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How are real numbers represented on a number line?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — algebra This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

Real numbers are represented on a number line by imagining a straight line that extends infinitely in both directions. To construct this representation:

Origin: An arbitrary point $O$ is chosen on the line to represent the number 0.
Unit Length: A second point $U$ is chosen to the right of $O$ . The segment $O U$ defines the unit length, representing the number 1.
Positive Numbers: Multiples of this unit length extend to the right of the origin. For example, the number 2 is represented by a segment twice as long as $O U$ , extending in the same direction.
Negative Numbers: These are represented by extending the line to the left of the origin $O$ .

This geometric model identifies every point on the line with a unique real number. While many mathematicians historically recognized this relationship, it was formally established as a postulate by Richard Dedekind in 1872.1