Math 2.0

example

The exponential map from the additive reals to the positive reals

Example

Let (R,+)(\mathbb{R}, +) be the real numbers with addition, and let (R>0,×)(\mathbb{R}_{>0}, \times) be the positive real numbers with multiplication. The exponential function gives a map exp⁡:R→R>0\exp : \mathbb{R} \to \mathbb{R}_{>0} with exp⁡(x+y)=exp⁡(x)exp⁡(y)\exp(x + y) = \exp(x)\exp(y). Thus it is a homomorphism. Its kernel is {0}\{0\}, because exp⁡(x)=1\exp(x) = 1 only for x=0x = 0. Thus the map is injective. Each y>0y > 0 is exp⁡(ln⁡y)\exp(\ln y), so the map is onto. Thus exp⁡\exp is an isomorphism.

Remarks

The inverse isomorphism is the natural logarithm.

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To read this node, read these 30 nodes in this order. The foundations come first, and the list ends with this node.

Group, concept, no evidence levelGroupconceptReal numbers, object, no evidence levelReal numbersobjectSequence, definition, no evidence levelSequencedefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionGroup isomorphism, definition, no evidence levelGroup isomorphismdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionGeometric sequence, definition, no evidence levelGeometric sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionSum of a finite geometric series, statement, no evidence levelSum of a finite geome…statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionSqueeze theorem, statement, no evidence levelSqueeze theoremstatementPartial sum, definition, no evidence levelPartial sumdefinitionPower series, definition, no evidence levelPower seriesdefinitionThe sequence r^n tends to zero for |r| < 1, example, no evidence levelThe sequence r^n tend…exampleRadius of convergence, definition, no evidence levelRadius of convergencedefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementSum of a series, definition, no evidence levelSum of a seriesdefinitionAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionComparison test, statement, no evidence levelComparison teststatementGeometric series, example, no evidence levelGeometric seriesexampleRatio test, statement, no evidence levelRatio teststatementRatio formula for the radius of convergence, statement, no evidence levelRatio formula for the…statementThe exponential series has infinite radius of convergence, statement, no evidence levelThe exponential serie…statementExponential function, definition, no evidence levelExponential functiondefinitionThe exponential map from the additive reals to the positive reals, example, no evidence levelThe exponential map f…example
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Group concept
  2. Real numbers object
  3. Sequence definition
  4. Group homomorphism definition
  5. Real sequence definition
  6. Group isomorphism definition
  7. Bounded sequence definition
  8. Geometric sequence definition
  9. Limit of a sequence definition
  10. Monotone sequence definition
  11. Series definition
  12. Sum of a finite geometric series statement
  13. Monotone convergence theorem for sequences statement
  14. Convergent sequence definition
  15. Squeeze theorem statement
  16. Partial sum definition
  17. Power series definition
  18. The sequence r^n tends to zero for |r| < 1 example
  19. Radius of convergence definition
  20. Convergent series definition
  21. A series of nonnegative terms converges iff its partial sums are bounded statement
  22. Sum of a series definition
  23. Absolute convergence definition
  24. Comparison test statement
  25. Geometric series example
  26. Ratio test statement
  27. Ratio formula for the radius of convergence statement
  28. The exponential series has infinite radius of convergence statement
  29. Exponential function definition
  30. The exponential map from the additive reals to the positive reals example

Sources

  1. Web https://en.wikipedia.org/wiki/Group_isomorphism, Examples
  2. Web https://en.wikipedia.org/wiki/Group_homomorphism, Examples

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