Math 2.0

Real functions Definition

Absolute value

Builds on The reals form an ordered field

Definition

Let xx be a real number. The absolute value of xx is the real number ∣x∣={xif x≥0,−xif x<0.|x| = \begin{cases} x & \text{if } x \ge 0, \\ -x & \text{if } x < 0. \end{cases}

Remarks

For each real xx: ∣x∣≥0|x| \ge 0; ∣x∣=0|x| = 0 only for x=0x = 0; ∣−x∣=∣x∣|-x| = |x|; and ∣x∣=max⁡(x,−x)|x| = \max(x, -x). The distance between two real numbers xx and yy is ∣x−y∣|x - y|. For r≥0r \ge 0, the inequality ∣x∣≤r|x| \le r holds exactly when −r≤x≤r-r \le x \le r.

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Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionRational numbers, object, no evidence levelRational numbersobjectReal numbers, object, no evidence levelReal numbersobjectRing of integers, statement, no evidence levelRing of integersstatementThe reals form a field, statement, no evidence levelThe reals form a fieldstatementThe rationals form a field, statement, no evidence levelThe rationals form a …statementThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementOrder on the reals, definition, no evidence levelOrder on the realsdefinitionThe reals form an ordered field, statement, no evidence levelThe reals form an ord…statementAbsolute value, definition, no evidence levelAbsolute valuedefinition
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  1. Construction of the reals from Cauchy sequences concept
  2. Integers object
  3. Ordered field definition
  4. Rational numbers object
  5. Real numbers object
  6. Ring of integers statement
  7. The reals form a field statement
  8. The rationals form a field statement
  9. The rationals form an ordered field statement
  10. Order on the reals definition
  11. The reals form an ordered field statement
  12. Absolute value definition

Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Absolute_value, Real numbers
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DefinesAbsolute value
UsesThe reals form an ordered field

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