This chapter explores numerals, divisibility & Euclid's fundamental theorem of arithmetic (prime factorization) from a Caveman's perspective.
Transcript
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[Music] first let's go away way back in time the day-to-day life of our distant ancestors was based on one thing survival and to survive is to hunt so with food we will begin our story let's imagine two Hunters preparing The Catch of the Day eating in groups requires them to intuitively agree on what toins a fair share a fair share requires a split into equal siiz pieces an unfair share occurs when someone obtains a larger piece at someone else's expense the idea of fair versus unfair is the basis of divisibility remember if we say some quantity is divisible by four then it implies that we can form four equal size pieces now in order to hunt successfully our ancestors needed a way to predict animal Migration patterns and plan future hunts how could they predict the most successful times to hunt this question leads to one of the oldest and most powerful human Technologies the clock all clocks are based on some repetitive pattern which divides the flow of time into segments to find repetitive patterns they look towards the heavens the most obvious cycle is the rising and the falling of the sun each day however they would have required longer Cycles to track longer periods of time for this they look to the Moon which seem to gradually grow and Shrink over many days and every so often a beautiful event would occur a full moon but was the period between one full moon to the next constant to answer this they needed a method of counting ancient artifacts show they counted this pattern using notches or numerals each numeral would represent one unit such as one day grouping these units together allowed them to build numbers numbers allowed our ancestors to calculate exactly 29 days between each full moon this is the origin of a month however when they tried to divide 29 into equal parts they ran into a problem it was impossible no matter how hard you try you canot split 29 into equal pieces we could say that 29 is unbreakable unbreakable numbers are known as prime numbers imagine that our hunter was mathematically curious as he continued his day he couldn't stop thinking about these prime numbers how many other numbers are prime how big do they get first he tries dividing all numbers into two categories he lists prime numbers on the left and other numbers on the right the first thing a mathematician would do is look for patterns at first they seem to dance back and forth in a strange pattern he was on to something now let's do a modern trick to see the bigger picture the trick is to use a spiral first we list all possible numbers in a growing spiral then we color all the prime numbers blue finally we zoom out to see millions of numbers here is one of the great examples of true mathematical Beauty this is a pattern of primes which goes on and on forever incredibly the structure of this pattern is still unsolved today this idea was finally Advanced sometime around 300 BC in ancient Greece with a philosopher known as uid of Alexandria uid understood that all numbers split into two distinct categories prime numbers which we cannot share equally and composite numbers which we can he begins by realizing that any number can be divided down until you reach a group of smallest equal numbers and by definition these smallest numbers are always prime numbers H to be clear imagine the universe of all possible numbers and ignore the primes now pick any composite number and break it down you were always left with prime numbers uclid knew that every number could be expressed using a group of smaller primes or building blocks no matter what number you choose it can always be built with addition of smaller primes this is the root of his Discovery known as the fundamental theorem of arithmetic as follows take any number say 30 and find all the prime numbers it divides into equally this is known as factorization this will give us the prime factor factors in this case 2 3 and 5 are the prime factors of 30 you could realize that you could then multiply these prime factors a specific number of times to build the original number in this case you simply multiply each factor once to build 30 2 * 3 * 5 can be thought of as a special key or combination for 30 so every possible number has one and only one prime factorization a good analogy is to imagine each number as a different lock the unique key for each lock would be its prime factorization no two locks share a key no two numbers share a prime factorization this idea marks one of the greatest advances in the history of mathematics and will return thousands of years later in our story