Exploring the Many Faces of Infinity: A Math Guide

Discover the concepts of infinity, from countable and uncountable sets to Cantor's diagonal argument and Hilbert's Hotel in this educational math guide.

#infinity#mathematics#calculus#cantor-diagonal-argument#set-theory#hilbert-hotel#education

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Infinity

Noah & Keenan

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What We Already Know

— Infinity shows up in limits

— It's not a number

— Means "grows without bound"

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Potential vs Actual Infinity

- Potential: keeps going

- Actual: complete infinite set

- Example: counting vs ℕ

Potential

1, 2, 3, ...

Never ends

Actual

N

Complete set

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Countable Infinity

— Can list elements

— Natural numbers, integers, evens

— All same "size"

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Uncountable Infinity

- Can't list all elements

- Real numbers

- Bigger than countable

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Cantor's Diagonal Argument

— Assume list of decimals

— Change nth digit

— New number not on list

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Hilbert's Hotel

— Infinite rooms

— "Full" hotel

— Still room for more

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Infinity in Calculus

— Limits → ∞

— Infinite series

— Improper integrals

Limit → ∞

Series

∑ 1/2ⁿ → finite ✔

∑ 1/n → ∞ ✘

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Gabriel's Horn

— Infinite surface area

— Finite volume

— "Fillable but unpaintable"

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Sizes of Infinity

— Countable < uncountable

— Continuum

— Even bigger infinities exist

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Why Infinity Matters

— Defines real numbers

— Used in physics

— Used in computer science

Real Numbers

E=mc²
∫ F dx

Physics

Computer Science

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A Question to Think About

— Which infinity does the universe use?

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