# From Natural Numbers to Absolute Value on the Real Line

Move from natural numbers used for counting to integers and rational numbers, enter the real-number system through completion language, and then express number-line distance with absolute value.

[← Return to results](/en/paths/)

This is a suggested learning sequence, not a universal logical dependency or a formal proof.


- Stage 1 · required



## [Natural number](/en/mko/mko-natural-number/?from=%2Fen%2Fpaths%2Fpath-number-system-extension-candidate%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Begin by fixing the convention for natural numbers and zero.

[Related questions](/en/explore/?mko=mko-natural-number) [Concept guides](/en/derivations/?q=mko-natural-number)


- Stage 2 · required



## [Integer](/en/mko/mko-integer/?from=%2Fen%2Fpaths%2Fpath-number-system-extension-candidate%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Add a negative direction to form the integers.

[Related questions](/en/explore/?mko=mko-integer) [Concept guides](/en/derivations/?q=mko-integer)


- Stage 3 · required



## [Rational number](/en/mko/mko-rational-number/?from=%2Fen%2Fpaths%2Fpath-number-system-extension-candidate%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Represent rational numbers as ratios of two integers while excluding a zero denominator.

[Related questions](/en/explore/?mko=mko-rational-number) [Concept guides](/en/derivations/?q=mko-rational-number)


- Stage 4 · required



## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-number-system-extension-candidate%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Enter the real-number system through the rational embedding and completion.

[Related questions](/en/explore/?mko=mko-real-number) [Concept guides](/en/derivations/?q=mko-real-number)


- Stage 5 · optional



## [Absolute value](/en/mko/mko-absolute-value/?from=%2Fen%2Fpaths%2Fpath-number-system-extension-candidate%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Use a piecewise function to express the nonnegative distance from a real number to zero.

[Related questions](/en/explore/?mko=mko-absolute-value) [Concept guides](/en/derivations/?q=mko-absolute-value)



Path source & statuspath-number-system-extension-candidate · candidate

[Original paths JSON](/data/architecture/learning-paths.json)