are irrational numbers.
| 4. Can irrational numbers be expressed as decimals? | |
Ans. Yes, irrational numbers can be expressed as decimals. However, unlike rational numbers, which either terminate or repeat, irrational numbers have non-repeating and non-terminating decimal representations. For example, the decimal representation of √2 is approximately 1.41421356, and it continues indefinitely without repeating any pattern.
| 5. Are all numbers that cannot be expressed as fractions irrational? | |
Ans. No, not all numbers that cannot be expressed as fractions are irrational. Some numbers that cannot be expressed as fractions are transcendental numbers, such as π and e. Transcendental numbers are real numbers that are not algebraic, meaning they are not the root of any non-zero polynomial equation with integer coefficients. Therefore, while all irrational numbers cannot be expressed as fractions, not all numbers that cannot be expressed as fractions are irrational.
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