Differences Between Natural and Real Numbers Explained

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Natural numbers are the counting numbers starting from 1, with some definitions including 0. Integers expand this set to include negative numbers and zero. Rational numbers consist of all fractions, encompassing both integers and non-integer values. Real numbers encompass all rational and irrational numbers, including those that cannot be expressed as fractions, such as the square root of 2 and pi. Understanding these distinctions is crucial for grasping the broader landscape of number theory.
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what is the differenae between natural and real numbers
 
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Natural numbers (N) are 1, 2, 3, 4, ... (some authors include 0, some don't)
Integers (Z) also include the negatives: 0, 1, -1, 2, -2, 3, -3, 4, ...
Rational numbers (Q) are all the fractions, like 0, 1/2, 3/5, 4/1, 2/239, ...
Real numbers (R) are "all" numbers, so all the fractions and all the other (irrational) numbers like \sqrt{2} and \pi. If you want, you can think about a real number as any number which can be written in a (infinite, repeating or non-repeating) decimal expansion, like 1.23495012398530913298...

(Sometimes, blackboard bold notation is used, for example \mathbb{N}, \mathbb{R} instead of N, R.
 

Thread 'My basic understanding of set theory'

If there are an infinite number of natural numbers, and an infinite number of fractions in between any two natural numbers, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and... then that must mean that there are not only infinite infinities, but an infinite number of those infinities. and an infinite number of those...
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