Understanding Irrational Numbers: Definition, Examples, and Proofs

Irrational numbers

Irrational numbers are numbers that cannot be expressed as a fraction of two integers

Irrational numbers are numbers that cannot be expressed as a fraction of two integers. They are real numbers that cannot be written in the form of a terminating or repeating decimal. In other words, they are numbers that do not have a finite or periodic decimal representation.

Some common examples of irrational numbers include:

1. √2 (the square root of 2)
2. π (pi)
3. e (Euler’s number)
4. √3 (the square root of 3)

To prove that a number is irrational, one can use proof by contradiction. Assume that the number can be expressed as a fraction, and then show that this assumption leads to a contradiction.

For example, let’s prove that √2 is an irrational number.

Assume √2 can be expressed as a fraction a/b, where a and b are integers with no common factors (other than 1), and b ≠ 0.

√2 = a/b
Squaring both sides:
2 = (a^2)/(b^2)

This implies that a^2 = 2b^2. From this equation, we can deduce that a^2 is even because it is divisible by 2. When a^2 is even, a must also be even (since the square of an odd integer is odd). So we can rewrite a as 2c, where c is also an integer.

Plugging in 2c for a:
(2c)^2 = 2b^2
4c^2 = 2b^2
2c^2 = b^2

This means that b^2 is even, which in turn implies that b is also even.

Since both a and b are even, they must have a common factor of 2. However, this contradicts our initial assumption that a and b have no common factors. Hence, our assumption that √2 can be expressed as a fraction is false, and we can conclude that √2 is irrational.

In summary, irrational numbers are real numbers that cannot be expressed as fractions of integers. They have infinite, non-repeating decimal representations and can be proven to be irrational using proof by contradiction.

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