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The Fundamental Theorem of Arithmetic

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    Imagine we are living in prehistoric times.
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    Now, consider the following:
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    How did we keep track of time without a clock?
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    All clocks are based on some repetitive pattern
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    which divides the flow of time into equal segments.
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    To find these repetitive patterns,
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    we look towards the heavens.
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    The sun rising and falling each day
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    is the most obvious [pattern].
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    However, to keep track of longer periods of time,
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    we looked for longer cycles.
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    For this, we looked to the moon,
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    which seemed to gradually grow
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    and shrink over many days.
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    When we count the number of days
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    between full moons,
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    we arrive at the number 29.
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    This is the origin of a month.
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    However, if we try to divide 29 into equal pieces,
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    we run into a problem: it is impossible.
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    The only way to divide 29 into equal pieces
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    is to break it back down into [29] single units.
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    29 is a 'prime number.'
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    Think of it as unbreakable.
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    If a number can be broken down into
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    equal pieces greater than one,
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    we call it a 'composite number.'
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    Now if we are curious, we may wonder,
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    "How many prime numbers are there?
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    – and how big do they get?"
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    Let's start by dividing all numbers into two categories.
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    We list the primes on the left
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    and the composites on the right.
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    At first, they seem to dance back and forth.
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    There is no obvious pattern here.
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    So let's use a modern technique
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    to see the big picture.
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    The trick is to use a "Ulam spiral."
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    First, we list all possible numbers in order
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    in a growing spiral.
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    Then, we color all the prime numbers blue.
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    Finally, we zoom out to see millions of numbers.
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    This is the pattern of primes
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    which goes on and on, forever.
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    Incredibly, the entire structure of this pattern
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    is still unsolved today.
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    We are onto something.
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    So, let's fast forward to
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    around 300 BC, in ancient Greece.
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    A philosopher known as Euclid of Alexandria
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    understood that all numbers
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    could be split into these two distinct categories.
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    He began by realizing that any number
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    can be divided down – over and over –
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    until you reach a group of smallest equal numbers.
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    And by definition, these smallest numbers
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    are always prime numbers.
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    So, he knew that all numbers are
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    somehow built out of smaller primes.
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    To be clear, imagine the universe of all numbers –
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    and ignore the primes.
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    Now, pick any composite number,
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    and break it down,
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    and you are always left with prime numbers.
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    So, Euclid knew that every number
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    could be expressed using a group of smaller primes.
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    Think of these as building blocks.
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    No matter what number you choose,
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    it can always be built with an addition of smaller primes.
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    This is the root of his discovery,
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    known as the 'Fundamental Theorem of Arithmetic' –
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    as follows:
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    Take any number – say 30 –
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    and find all the prime numbers
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    it [can be divided into] equally.
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    This we know as 'factorization.'
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    This will give us the prime factors.
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    In this case 2, 3, and 5 are the prime factors of 30.
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    Euclid realized that you could then multiply
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    these prime factors a specific number of times
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    to build the original number.
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    In this case, you simply
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    multiply each factor once to build 30.
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    2 × 3 × 5 is the prime factorization of 30.
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    Think of it as a special key or combination.
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    There is no other way to build 30,
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    using some other groups of prime numbers
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    multiplied together.
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    So every possible number has one –
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    and only one – prime factorization.
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    A good analogy is to imagine each number
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    as a different lock.
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    The unique key for each lock
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    would be its prime factorization.
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    No two locks share a key.
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    No two numbers share a prime factorization.
Title:
The Fundamental Theorem of Arithmetic
Description:

The Fundamental Theorem of Arithmetic

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Video Language:
English
Duration:
03:52

English subtitles

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