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What does the same mean?

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[NOISE]

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Equality is a subtle topic.

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The idea of four could just be conveyed by
four dots, it could also be conveyed by.

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Upper case roman numerals, lower case
roman numerals or a ton of other symbols

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that of been used in various times

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in and places throughout the world,
throughout history.

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All of these mean the same thing.

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They all mean four.
They're all in some sense equal.

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But that's just equality of, of symbols
of, of shapes.

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It's something much more subtle happening
for sequences.

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When are two sequences the same?
Two sequences a and b are equal, they're

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the same, if they start at the same index,
which I'll call big N and coresponding

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terms are the same.
So that a sub n equals b sub n.

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Whenever n is bigger than or equal to that
starting index.

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Let's see how this works out in practice.

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Well, here's one sequence, a sub n.
Here's the sequence that starts

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with a 0th term, and is defined by the
rule that its nth term is 2 to the N.

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And here's another sequence,

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B sub N.
The sequence B sub N whose 0th term is

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defined to be 1 and subsequent terms will
be calculated by referring

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back to previous terms so that the Nth
term is twice the preceding term.

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These two sequences a sub n, and b sub n,
they're the same.

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They're equal, but they're written down
really differently.

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Alright, this sequence b sub n, is defined
recursively and the sequence a

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sub n is just defined by a single formula
in terms of n.

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They both start with a term labelled 0.

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And corresponding terms have the same
value.

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A sub zero is two to the 0 using this
formula which is one.

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And that's the same as b sub 0.

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B sub 1, is using this for a cursive
formula, twice b sub

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0, b sub 0 is 1, so b sub 1 is 2 times 1
which is 2.

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And that's the same as a sub 1 which is,
using this formula, 2 to the first power.

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And that patterns continues.

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These two sequences both start with a 0
term.

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And each term of a sub n is twice the
preceding term.

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Which is exactly the recursive definition
that I'm giving for b.

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So a sub a thousand equals b sub of a
thousand.

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A sub a million b sub of a million.

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A sub anything equals b sub the
corresponding thing.

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So the sequence a sub n and the sequence b
sub n, these two sequences.

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[SOUND]

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Are equal as sequences.

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Quality isn't about outside appearances,
it's what's inside that matters.

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It's the same for sequences.
Two sequences are equal not if they've got

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the same outside form, but if their
corresponding terms have the same value.

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You.

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[NOISE]