List all numbers from each set that are the following. (a)...

List all numbers from each set that are the following.
(a) natural numbers
(b) whole numbers
(c) integers
(d) rational numbers
(e) irrational numbers
(f) real numbers
$\left\{\frac{7}{9},-2 . \overline{3}, \sqrt{3}, 0,-8 \frac{3}{4}, 11,-6, \pi\right\}$

Key Concepts

Irrational Numbers

Irrational numbers are numbers that cannot be written as a simple fraction, meaning they cannot be exactly expressed as a ratio of two integers. They are typically non-repeating, non-terminating decimals and include values such as the square roots of non-perfect squares and transcendental numbers like pi. They are crucial for representing quantities that are not neatly represented as fractions.

Natural Numbers

Natural numbers are the set of positive integers used primarily for counting. They start typically with 1 (or sometimes 0, depending on convention) and do not include fractions, decimals, or negative numbers. These numbers are the basic building blocks for elementary arithmetic and form the foundation for number theory.

Whole Numbers

Whole numbers include all natural numbers along with zero. They are a complete set of non-negative integers and are used in scenarios where a count begins at zero, such as counting objects that may include none. This set excludes negative numbers as well as fractions and decimals.

Integers

Integers are the collection of whole numbers and their negative opposites. This set includes positive numbers, negative numbers, and zero. Integers are essential in mathematics for representing quantities that may fall below zero, such as in temperature scales or financial contexts.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. They include fractions, terminating decimals, and repeating decimals. These numbers are important because they form a dense subset of the real numbers, meaning between any two rational numbers there is another rational number.

Real Numbers

Real numbers consist of all the numbers that can be found on the number line, including both rational and irrational numbers. They form the basis for much of calculus and analysis, and they represent all possible magnitudes and values that can be measured continuously.

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