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0:00
The beginning of mathematics starts with numbers and counting which at one time humans did not know anything about One of the age-old questions actually is did we invent numbers or are they already there? Now, of course we use a ten digit number system, but there's no mathematical reason for that There's no reason we could not have nine digits which means after eight to the next number would be one zero because you'd be out
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of digits and you would need to reset the ones place to zero and then add a 1 This would represent what we know of as 9 if there were nine digits We believe that the 10 digits comes from the fact that we have 10 fingers simple as that In fact, there are many times. We only use two digits It's known as binary The mathematicians actually say had we pick 12 digits that arithmetic would be much easier considering 12 is divisible by more numbers than 10
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now one of the oldest foundations of mathematics which has lost on many people today is Logic, although a lot of advancements in this field were done in the 19th and 20th century it started around 2,500 years ago in a logic class nowadays You'll notice some odd examples of problems that involve just thinking for example take the sentence if it is raining then the ground is wet Let's say that is definitely a true statement. Now, is this next statement also true
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If the ground is not wet then it is not raining
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Anyone who's learned basic logic, which even includes various geometry students can answer this in a second others may need a little more time But these two statements are in fact Logically the same in fact whenever you say if something then something else it is always the same as saying if not the second thing then not the first thing if It angles 40 degrees then it is acute and if an angle is not acute than it is not 40 degrees are Logically the same thing
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Concepts like this may seem weird both in mathematics we need very formal language that we all agree on in order to prove theorems that are extremely difficult to understand if you're asked to prove the statement if x squared is even than X is even You now know you could also prove if X is not even then x squared is not even Proving one automatically proves the other and although this is an easy example when dealing with more advanced mathematics
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We need a very clear understanding of the language and logic behind every statement Then later just over 2,000 years ago, the mathematician Euclid published a series of 13 books known as elements Which main regard is the most influential textbook of all time? Well, there's a lot discussed in these books one important concept is the Euclidean algorithm or Euclid's algorithm one of the first mathematical algorithms ever Discovered an algorithm is simply a series of steps that solves a problem
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Executed often but not always by a computer this algorithm calculates the greatest common divisor of two numbers in an efficient way. So if you want to know the largest number that goes into let's say 714 and 1054 it would not be that hard, but may take a little bit of time to find the answers 34 Instead we can use the algorithm I'm not going to explain how it works
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But just involves basic elementary school arithmetic and usually you can probably figure out the answer by hand in less than a minute. I Mentioned this algorithm because it extends the field of cryptography cryptography is really about techniques that ensure secure communication Especially in the presence of adversaries such as hackers when putting sensitive information online let's say Normal text is encrypted using
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mathematical techniques that turn it into text that cannot be understood by a third party or really anyone at least until it is decrypted back to original text which you need a secret key to do that only the receiving party should have although there are various methods of encryption certain types of RSA rely on the fact that is very difficult to break large numbers up into prime factors
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Cryptography heavily involves number theory or the study of integers, which is why that Euclidean algorithm from before has applications in this field This is what protects our passwords credit card numbers and any type of sensitive online information in fact The NSA or National Security Agency is the largest employer of mathematicians in the United States as they need those mathematicians to create and break these codes
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You're may think cryptography is a recent development as computers have not been around for even a century yet However, it dates back hundreds of years written letters were encrypted for privacy purposes. In fact tech needs to decipher These encrypted messages were used on messages from Mary Queen of Scots Which revealed that she had sanctioned an attempted assassination of Queen Elizabeth, and she was executed shortly after that in 1587
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Moving on there were of course several hundred years of discoveries that we all know very well For example the discovery of Pi which actually took a lot of work to pinpoint exactly It was approximated as three point one two five and also three point one six for awhile until several years later better approximations were made Then logarithms geometry algebra basic artesian coordinates and complex numbers
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Which I made an entire video on and so on were all discovered hundreds even thousands of years ago So I'm not going to cover those main detail But if you want a random fun fact the Rays of the earth was approximated in the third century BC to nearly 99% Accuracy using basic geometry deductive reasoning and just measuring a shadow length But with that said now I'm going to jump ahead to the 17th century when calculus was introduced to the world Which would change math and physics forever?
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Newton was concerned with analyzing rates have changed not as an average like average slope, but at one instant This instant rate of change became known as a derivative Telling us the change in some parameter with respect to another at again one specific instant. For example, if your speed is constantly changing Algebra can tell you the average speed but calculus can tell you the exact speed at any moment in time Kind of like what a radar going to read?
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This gave us much more powerful insights in the motion of planets how they change their speed throughout in orbit Then the motion and behavior of electromagnetic waves is expressed through calculus moment of inertia or essentially an object's resistance to rotation is calculated using calculus techniques and Calculating the work done on a particle moving through a complex vector field requires calculus
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Calculus is used within economics and maximizing profit chemistry and calculating diffusion rates and so much more there's no way to do justice for all the applications of calculus, but whether you study engineering physics math chemistry Bio or even business and college unless you have all the needed credits College will start with a series of calculus courses to be used later on
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Next in 1736 Leonard Euler published a paper on the seven bridges of königsberg, which is regard as the first paper and graph theory The question was quite simple given this network of bridges. Can you cross each bridge exactly once of course without going into the water you
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Can try it for yourself, but spoiler Euler eventually proved it to be impossible Graph theory is not about the graphs You know from middle school in high school a graph here is made up of nodes and edges that connect them The screw to have a wide variety of applications many of which lie in the field of computer science Now a graph can have nodes that represent people and their connections could represent Compatibility in which dating sites need to use algorithms to create best matches
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Nodes could be cities and edges are routes that connect them. We need to figure out the shortest path one point to another Aura graph could represent how we're all connected through social media or another social structure This investigation is known as social network analysis Which is used for example in security applications to map out information on street gangs terrorist organizations and more
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To end this subfield and emphasize how powerful graph theory can be is made to computer science graduates, Larry Page and Sergey Brin Billions of dollars in the 90s they use graph theory in such a way that note Represented websites on the Internet and if one website linked to another that would be represented by an edge The more links going to a website the quote better that web page would be considered this algorithm to rank web pages Became the backbone of what is now Google
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Out of Euler seven bridges of königsberg paper now only came graph theory But also the field of topology one of the most advanced math courses you would take if you studied math as an undergrad in topology You have to forget about the notion of lengths and angles like in geometry in this field. Who care more about Connectedness and holes, for example where bending and stretching is totally fine and does not change the properties of the space that we care about
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For this reason in topology, they famously say that a doughnut and a coffee mug are the same thing
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The questions of can you morph one object into another or whether the two are homeomorphic is important within topology? You'll analyze complex shapes higher dimensional objects knots and so on this class allows you to understand the mathematics of things like cutting a mobius strip in half, which I did in a previous video or Reveals mathematically how you can turn a sphere inside out without cutting or tearing it or making any creases while yes allowing self? intersections is this
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Is this a sphere turning inside out topology applies to fields of physics such as quantum field theory or cosmology in general relativity? states that space-time is a four-dimensional Lorentzian manifold and analyzing this involves concepts within topology in Robotics and motion planning all the possible states that the robot can be in or the configuration space can be modeled using concepts taught in topology in
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computer science topology can be used to model how networks are connected and how data flows and Even not theories used in biology to analyze how enzymes cut and reconnect DNA
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next up in 1822 Joseph Fourier published a paper on heat flow and while working on that he made a discovery that would go on and have a wide range of Applications this discovery. We now call the Fourier transform he determined that any function no matter how weird it looked could be broken up into just a sum of sine and cosine functions as In if you just take a bunch of these sine and cosine functions and pick just the right frequencies and amplitudes
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You could add them up to make any function you wanted this has lots of applications But the main ones are in quantum mechanics and signal processing when looking at real-world signals whether it be a radar signal a signal from a digital image sound or even light the Physical signal itself can be quite complicated and not tell us much by using Fourier analysis We can break that signal up into those trig functions that make up the signal in question
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Which reveal properties that cannot visually be seen originally Like whether the signal is made up of much higher frequency sources or lower frequency sources Then although difficult to trace back to a specific date the fundamentals of a field known as group theory began in the early 1800s This is all about the study of groups and a group is basically a set of elements that along with some operation satisfy certain conditions the set of integers under addition is a group for this reason if
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You pick any two integers and add them up you get another integer something that's in the set then there also exists an integer in the set in this case 0 Where if you add any number in the set to it, you get the same thing out
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Next for any number in the set there exists another such as that if you add them you get that identity from before which was 0 in this case and Lastly how you group certain numbers under addition does not change the result These four properties of closure an identity element an inverse element and associativity mean the set is a group Yes, this seems very weird and random, but group three has given us a lot of insight into the mathematics of symmetry
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This math can apply two sets of numbers, but it applies to other sets such as the manipulations of a rubik's cube All the ways that a rubik's cube can be altered form a group with its own unique properties In our last example with integers in addition you can easily see that the order of the integers does not matter when you add them The fact that you can swap two numbers and get the same result means the group is commutative. Otherwise known as an abelian group
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Knowing that is the group of all possible Rubik's cube manipulations in a billion group as in if I turn the bottom 90 degrees in the front face Is that the same thing as turning the front face and then the bottom? May take a little thought but the answer is no this is not an abelian group Well, you don't need group theory to solve a Rubik's Cube group Theory does give you insight into the mathematics behind the rubik's cube
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This mathematics of symmetry as applications in chemistry, for example, as groups can classify certain crystal structures and symmetries within molecules It can apply to public key cryptography and it has various applications in physics for example anothers theorem explains how symmetry within a system Corresponds to a conservation law and this gave us a better understanding of einstein's general theory of relativity
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Then real quick boolean algebra, which was discovered in the 1800's is essentially algebra using just ones and zeros Which can be used in computer applications. For example using boolean algebra you can simplify the amount of logic gates within a circuit where these logic gates are what flip and transmit ones and zeros within a Computer the less of these there are the faster computer can run
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Moving on in 1874 mathematician Georg Cantor published a paper titled on a property of the collection of all real algebraic numbers which is where the branch of math known as set theory began a Set is basically a collection of objects It could be a set of colors people the set of prime numbers even numbers and so on Reach member is known as an element of the set Set theory is concerned with the intersections of sets unions
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Subsets or sets within sets and what I'll cover now whether a set is countable or not for example the set of all integers which includes negatives positives and zero is Countable now that does not mean that you can count every element in the set It does go on forever and therefore is an infinite set, but it is countable well actually countably infinite and That's because I can order all the elements such that nothing ever repeats and I don't skip a number I could put these in order
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Such that I have 0 then 1 the negative 1 to the negative 2 3 then negative 3 and so on If I keep going I would eventually hit any integer you could name and I would not skip anything Another way to think about this is I can assign a natural number to every element of the set without skipping one
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Now a tougher question is is the set of rational numbers countable Rational numbers are either decimals that end or decimals that go on forever. But repeat 2 2.2 2.3 for repeating 2.5 9 4 etc. These are all rational now could I line these up such that I have a first rational number a Second a third and so on so it's that I don't skip anything
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Amazingly it was proved that the answer is yes. The set is countably infinite First remember that any rational number can be expressed as a ratio of two integers
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Now the most classic proof is ordering the rational numbers in a clever way Such as they are in this form you have rows and columns of integers and every entry is the ratio of those two numbers We can start at the top left corner Where one is our first rational number then we go down to the next for the second then Diagonal and keep going while skipping the rational numbers that repeat
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doing this we would eventually reach any rational number you could pick even this repeating decimal and therefore rational number is just 67 over 241 and if you continue this pattern forever, you'd eventually find that 67 and 241 column and row If you study math or maybe computer science in college, you will learn how the set of irrational numbers is actually not countable There's no way to line them up such that you list every single one without skipping
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This means you could say that there are more irrational numbers than rational Even though there are an infinite amount of Beach You'd also see that the set of real numbers which includes integers decimals PI square root of 2, etc is also not countable Set theory now has applications in several topics. We talked about including graph theory topology group theory and more Now when you flip a coin or play a game of roulette, every single event is an independent one
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It does not matter what your last spin was the odds of getting red on this next one are still the exact same as before Well in the early 1900's work was done on a statistical model describing events in which probability depends only on the previous event This model became known as a Markov chain Markov chains include a state space and transition probabilities Imagine you want to analyze the population of Los Angeles and New York for simplicity
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we can say people only can move from one to another or they can stay put So maybe every year 10% of people who live in LA moved to New York meaning that 90% of the people stay And at the same time every year 15% of people in New York moved to LA meaning 85% of stay in this example where you are now matters based on if you're in LA or New York We can predict whether a person may move or not
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But saying that a person took this specific series of moves really does not matter in terms of what comes next Using certain techniques in this case linear algebra You can determine how the system will evolve over time and whether it will reach some steady state or not These apply to thermodynamics and representing certain unknown details of the system They apply to modern speech recognition systems and they're what defines the page rank of a website as used by Google
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Then also in the early 1900s work was being done on the existence of a strategy in two-person zero-sum Games or games in which one person's gain is balanced out by another's loss this field started by john von neumann in 1928 became known as game theory and quickly expanded throughout the mid 1900s to Summarize it's about the study of logical decision making and strategy within competitive situations
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One of the most famous quote games that came up in the early days of this field was the prisoner's dilemma where you and a co-conspirator have been in prison for committing a crime you're separated and each have the option either stay silent or Rat on the other and say they did it and based on what each of you do You may go to jail for a certain amount of time or you may be set free
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The question here is what is in your best interest to do especially while not knowing what the other person will say This field grew to have applications in both economics and computer science as well as a few other fields in computer science There's cloud computing in which game theory can be used for modeling interactions between cloud providers where cost should be minimized while other factors? maximized and
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Economics, it applies to auctions mergers and acquisitions rage party wants the most benefit financial analysis and a new product release and much more Then in the 1880s the mathematician henri poincaré a was studying the three-body problem Which deals with studying the motion of three point masses for example, the motion of the moon earth and sun under each other's gravitational pull Analyzing the motion of two orbiting bodies is very doable using newton's laws of motion
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But throw a third in and it actually gets very difficult Now Poincare, I found that small changes in the position and initial velocity of these masses would cause for way different behaviors over time This was the beginning of chaos theory a field of math dealing with dynamic systems that are very sensitive to initial conditions
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Now after his contributions the field slowed down a lot due to the lack of computational power back then eventually a man named Edward Lorenz Accidentally found his interest in this subject through his work in weather prediction in the early 60s He realized that when he made a small change to initial conditions in his computer simulation It produced huge changes in the long term final outcome you may have heard of the butterfly effect a
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metaphorical example where if a butterfly flaps its wings on one side of the Atlantic Ocean Let's say it could cause a tornado on the other side weeks later Will this very minor change in the environmental conditions causing a large difference is a concept of chaos theory and its name the butterfly effect Was coined by that man Ellard Lorenz after his findings with the weather simulation Now we can use chaos theory to analyze systems like a double pendulum
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We're changing its starting high even a little greatly changes its path It's using robotics to predict the motion of a robot and how it will develop over time and it can even be using cryptography
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Next geodesics are curves representing the shortest path between two points on a curved surface These are what an aunt living on the surface would perceive to be straight and explains Why airplane routes on a map are curved a little these are of huge importance for Einstein's general theory of relativity
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Moving on Fermat's Last Theorem, which is easy to understand on the surface was unsolved for 300 years The theorem says that there are no three integers a B and C such that this is true Where n is any integer greater than 2 no one could prove whether this was true until just a few decades ago in 1994 where Andrew Wiles finally proved the theorem and found that in fact There are no integers that make this true and although the theorem is easy to understand the proof involves
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very rigorous math taking up over a hundred pages and Last of the millennium prize problems which are seven very difficult problems within mathematics These seven were decided on at a conference about two decades ago and each question comes with a 1 million dollar prize if you solve It but so far only one of the 7 has been solved which was the Poincare a conjecture proven in 2003 and although there are many more things I could talk about. I'm gonna end that video there
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For anyone looking to learn more about any of these subjects I will leave some resources down below including books and lecture series Some of which will be affiliate links if you want to support the channel Don't forget to Like and subscribe Follow me on Twitter and join the major Facebook group for updates on everything and I'll see you all in the next video
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