Introduction of Real Numbers

First, let's start with the basics: what are real numbers? Real numbers are a set of numbers that include all the rational numbers (numbers that can be expressed as a fraction of two integers) and all the irrational numbers (numbers that cannot be expressed as a fraction of two integers).

The real numbers are denoted by the symbol R, and they can be visualized on a number line, where the rational numbers are represented by dots, and the irrational numbers are represented by points on the line.

Now let's move on to Euclid's division lemma. Euclid's division lemma states that given two positive integers a and b, there exist unique integers q and r such that:

a = bq + r, where 0 ≤ r < bI

n other words, any positive integer a can be divided by a positive integer b in such a way that the remainder r is between 0 and b-1, and this division is unique. This lemma is the basis of the Euclidean algorithm, which is used to find the greatest common divisor of two positive integers.