 Hi, I'm Zor. Welcome to a new Zor education. I would like to expand discussion about real numbers into something which is called approximation, approximation of real numbers. Now this lecture is part of the course Math for Teens. It's presented on website unizor.com Now it also has a physics for teens as a continuation course now, but today we are talking about math and about real numbers and approximations. I suggest you to watch this lecture from the website. You go to unizor.com and then choose Math for Teens course Algebra, real numbers, and that's where you will find this lecture because there is a link to the video part of this and also there is a textual explanation of everything whatever I'm talking about Maybe with some pictures in some cases In any case, it's just a dual representation video and textual. Textual part is basically like a textbook, which is completely Paired with the video presentation. Plus the website contains the course, which means there is a menu, there is a sequence of Chapters or topics, etc. It's a logical sequence I'm always using something which I have already addressed before into subsequent in the subsequent Lectures, so it's very important to take it as a course Also, there are exams for those who would like to challenge themselves and Everything is totally free. There are no advertisement No strings financial or any other strings are attached So just pure knowledge used Now let's talk about Approximation of the real numbers so Real numbers obviously are quantitative representations of everything whatever we have right now We have integer numbers when we are counting one two three, etc We have rational numbers, which is basically like a ratio between two integer numbers We have irrational numbers like square root of two or pi or something now we would like to do something with these numbers and It's not always possible in their exact value for example, what's the distance between? The earth and the moon well, it's changing obviously, so there is no such thing as a exact number Well, we can say, okay What's the distance between the closest part of the surface to a closest part of the moon on certain time? And even that is not really exact How many people live on the planet earth? Well, we don't know I mean approximation is the way to address those numbers, which we Don't know or even if we do know it's kind of inconvenient to do anything with them in their exact value for example, we would like to For instance evaluate what's the density of the population Of United States of America. Well, we have to have the whole number of people and divide by the square Kilometers or whatever Measurement we're using but none of them is exact So approximation is the way to approach certain problems Which we cannot really address with their exact values, but even within the pure mathematics And we would like to do something with certain numbers which have Which do not have a finite Representation for example pi pi is not 3.14 pi is actually irrational number with infinite Number of digits after the decimal point so we cannot really do anything with exact number pi Well, obviously we can do some formula where pi is really written as a Greek letter But that's not the quantitative approach quantitatively. We have to really put some kind of a Boundaries we have to approach this not as an exact number But as as I was saying Approximation so approximation is a replacement of exact number with approximate number Well, approximation actually has two sides First if we do know exact number How to approximate it and there are certain rules which we will talk about The second part is what if we don't approximate number and you would like to do some calculations as if it is Exact number question is by how much we are mistaken What kind of an error is introduced if we're using approximation? Instead of the exact number so that will be the second part of this lecture Okay, so the first part is what if we know exact number? How to approximate it, okay Now we approximate pi with 3.14 Right In reality, it's infinite number of digits. We approximate number of population Po Population Is approximately seven billion People on earth This is approximation and this is approximation none of these are exact Now what's the difference? I mean how can we define what exactly the rules of approximation? Well in this particular case, we know that approximation has The precision of one hundredth, right? In this case precision is seven billion billion is Nine zero so the precision is So if we are talking about an integer number of billions It means its approximation is with this particular precision If you would like to make a slightly more precise for instance seven point five billion then the precision of this would be One zero less So this is nine zero this is Eight zero it's up to one hundred million basically because what is seven point five? uh Billion it's seven point five hundred and five hundred So that's what it is, right? Or more no one one more one more two right Thousands million so it's five hundred right So this is basically the precision of this number. This is precision of this number. So every approximate number has certain precision Up to which it defines or represents actually the exact number Okay, so the first rule of approximation is First choose precision We have chosen this one this one for this one for whatever purpose we have for whatever need We would like actually to use this approximation And after we have chosen the precision we choose the base values our Number which we would like to represent approximately should actually take right so it can be zero Now let's say precision is delta. It can be delta. It can be two delta It can be three delta It can be any n delta because for instance We are talking about seven billion but it can be six billion or seven billion or two billion If we are talking about billions It means we have to count in billions in this case billions is delta If we are talking about five zero point zero one one hundreds That means in this particular case The value which we approximate with a precision zero point zero one one hundreds What kind of values can be zero one hundreds two hundreds three hundreds five hundreds Well, actually minus delta as well and minus two delta as well and minus three delta as well So negative numbers are also on the same Set of base numbers. So we have a precision and we have a set of base numbers As the beginning of our approximation process fine so in this particular case Our approximate numbers our base is zero billion two billion three billion Minus billion minus two billion minus three billion. Okay, these are the base numbers The values of which can represent the exact number Okay Now we do have an exact number. For example, we do have we do have an exact number. We would like to represent it approximated Using this precision Well, let's do it on a Y, okay, this is zero This is delta this is two delta This is three delta This is minus delta minus two So if we have a number it falls on the real line somewhere for instance here So which number should represent? Which approximate number in this case? It's two delta or three delta Or in this case it's seven billion or eight billion or six billion Which number should represent the exact number if we know the exact number? So the answer is obvious the one which is closest. That's the most reasonable solution, right? So first we we define the Base numbers using the precision and all the multiples of this precision and second we choose Closest base number To exact number which we basically have So we are talking about from exact number to approximate number Now sometimes we don't have exact number, but we have more precise I would say approximate number for instance We do know that there is actually 7.5 for 7.5 4 7 etc. We have some numbers not exact numbers but You know plus or minus one hundred let's say Well, we can still approximate it by cutting whatever number of digits to the one which which we would like actually to have If it's something like this Whatever we cut it to a closest closest is Eight to this right so we always go from exact number or Relatively precise or almost exact number To the approximation precision which we actually would like to use now Same thing with here. It's actually 3.1415 blah blah blah. Don't remember Well, if we have chosen precision 0.01 100 It means that this number is between 3.14 and 3.15 Right this number falls somewhere between and it's closer to this one because it's one So approximation is this if we would like to approximate let's say with two Thousands let's say one one thousands we have to really cut only three And decide is it closer to three point one for one or three point one for two And actually it seems to be closer to one for two So that would be approximation to one hundredths precision base base would be Base numbers should be separated by one thousands all right, so Great. I mean it looks very Rational very logical. We choose precision. We choose this integers Multiples of this precision as a base number and choose the base number which is closest very easy very logical Well, there is one little detail. What if it falls in the middle? It happens right in the middle for example Let's say our exact number is 3.575 And I would like to Approximated to 0.01 So it's between 3.57 and 3.58 and right in the middle, right So which one should I choose? Well, there is no kind of logical solution in this particular case We just have to say okay do it this way and that's the rule. That's it So the rule which is most likely Used in cases in most frequently I should say used for approximation in this case when exact value Is right in the middle between two base values is go to the value which has a Larger absolute value away from zero It's either this way on the positive or this way on the negative So for instance If okay, let's say delta is 0.001, okay and our number X is equal minus 5.757314 I would like to approximate it with a precision of 1000s. Well 1000 is this, right? So it's between minus 5.758 X minus 5757 It's negative. That's why this one is to the left and this one is to the right Zero is somewhere there, right? So in this case In this case, there is no problem to approximate because it's definitely closer to 757 You see this is three So it's closer to this one But if it's not three if it's five It goes right in the middle between these two And in this case as I was saying before you go to the one with absolute value greater So which is this one further from zero to the left or to the right And obviously if after five you have some other numbers Like one Three seven, etc. It will be even closer to the five. So everything which starts with five In the next digit after our cut off So our precision is 1000. So the cutoff is after the third After the third digit after the decimal point So if it's five or four or more Then you go to Upper to the greater absolute value If it's four or any four and anything after that as long as it's less than five You go to the smaller one. So that's the rule now Unfortunately, this is not the only like all encompassing rule In certain cases people are using other rules. For example if I remember about one particular rule if these are Two numbers now if this is for instance odd number of deltas and this is even number of deltas go to even but if it's something between Odd and even if even is greater than you go to even number So these number which is in the middle will go to This one and this one will go to this one This one to the left. This one to the right. It's just one of the rules And it's rarely used. I mean my my personal opinion is always use the rule go To the greater absolute value away from zero As a rule unless it's specifically Said that you have to use another rule for approximation Which is rare condition. So I think you should remember one rule Which is go away from zero to the greater absolute value if it falls right in the middle between two integers of delta If it's like let's say three thousands and four thousands You go to four thousands if it's right in the middle That's it about how to get from exact number to approximate number These are the rules again first choose the precision The precision in this case is a billion in this case is one hundredths In this case, it's a hundred million. I mean whatever the precision is Choose the precision which defines this set of countable obviously set of base values It's zero precision double precision triple precision etc to infinity and then You just think about where exactly your number falls between which two base numbers So it's always well it can be either exactly on the base numbers, which means That's basically the approximation is equal to equal values But in most cases it's in between and you choose the one which is closest With this little correction about if it falls in the middle So that's it now. Let's talk about the reverse In theory reverse is more important because all numbers which we are dealing with Well, except number of people in this room I I mean that that's exact number But in most cases we are dealing with numbers like we are measuring something Okay Now whenever we are measuring we are using some kind of a ruler or whatever else And the result is approximation It's not exact None of these real life numbers except some you know integer numbers Okay, let's say most of them most of them are not precise. They are always Approximations of some real exact number, which we really don't know So if the length of this is let's say, I know 18 centimeters for example, okay, is it really 18 centimeters? Of course not First of all, it's different between this point and this point and this point because there is some kind of slope here So which which one is the length? Okay, we can say the maximum length But even that this is kind of a soft thing And whenever it's soft thing God only knows where it ends I mean, it's always approximation So how to deal with approximate value but still have a good feelings About what's the result of calculations on approximate value? and unfortunately We do not have A good feeling about this Let me give an example For example, we are approximating p to 3.14 Well, there is certain error obviously. I mean, we know this is not 314. It's 314 Etc etc etc. So pi is actually between 3.14 and 3.15 Closer to 3.14. Yes, we know that But in many cases, we don't even know I mean, if it's not like a pi, we do know from other sources what exactly okay Let's say this is not a pi. Let's say this is the length of certain piece of wood and we have calculated in meters and centimeters Because we have a ruler which has meters and centimeters. We don't have anything else So obviously the result of this is Number of meters and centimeters Let's use centimeters or better centimeters 314 centimeters Does it mean that this is exact number? No, but we do know that this real length is between this and this centimeters Okay Great. Now, let's say we have to have two pieces of wood one and another and one of them has this length and another has approximately let's say six point 13 centimeters What does this mean? Well, it's in between 6.13 This is L1 and this is L2 and 6.14 right So if we measure this It means it's somewhere in between these We measure this it's between these Now we would like to add them together and have a combined length What can we say? What actually should we say? What should we say about this? Well, obviously we can calculate it and Well, I meant Not six point just six hundred. So that's just easier So we add them up with 929 And here we have 900 and 27 Now let's talk about precision now, we know that if this is Approximation with a precision delta it means that this difference between base Various is exactly delta right so we can say that our actual length has a precision of one centimeter here Because the difference between these two base values is one centimeters. So it's here How about here? The difference between this and this is two centimeters So you see my error Which was within one centimeter here and one centimeter here whenever i'm adding together my real arrow is summing up So the arrow is increasing so any calculation you do would increase the arrow so our exact number which is Which is sum of two exact numbers Compared with approximate number which is sum of two approximate numbers Now approximation of l one plus l two to exact sum of real length is worth any operation on approximate numbers decreases The quality of approximation If the quality of this approximation is one centimeter and this as well the quality of Approximation of their sum is two centimeters in this case Or whatever these maybe they have different Delta's maybe this is in millimeters and this is in centimeters, but whatever it is Errors are always accumulated In any operation on approximate number errors are accumulated here is in more mathematical Representation let's say we have number x And it's approximately a we have number y which is approximately b What does this mean? It means that x is Between a minus delta x Whatever the delta whatever the precision which we are measuring and a plus delta x and For y we can say exactly the same plus delta y And let's say I would like to Divide x by y Now what is the range of x by y? well If I know that x is between this and y is between this what is the smallest value x by y Should have When x is the smallest and y is the largest right and here we will have The biggest x and the smallest y Which is this And again if you will have the difference between these It will be bigger And if you will compare this minus this with either delta x or delta y it will be bigger than both of them That's basically the point which I would like to convey whenever you do some approximations It's the error is growing And as an example, let me just give you a very simple practical example Okay The sea level sea level in 1880 was zero That's just we decided this is zero now By 1994 This level was 160.192 um millimeters so it's um 16 centimeters above now from By 2019 It was 240.775 Well one would say okay the level the sea level is growing fine By how much well we have measured all the intermediary Growth every year here, okay At certain dates probably I don't know at certain time at certain date at certain place whatever it is So they have measured so we have all the numbers between these Now how can we predict what will be in the future? Okay, here is an algorithm which is which sounds really good Let's take the ratio between One year in the previous year it will be if it's growing it will be one point something right If it goes down it will be 0.9 something Now in most cases it it's really greater than one because there is a growth So most of these these will become something like 1.0001 1.003 etc a little bit fact a little factor a little greater than one which increases The absolute value these factors We are assuming this is an assumption That there is some something like a objectively exact value of this factor The factor of growth the factor of Sea level growing every year It can be less or more than this exact number, but we can actually consider This exact number as the real factor and starting from this we can apply it for the subsequent years So if we will take one of them, which is something like you know 1.003 If we found that this is exact number Based on these actual which we have received and then we will apply this factor To all the subsequent years we will get whatever we will have the sea level in 100 years from now, right? How to find this number? Well one of the Logical way is okay. Let's just have the average of these Fine you take the average and The average was 0.16 So I took these numbers Every year I calculated so I have I have the level sea level every year So I divided every year to a previous year got this number got all these numbers and got the average well, if I will apply this average to this and Do it 100 times multiply this by this 100 times. I will get some number Right. This is the number which will represent the sea level in 100 years Sounds logical, right? But let's be a little bit more precise now I've obviously these numbers are different And I have calculated just the average Does average really represent the future growth? Well, no, it's an approximation of of the growth the real growth will be probably somewhere around this Number but who knows what it is exactly Maybe it's 1.016 or maybe it's 1.012 and the difference will be Relatively relatively big in 100 years if I will multiply it by one number It will be greater than another and maybe significantly greater how significantly okay For this purpose statisticians have something which is called Standard deviation. So if you have a certain number of Values which are more or less around this one, but they are around and then can be Distributed If this is the average value all the numbers can be here very close to this one or they can be Very far away from this still giving the same number as an average How can we measure? The quality of using this as the future calculations based for the future calculations Well, if it's a very large extent of spreading of numbers around the average The validity of this will be less right If they are very close to the uh to to to this average people these numbers are very close to this one Then the precision of our evaluation is better So how can we measure it? Well the standard deviation actually is Average of the of squares of deviations So if you have x i as this particular number new means Average they take this square They have sum from i is equal to 1 to n and this number of occurrences, right Now you divide it by n that's average Square deviation and then have a square root of it. That's the standard deviation Now why square because this would give you a positive Number no matter whether it's uh deviation to the left or to the right from the from the from the average Okay, fine. So this basically qualifies our Um values these values as being close or far from the average From this number Okay So Let's take the standard deviation in this particular case And standard deviations of mu i have 0.016 And sigma which is a standard deviation is equal to zero point zero twelve Well If If this is a standard deviation Well, you can say it's average deviation average square deviation something like this So if this is a standard deviation Um It kind of qualifies Um How valid to use this particular number for future calculations The greater this thing is The less precise this number is As an approximation of the real factor of growth And but and right now we are assuming that the factor of growth is actually constant throughout the hundred years Which is wrong as well, but anyway, that's an assumption Now the in reality The real value of This factor if this is Average and this is a standard deviation It's it's a range now I can definitely tell you that Uh, the real value should be between Something like mu minus three sigma and mu plus three sigma That's something which they say the probability like 95 percent The real value will be in between these this probability 95 Would be two sigma two sigma notion saying X whatever x is This my real factor mu plus two sigma That's probability 95 percent now this Uh bell shaped curve This is the frequency. This is a mu and this is the number Of points the area of this is number of points, which is closest to the mu So the wider the boundary we will have the more real numbers will be inside that particular segment I mean, we will talk about this in statistics, but I'm just very very briefly trying to introduce you now What is like in this particular case mu minus two sigma two sigma is 12 So two sigma is 12 so two sigma is 24 so Mu minus two sigma would be even less than one which means it will be Factor will be growing down The level of the sea level It's 95 probability. Okay. Let's do it even less than that Let's have even a more narrow Mu minus one sigma What's the probability of this? It's something like 68 percent or whatever something like this So let's choose only those numbers. So we basically ignore the numbers which are far away from the average I'm saying these are some kind of Abnormal Factors of growth normal factors of growth are within these boundaries Hopefully Which means it's between zero one point zero zero four like sixteen minus twelve And the real number is one point zero twenty eight so Yes, I can use this and multiply this a hundred times by this number But if I would like to really make some kind of a variation How good the number which I have received is I have to really check the boundaries Because I don't know exact number. This is approximation Exact number is somewhere between these with the probability of 68 also not a hundred Okay, let's just do the calculation. Let's multiply this 100 times by this and see the result And multiply by 100 times this and see the results if results are closer to each other Then we don't have to worry about it means our approximation is okay Well, I did it and the results were Estonishing let's put it this way because multiplying by 100 times Really is significant so Two forty times seven seven five times one point zero zero four To a power of one hundred That's multiplying hundred times Gives me approximately three hundred and fifty nine millimeters So in a hundred years if I'm using this factor My c will rise by Well, it's about one-third of a meter approximately. It's about like this Even less something like this No big deal quite frankly But if I will use my upper boundary One point zero twenty eight to a power of one hundred I will have Three eight one ten millimeters, which is three point eighty one meters Three point eighty one meters. That's significant That's More than double my height so What what what is my point? My point is that whenever you're dealing with approximate numbers To evaluate something Which involves calculations and these are calculations, especially calculations Which are Especially complex calculations and this is Multiplying by the same number one hundred times. That's complex in terms of quantity I mean obviously on the calculator. It's simple but From the from the position of accumulating error It's significant. So error is accumulating. That's my point And you see the difference So be very very careful when you are using approximate numbers Do some calculations And use the result of these approximate numbers as kind of a Variation of exact numbers if the same calculations were done with exact numbers Big difference maybe big difference and the more complex the calculations are The Verse is the quality of your approximation of the result. So again calculations of approximate value Result in Verse quality the more complex calculations are And that's the point which I would like to make today That's it Uh, I would I do suggest you to read The notes for this lecture They are basically contained the same material as as I'm just talking about There are some numerical examples more precise than whatever I was just talking about here Um, and I do suggest you to use unizord.com as an entrance point to the whole course Rather than you might find something like on youtube or somewhere else In the individual lecture like this one for instance Just don't forget. This is just the part of the course and I do suggest you take the course And there are some other courses like physics for jeans for instance on the same website All right. Thank you very much and good luck