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What is an intuitive, calculus-grade way to explain that rationals have the same cardinality as N? (Same question for perfect squares.)
|Q| = aleph zero is a mathematical concept that refers to the cardinality, or size, of the set of rational numbers. It is the same size as the set of natural numbers (denoted as |N|) and is equal to the infinite number aleph zero.
The intuitive explanation of |Q| = aleph zero can be understood by imagining that the set of rational numbers can be paired up with the set of natural numbers. This shows that the two sets have the same size, and therefore, |Q| = |N| = aleph zero.
Understanding |Q| = aleph zero is important in the field of mathematics as it provides a deeper understanding of the concept of infinity and the different sizes of infinite sets. It also has practical applications in areas such as computer science and cryptography.
Some examples of rational numbers include 1/2, 3/4, -2/5, 0.25, and 2.5. These are numbers that can be expressed as a ratio of two integers, with the denominator not equal to 0.
|Q| = aleph zero is smaller than the set of real numbers (denoted as |R|), which is considered a larger infinite set. It is also larger than the set of integers (denoted as |Z|), which is considered a smaller infinite set.