Irrational Numbers Quiz

Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as the ratio of two integers. They are different from rational numbers, which can be expressed as a ratio of integers, and are characterized by their non-repeating and non-terminating decimal expansions.

Properties

Irrational numbers have several unique properties. They are not expressible as a finite decimal or fraction, and their decimal expansions are either repeating (periodic) or non-repeating (non-periodic). The set of all real numbers is the union of the set of rational numbers and the set of irrational numbers.

Examples

Some common examples of irrational numbers include:

• Square Root of 2: Also known as √2, this number has a non-repeating decimal expansion.
• π (pi): This is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number with a non-repeating decimal expansion.
• e: This is another mathematical constant, also known as Euler's number, which is approximately 2.71828. It is an irrational number with a non-repeating decimal expansion.

Approximations

When dealing with irrational numbers, it is often necessary to approximate them using rational numbers. This is because most calculations are performed using rational numbers, and computers can only store a finite number of decimal places.

One common method for approximating irrational numbers is to use their finite decimal expansions. For example, the square root of 2 can be approximated using the first few digits of its decimal expansion:

√2 ≈ 1.41

However, this approximation is not exact, and the difference between the approximation and the actual value is called the error. In this case, the error is:

√2 - 1.41 ≈ 0.41

Another method for approximating irrational numbers is to use continued fractions. A continued fraction is a way of expressing a number as an infinite sum of finite fractions. For example, the square root of 2 can be expressed as a continued fraction:

√2 = 1 + 1/(2 + 1/(2 + 1/(2 + ...)))

This continued fraction can be used to calculate approximations of √2 with increasing precision, by truncating the continued fraction after a certain number of terms.

In conclusion, irrational numbers are real numbers that cannot be expressed as the ratio of two integers. They have non-repeating and non-terminating decimal expansions, and are characterized by their unique properties. When working with irrational numbers, approximations are often necessary, and can be obtained using finite decimal expansions or continued fractions.