Understanding Irrational Numbers and Their Significance in Mathematics

Irrational Numbers (I)

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An irrational number is a number that cannot be written as a fraction of two integers. In other words, an irrational number is a number that cannot be expressed as a terminating or repeating decimal.

Examples of irrational numbers include:

– √2
– π (pi)
– e
– √3
– √5

Irrational numbers are important in mathematics because they play a crucial role in geometry, calculus, and many other areas of mathematics. For example, the value of pi (π) is necessary for calculating the circumference and area of a circle, while the value of e is used in calculus to calculate the slope of a curve at a given point.

One way to prove that a number is irrational is by contradiction. Suppose that a number is rational, and then write it as a fraction in lowest terms. If the numerator and denominator have a common factor, then we can divide both by this factor to obtain a new fraction in lowest terms. Continuing in this way, we eventually arrive at a contradiction, showing that the number cannot be rational.

For example, to prove that √2 is irrational, we suppose that √2 is rational and write it as a fraction in lowest terms, say √2 = a/b, where a and b have no common factors. Then, squaring both sides gives us 2 = a^2/b^2, which implies that 2b^2 = a^2. But this means that a^2 is even, which implies that a is even. Therefore, we can write a as 2c for some integer c, which gives us 2b^2 = (2c)^2 = 4c^2, or equivalently, b^2 = 2c^2. Thus, b^2 is even, which implies that b is also even. However, this contradicts the assumption that a and b have no common factors, so our original assumption that √2 is rational must be false. Therefore, √2 is irrational.

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