Math 2.0

example

Existence of nth roots by the intermediate value theorem

Example

Let n≥1n \ge 1 be a natural number and let y≥0y \ge 0 be real. Then there is a unique real x≥0x \ge 0 with xn=yx^n = y. For existence, let b=max⁡(1,y)b = \max(1, y). The function p(x)=xnp(x) = x^n is continuous, with p(0)=0≤yp(0) = 0 \le y and p(b)≥b≥yp(b) \ge b \ge y. The intermediate value theorem gives x∈[0,b]x \in [0, b] with xn=yx^n = y.

Remarks

Uniqueness holds because x↦xnx \mapsto x^n is strictly increasing on [0,∞)[0, \infty). The number xx is the nn-th root yn\sqrt[n]{y}. For example, it proves that 2\sqrt{2} exists as a real number.

Explanations

L1 human

The graph of xnx^n starts at 00 and grows past yy. Since the graph has no gaps, it reaches the height yy at some point, which is the root.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionEpsilon-delta definition of continuity, statement, no evidence levelEpsilon-delta definit…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEpsilon-N definition of the limit of a sequence, statement, no evidence levelEpsilon-N definition …statementSequential continuity, definition, no evidence levelSequential continuitydefinitionIntermediate value theorem, statement, no evidence levelIntermediate value th…statementContinuity is equivalent to sequential continuity, statement, no evidence levelContinuity is equival…statementConvergent sequences are bounded, statement, no evidence levelConvergent sequences …statementLimit of a product of sequences, statement, no evidence levelLimit of a product of…statementProduct of continuous functions is continuous, statement, no evidence levelProduct of continuous…statementPolynomial functions are continuous, statement, no evidence levelPolynomial functions …statementExistence of nth roots by the intermediate value theorem, example, no evidence levelExistence of nth root…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.
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  1. Limit point of a set definition
  2. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. Continuity at a point definition
  6. Bounded sequence definition
  7. Limit of a sequence definition
  8. Continuous function definition
  9. Epsilon-delta definition of continuity statement
  10. Convergent sequence definition
  11. Epsilon-N definition of the limit of a sequence statement
  12. Sequential continuity definition
  13. Intermediate value theorem statement
  14. Continuity is equivalent to sequential continuity statement
  15. Convergent sequences are bounded statement
  16. Limit of a product of sequences statement
  17. Product of continuous functions is continuous statement
  18. Polynomial functions are continuous statement
  19. Existence of nth roots by the intermediate value theorem example

Sources

  1. Web https://en.wikipedia.org/wiki/Nth_root, Definition and notation

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