Understanding Mathematics

What is a number? Different people seem to answer differently to this simple question. For example, a Russian will say: odin, dva, tri, and write digits 1, 2, 3 And a German will say Ein, Zwei, Drei ... but he will write the same digits. A Japanese would say: iti, ni, san ... and draw three hieroglyphs. But of course they mean the same thing by a number: it is something that can be added and multiplied, obeying certain rules. In this video first knowledge from the Greek word "μαθημα" which means "knowledge" 16th century is a breakthrough in mathematics 17th century. Fermat and Descartes School and advanced math As soon as you open a textbook on advanced math, the letters seem to be the same, but nothing is clear The tip of the iceberg and cryptography paradoxes of advanced math and this number is not positive benefits of advanced math bringing your mobile phone to the payment terminal in the store Mathematics is not just science, it is a language of science. For example, if I write such a complicated expression then anywhere in the world everyone will understand me the same way. Namely, there is such a number zero, and if we add zero to any number, this number does not change. And there is such a number one, and if we multiply any number by one, this number does not change. If you didn’t know this, then right now you have become a little bit smarter. Mathematician MSU Mathematics comes from the ancient Greek word "ΜΑΘΗΜΑ", which means "knowledge". Therefore, it is believed that mathematics, as a systematic science, appeared precisely in Ancient Greece, but this is certainly not true; and the history of mathematics is much richer. Ancient Babylon, Ancient Egypt, China and India, Arab countries. The mathematicians of these countries had had much to be proud of before studying and the development of geometry in ancient Greece. And in fact, it all starts with the concept of a natural number and the simplest elementary geometric figures. The mathematical knowledge, the very first one, is obtained in practice. Let's take an apple and another apple, add them up and we get two apples. And then an apple is cut into pieces and the apple is put back from these parts, the whole comes out. From the parts the whole is obtained. All these arithmetic operations has arisen from practical needs. Until about the 16th century the period of elementary mathematics had continued. This is the knowledge that modern students of secondary schools receive nowadays. These are arithmetic, methods for solving equations of the first and second degree, simple problems of plane and solid geometry. But the 16th century is a breakthrough in mathematics. Having comprehended everything that had been done before them by their predecessors, mathematicians of that time made a qualitative leap forward First, they learned to solve equations of the third and fourth degree. Solving them, they faced with an unpleasant surprise - they had to extract square roots from negative numbers. The mathematicians has not known how to do anything like this before. These numbers were given the name "imaginary." And then their arithmetic was developed and it led to appearance of complex numbers. Then, François Viète introduced algebraic notation. Before him mathematical works had been just literary essays. He also introduced letters to indicate arbitrary numbers. And although his notation was not very convenient yet, it was already much easier to work with algebraic expressions and formulas. Descartes then finalized his notation and, in fact, the period of mathematics of variables began. The third important discovery was logarithms. With their discovery complicated calculations became much simpler. At the end of the century, Simon Stevin published his book “De Thiende”, which showed how to work with decimal fractions. In fact, he legalized the decimal system Before this publication everyone worked with hexadecimal fractions. Actually, we still use them sometimes, have a look at your wrist watch. An hour consists of 60 minutes and a minute consists of 60 seconds. And simultaneously with these discoveries, the authority of mathematics was growing. Mathematicians of that period learned how to solve a lot of important practical problems. These were tasks related to artillery, navigation, construction, optics, hydraulics, and so on and so forth. The 17th century. Fermat and Descartes developed and introduced a coordinate system, thereby connecting two branches of mathematics - algebra and geometry. Solution of many problems was greatly facilitated, thus creating an analytical geometry. Further Fermat, Huygens and Bernoulli created probability theory based on the odds calculation at gambling. And finally, another deepest idea was to represent smooth curves by a set of infinitesimal segments of straight lines. At the end of the century, this idea was embodied in works of Newton and Leibniz, who created a powerful tool - calculus. In essence, it is an “analysis of the infinitesimal”, which combines differential and integral calculus. Newton said the following on his works “If I have seen further it is by standing on the shoulders of Giants,” thereby emphasizing that with no works of predecessors there would have been no modern discoveries. What is the difference between advanced and elementary mathematics? The fact is that when you open a textbook on elementary mathematics, everything seems to be clear, but as soon as you open a textbook on advanced mathematics, the letters seem to be the same, but nothing is clear. Of course, this is a joke. But seriously, the main difference between advanced mathematics and elementary one is a concept of limit transition and infinity. Properties familiar to us from school time may turn out to be wrong as soon as we rush to infinity. For instance, a limit of a sequence of positive numbers may be a non-positive number. A permutation of the factors may lead to change of the product. And if we rearrange the terms, then under certain circumstances the sum can be made equal to any predetermined number. And despite of the fact that it sounds more than strange, mathematics remains the standard of the most accurate and reliable knowledge that we are only able to achieve. Let us remember that at elementary school for six years we study math. And in the seventh grade, unexpectedly we are told that it is divided into algebra and geometry, and geometry, in turn, is divided into plane geometry and solid geometry. And in the tenth grade we start studying the beginnings of analysis. And this are three foundations of mathematics - algebra, geometry and calculus. Let's write it. algebra - geometry - analysis But of course, when we go to a higher school, that is, a university, and study advanced mathematics, then algebra turns into higher algebra, geometry into analytical geometry, and analysis, in turn, is mathematical analysis (calculus). They at once generate new branches of advanced mathematics. Algebra and geometry, when combined, give linear algebra. In fact, it is a geometric algebra But you can swap words and then get algebraic geometry. Further, if we add complex numbers to calculus, we get a complex analysis. And if we want to study functions, we get a functional analysis. Another important branch of calculus is differential equations. Let us go on. If we can solve equations, and now we face with hundreds, thousands, tens of thousands of equations, we immediately get computational mathematics and linear programming. Algebra and complex analysis generate a very important branch of mathematics called number theory. And it turns out that without a complex analysis, there’s practically nothing to do there, because modern number theory is a foundation of cryptography. In turn, calculus generates a very important branch of mathematics - probability theory. Two branches adjoin to probability theory, they are mathematical statistics and theory of stochastic processes And cryptography is associated with such sections as discrete mathematics, graph theory, and so on and so forth. That is, what I'm showing you is the tip of the iceberg called modern mathematics. A piece of information for students. Right now I am working on creation of training courses at several sections of advanced mathematics. Including analytical geometry, calculus, probability theory, differential equations. Details can be found via a link in the description under this video. We begin learning process this particular year. Let me show you few examples. Let us take a sequence of positive numbers and make sure that it converges to a non-positive number. Consider a very simple example. Numbers like one divided by n, where n is a natural number. We write out the first few: one first, one second, one third, one fourth, one fifth. It’s clear what’s going on. The numerator is always one, and the denominator is constantly increasing. Let's assume that the sequence of these numbers tends to some positive number a. Then let's have a look at the reciprocal number - one divided by a. Of course, this number should be large, since the number a is small. Now, take this number and increase it ten more times. Fraction of ten divided by a. And then, if we say that our number n is greater than this fraction, then the reciprocal number, that is, a quotient of one by n is less than a tenths. But the number a of tenths is certainly much less than the number a. So there is no such positive number. And if there is no positive number, then our sequence converges to zero. And this number is non-positive. Now let's look at this series. One minus one plus one minus one plus one minus one plus and so on. That is, what happens? First a positive unit comes, then a negative one, and we sum them up. What could a sum be equal to? Let's count. The sum of the first two terms is just zero The sum of the next two terms is also zero. The sum of the next two terms is zero again. But the sum of all zeros is of course zero. That's right, but let's try to calculate differently. Let us write out again a set of our terms with plus and minus signs and see what happens. Let's not touch the first one, but take the next two. Minus one and plus one is of course zero. Again minus one and plus one, this gives zero also Further, minus one and plus one, it is zero again. And it turns out that a lot of zeros are added to the first number one Of course we get one. But it turns out that the results are different. Have we calculated correctly? Or maybe we are grouping the terms incorrectly? Let us then simply denote the sum of the series by some number X. And find what this X is equal to. Let's not touch the first term, leave it unchanged for now, and then say the following - we factor out minus one. Then one minus one plus one minus one will remain in brackets, that is, all terms change their signs to the opposite ones But then in parentheses we get the number X again. We denoted the sum of the series by X. Then we get that this sum equals to one minus X. But, before we put the brackets it was just X. So we get from here the equality X=1-X. And transferring X to the left side, we get that two X are equal to one. And in turn, X is equal to one second. Could you see how interesting it is? We were able to get both zero and one, and even a non integer, that is a fraction one second. Now consider an even more interesting example, when the sum of positive numbers can turn out to be a negative number. Let's look at this series: one plus two plus four plus eight plus sixteen and so on. What is it? Each subsequent term is two times larger than the previous one. Well, but we do not know what the sum of the series is equal to. Let's denote it by X. Now we will do the following. We leave the first term unchanged, and then note that all the other terms are even. So the two can be factorized In brackets there will be one plus two plus four plus eight plus and so on. But what do we have in parentheses? There is indeed X. That is exactly the sum of the original series. That is, we get that it equals to one plus two X. But if now we multiply out the brackets, then we get an X on the left. That is, we get the equation: X is equal to one plus two X. And from here we get that X minus two X equals one. That is, minus X is equal to one. So X equals to minus one. Indeed, the sum of positive numbers turned out to be a negative number. So where is advanced mathematics applied nowadays? In fact, wherever you look at. Space. Should we create a new channel "Artist of Moscow State University"? Medicine. Construction. Programming. Our century is an era of information, and to get this information quickly, and most importantly to find what the user needs, you should have good and fast search algorithms. And if we add here operating systems, virtual services, social networks, NFC chips, and so on, then we discover that mathematics is a foundation of all of this. The purpose of this video is not to scare you with advanced mathematics, especially with pictures like these ones, but to show you that advanced mathematics is used literally at every step you make and even if you do not know it. For example, bringing your mobile phone to the payment terminal in a store. At the moment, encryption algorithms - and this is discrete mathematics - do the work for you, and you calmly pick up your purchase. Or another example. No cashier knows what the Lebesgue integral is, but they all know perfectly how to use it on practice when they count bank notes. Surely, not everyone needs to know advanced mathematics, but for many of you it will come in handy in future work. Soon there will be useful videos on advanced mathematics, and by the way in comments you can write what kind of video you would like to watch on my channel. Subscribe to my Instagram channel, there will be unique content that is absent on YouTube. About life, about me and a little different mathematics. Those who watched the video up to this point, write a word “integral” in comments and what do you think of this video, I will find out who watches my videos till the end. Think with your own brain and see you on the channel!

What is a number? Different people seem to answer differently to this simple question. For example, a Russian will say: odin, dva, tri, and write digits 1, 2, 3 And a German will say Ein, Zwei, Drei ... but he will write the same digits. A Japanese would say: iti, ni, san ... and draw three hieroglyphs. But of course they mean the same thing by a number: it is something that can be added and multiplied, obeying certain rules. In this video first knowledge
from the Greek word &quot;μαθημα&quot; which means &quot;knowledge&quot; 16th century is a breakthrough in mathematics 17th century. Fermat and Descartes School and advanced math As soon as you open a textbook on advanced math, the letters seem to be the same, but nothing is clear The tip of the iceberg and cryptography paradoxes of advanced math
and this number is not positive benefits of advanced math
bringing your mobile phone to the payment terminal in the store Mathematics is not just science, it is a language of science. For example, if I write such a complicated expression then anywhere in the world everyone will understand me the same way. Namely, there is such a number zero, and if we add zero to any number, this number does not change. And there is such a number one, and if we multiply any number by one, this number does not change. If you didn’t know this, then right now you have become a little bit smarter. Mathematician MSU Mathematics comes from the ancient Greek word &quot;ΜΑΘΗΜΑ&quot;, which means &quot;knowledge&quot;. Therefore, it is believed that mathematics, as a systematic science, appeared precisely in Ancient Greece, but this is certainly not true; and the history of mathematics is much richer. Ancient Babylon, Ancient Egypt, China and India, Arab countries. The mathematicians of these countries had had much to be proud of before studying and the development of geometry in ancient Greece. And in fact, it all starts with the concept of a natural number and the simplest elementary geometric figures. The mathematical knowledge, the very first one, is obtained in practice. Let&#39;s take an apple and another apple, add them up and we get two apples. And then an apple is cut into pieces and the apple is put back from these parts, the whole comes out. From the parts the whole is obtained. All these arithmetic operations has arisen from practical needs. Until about the 16th century the period of elementary mathematics had continued. This is the knowledge that modern students of secondary schools receive nowadays. These are arithmetic, methods for solving equations of the first and second degree, simple problems of plane and solid geometry. But the 16th century is a breakthrough in mathematics. Having comprehended everything that had been done before them by their predecessors, mathematicians of that time made a qualitative leap forward First, they learned to solve equations of the third and fourth degree. Solving them, they faced with an unpleasant surprise - they had to extract square roots from negative numbers. The mathematicians has not known how to do anything like this before. These numbers were given the name &quot;imaginary.&quot; And then their arithmetic was developed and it led to appearance of complex numbers. Then, François Viète introduced algebraic notation. Before him mathematical works had been just literary  essays. He also introduced letters to indicate arbitrary numbers. And although his notation was not very convenient yet, it was already much easier to work with algebraic expressions and formulas. Descartes then finalized his notation and, in fact, the period of mathematics of variables began. The third important discovery was logarithms. With their discovery complicated calculations became much simpler. At the end of the century, Simon Stevin published his book “De Thiende”, which showed how to work with decimal fractions. In fact, he legalized the decimal system Before this publication everyone worked with hexadecimal fractions. Actually, we still use them sometimes, have a look at your wrist watch. An hour consists of 60 minutes and a minute consists of 60 seconds. And simultaneously with these discoveries, the authority of mathematics was growing. Mathematicians of that period learned how to solve a lot of important practical problems. These were tasks related to artillery, navigation, construction, optics, hydraulics, and so on and so forth. The 17th century. Fermat and Descartes developed and introduced a coordinate system, thereby connecting two branches of mathematics - algebra and geometry. Solution of many problems was greatly facilitated, thus creating an analytical geometry. Further Fermat, Huygens and Bernoulli created probability theory based on the odds calculation at gambling. And finally, another deepest idea was to represent smooth curves by a set of infinitesimal segments of straight lines. At the end of the century, this idea was embodied in works of Newton and Leibniz, who created a powerful tool - calculus. In essence, it is an “analysis of the infinitesimal”, which combines differential and integral calculus. Newton said the following on his works “If I have seen further it is by standing on the shoulders of Giants,” thereby emphasizing that with no works of predecessors there would have been no modern discoveries. What is the difference between advanced and elementary mathematics? The fact is that when you open a textbook on elementary mathematics, everything seems to be clear, but as soon as you open a textbook on advanced mathematics, the letters seem to be the same, but nothing is clear. Of course, this is a joke. But seriously, the main difference between advanced mathematics and elementary one is a concept of limit transition and infinity. Properties familiar to us from school time may turn out to be wrong as soon as we rush to infinity. For instance, a limit of a sequence of positive numbers may be a non-positive number. A permutation of the factors may lead to change of the product. And if we rearrange the terms, then under certain circumstances the sum can be made equal to any predetermined number. And despite of the fact that it sounds more than strange, mathematics remains the standard of the most accurate and reliable knowledge that we are only able to achieve. Let us remember that at elementary school for six years we study math. And in the seventh grade, unexpectedly we are told that it is divided into algebra and geometry, and geometry, in turn, is divided into plane geometry and solid geometry. And in the tenth grade we start studying the beginnings of analysis. And this are three foundations of mathematics - algebra, geometry and calculus. Let&#39;s write it. algebra - geometry - analysis But of course, when we go to a higher school, that is, a university, and study advanced mathematics, then algebra turns into higher algebra, geometry into analytical geometry, and analysis, in turn, is mathematical analysis (calculus). They at once generate new branches of advanced mathematics. Algebra and geometry, when combined, give linear algebra. In fact, it is a geometric algebra But you can swap words and then get algebraic geometry. Further, if we add complex numbers to calculus, we get a complex analysis. And if we want to study functions, we get a functional analysis. Another important branch of calculus is differential equations. Let us go on. If we can solve equations, and now we face with hundreds, thousands, tens of thousands of equations, we immediately get computational mathematics and linear programming. Algebra and complex analysis generate a very important branch of mathematics called number theory. And it turns out that without a complex analysis, there’s practically nothing to do there, because modern number theory is a foundation of cryptography. In turn, calculus generates a very important branch of mathematics - probability theory. Two branches adjoin to probability theory, they are mathematical statistics and theory of stochastic processes And cryptography is associated with such sections as discrete mathematics, graph theory, and so on and so forth. That is, what I&#39;m showing you is the tip of the iceberg called modern mathematics. A piece of information for students. Right now I am working on creation of training courses at several sections of advanced mathematics. Including analytical geometry, calculus, probability theory, differential equations. Details can be found via a link in the description under this video. We begin learning process this particular year. Let me show you few examples. Let us take a sequence of positive numbers and make sure that it converges to a non-positive number. Consider a very simple example. Numbers like one divided by n, where n is a natural number. We write out the first few: one first, one second, one third, one fourth, one fifth. It’s clear what’s going on. The numerator is always one, and the denominator is constantly increasing. Let&#39;s assume that the sequence of these numbers tends to some positive number a. Then let&#39;s have a look at the reciprocal number - one divided by a. Of course, this number should be large, since the number a is small. Now, take this number and increase it ten more times. Fraction of ten divided by a. And then, if we say that our number n is greater than this fraction, then the reciprocal number, that is, a quotient of one by n is less than a tenths. But the number a of tenths is certainly much less than the number a. So there is no such positive number. And if there is no positive number, then our sequence converges to zero. And this number is non-positive. Now let&#39;s look at this series. One minus one plus one minus one plus one minus one plus and so on. That is, what happens? First a positive unit comes, then a negative one, and we sum them up. What could a sum be equal to? Let&#39;s count. The sum of the first two terms is just zero The sum of the next two terms is also zero. The sum of the next two terms is zero again. But the sum of all zeros is of course zero. That&#39;s right, but let&#39;s try to calculate differently. Let us write out again a set of our terms with plus and minus signs and see what happens. Let&#39;s not touch the first one, but take the next two. Minus one and plus one is of course zero. Again minus one and plus one, this gives zero also Further, minus one and plus one, it is zero again. And it turns out that a lot of zeros are added to the first number one Of course we get one. But it turns out that the results are different. Have we calculated correctly? Or maybe we are grouping the terms incorrectly? Let us then simply denote the sum of the series by some number X. And find what this X is equal to. Let&#39;s not touch the first term, leave it unchanged for now, and then say the following - we factor out minus one. Then one minus one plus one minus one will remain in brackets, that is, all terms change their signs to the opposite ones But then in parentheses we get the number X again. We denoted the sum of the series by X. Then we get that this sum equals to one minus X. But, before we put the brackets it was just X. So we get from here the equality X=1-X. And transferring X to the left side, we get that two X are equal to one. And in turn, X is equal to one second. Could you see how interesting it is? We were able to get both zero and one, and even a non integer, that is a fraction one second. Now consider an even more interesting example, when the sum of positive numbers can turn out to be a negative number. Let&#39;s look at this series: one plus two plus four plus eight plus sixteen and so on. What is it? Each subsequent term is two times larger than the previous one. Well, but we do not know what the sum of the series is equal to. Let&#39;s denote it by X. Now we will do the following. We leave the first term unchanged, and then note that all the other terms are even. So the two can be factorized In brackets there will be one plus two plus four plus eight plus and so on. But what do we have in parentheses? There is indeed X. That is exactly the sum of the original series. That is, we get that it equals to one plus two X. But if now we multiply out the brackets, then we get an X on the left. That is, we get the equation: X is equal to one plus two X. And from here we get that X minus two X equals one. That is, minus X is equal to one. So X equals to minus one. Indeed, the sum of positive numbers turned out to be a negative number. So where is advanced mathematics applied nowadays? In fact, wherever you look at. Space. Should we create a new channel &quot;Artist of Moscow State University&quot;? Medicine. Construction. Programming. Our century is an era of information, and to get this information quickly, and most importantly to find what the user needs, you should have good and fast search algorithms. And if we add here operating systems, virtual services, social networks, NFC chips, and so on, then we discover that mathematics is a foundation of all of this. The purpose of this video is not to scare you with advanced mathematics, especially with pictures like these ones, but to show you that advanced mathematics is used literally at every step you make and even if you do not know it. For example, bringing your mobile phone to the payment terminal in a store. At the moment, encryption algorithms - and this is discrete mathematics - do the work for you, and you calmly pick up your purchase. Or another example. No cashier knows what the Lebesgue integral is, but they all know perfectly how to use it on practice when they count bank notes. Surely, not everyone needs to know advanced mathematics, but for many of you it will come in handy in future work. Soon there will be useful videos on advanced mathematics, and by the way in comments you can write what kind of video you would like to watch on my channel. Subscribe to my Instagram channel, there will be unique content that is absent on YouTube. About life, about me and a little different mathematics. Those who watched the video up to this point, write a word “integral” in comments and what do you think of this video, I will find out who watches my videos till the end. Think with your own brain and see you on the channel!

Transcript for:Understanding Mathematics

Transcript for:
Understanding Mathematics