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30Sep2011, 07:56 PM
(This post was last modified: 30Sep2011, 08:07 PM by Lije.)
(30Sep2011, 07:34 PM)praty Wrote: Two negatives cancel out each other guys. It's as simple as that!
Interestingly thats not always so. http://opinionator.blogs.nytimes.com/201...myenemy/ (link given by karatalaamalaka) cites examples where it isn't so. In Maths its defined that way to make addition/subtraction/multiplication rules simpler and more consistent as correctly mentioned in this link too..
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30Oct2011, 04:38 PM
(This post was last modified: 30Oct2011, 04:40 PM by Alan D'Souza.)
(08Aug2011, 09:27 PM)karatalaamalaka Wrote: Folks, I would like to direct you to this wonderful article by Steven Strogatz in NY Times. I also urge you to peruse the entire series of articles on mathematics by Strogatz.
http://opinionator.blogs.nytimes.com/201...myenemy/
He puts it awesomely. Consider multiplying 1 with a decreasing sequence of numbers. Logically, the product should also have a particular sequence, which, empirically turns out to be an increasing order. This is an inductive approach to understanding multiplication of negative integers.
1 X 3 = 3
1 X 2 = 2 = 3 + 1
1 X 1 = 1 = 2 +1
1 X 0 = 0 = 1 + 1
1 X 1 = ? = 0+1 = 1
@Alan Once the mathematics of integers is understood, it can be generalized to the mathematics of real numbers. A major breakthrough, historically, was made by Georg Cantor who described two orders of infinities one applying to rational numbers and another to irrational numbers. Before we jump from integers to pi, we have to first describe rational numbers. Rational numbers can simply be written in a direct (onetoone) correspondence with integers. This was first articulated by Georg Cantor. Now, rational numbers would suffice to understand most of the world. But you routinely encounter numbers which cannot be expressed as rational numbers, starting with sqrt(2).
Let us suppose, with p/q being the SIMPLEST form of the fraction (e.g. 2/4 is not the simplest, 1/2 is),
sqrt(2) = p/q
2 = p^2/q^2
2q^2 = p^2
Since only an even number can square to an even number, this means that p must be even, so we can write p = 2n
2q^2 = 4n^2
q^2 = 2n^2
This means that q is also even. If p and q are even, then p/q = 2n/2m = n/m, which contradicts our claim that p/q is the simplest form. Therefore, sqrt(2) cannot be written as p/q.
These numbers fill up the spaces on the number line that are not occupied by rational numbers or integers. And they are infinitely more numerous too! And cannot be counted, because counting requires you to put them on your fingers, or map them on to the integers, which is impossible to do. Hence, such infinities are called "uncountable" infinities.
How do you define pi*pi? You really cannot multiply it in the traditional sense of adding a finite number of objects. Hence, one needs to resort to tricks and use various properties of pi, like its approximation. Multiplying irrational numbers like pi and 1/pi or sqrt(2)*sqrt(2) are done quite simply.
Breaking down multiplication as adding numbers is only a *computational* tool. So, computationally, one cannot achieve pi*pi to full precision, just as you cannot achieve pi to full precision. "Full precision" doesn't make sense. However due to interesting properties you can achieve pi*(1/pi) and (sqrt(2)*sqrt(2)) to full precision.
I am replying to the post highlighted in bold.Cantor Ineffect discovered not just two orders of infinities but infinitely many. Taking the power set of any infinite set and repeating the process led to a countably infinite number of infinities.
I was simply displaying that the logic of considering multiplication as a sequence of repeated addition fails for simple numbers like fractions. While the idea is good for children, such attempts should be abandoned after a certain stage. Because in higher mathematics, trying to have a picture as to how it is applied is often detrimental to understanding the mathematics as it is meant to be. The generalization of abstract objects. This is especially true of pure math.
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My recent reading on real numbers introduce me to the following real number axioms (x, y, z are real numbers in all the axioms)
1. + and * is an operation such that (x + y) and (x*y) is exactly one real number.
2. x + y = y + x, x*y = y*x : commutative law
3. (x + y) + z = x + (y + z), x*(y*z) = (x*y)*z : associative law
4. x*(y+z) = x*y + x*z : distributive law
5. There exists a real number 0 such that x + 0 = x. There exists a number 1 such that x*1 = x: called as existence of identities
6. For any real x exists an additive inverse y such that x + y = 0. For any real x exists a multiplicative inverse y such that x*y = 1
From these seemingly innocent axioms it can be proved that (x) * (y) = x*y
(where x is additive inverse of x)
The proof can be very long, the intermediate steps are as follows
i. From the axioms it can be proved that if x+y = x+z, then y=z.
ii. From (i) and axioms it can be proved that x*0 = 0
iii. From (i) & (ii) it can be proved that (x)*y = (x*y) and that (x) = x.
iii. Using all the above now it can be proved (x) * (y) = x * y
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studying analysis?
The following 1 user Likes Alan D'Souza's post:1 user Likes Alan D'Souza's post
karatalaamalaka
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(02Dec2011, 12:28 AM)Alan DSouza Wrote: studying analysis?
Calculus, actually. (But this is hobby reading. I am too old to study anything
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I see, best of luck with studies.
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I have read discussions going on here until now since 10052010. A few could not digest the approach presented by the initiator of this topic. However a detailed analysis of their new approaches shows that they are not fundamentally differing at all. The concept of negative can be best understood by the analogy such as opposite direction in space, movement in reverse direction, going below a reference (positive is going above), etc. However, if you do not want to deviate much from generic/abstract approach while explaining the concept simple, the initiators approach was the best. The only challenge to the concept was the question raised for multiplication of ‘PI’. An answer was given later, but not with clarity (emphasized unnecessarily on approximation).
The existence irrational number does not challenge the validity of concept, as it appears. It will be clear, if we correlate numbers with physical quantities. Consider two different units for length – inch, meter. While USA measures the length in terms of inch, they are comparing the length w.r.t a standard length which we defined as 1 inch. Similarly, while Europeans measure the length in terms of meter, they are comparing the length w.r.t a standard length which we defined as 1 meter. Since 1meter is not an integral multiple of 1 inch, the length measured by US person and expressed as a perfect integer inches will be measured and expressed as fractional number with uncountable number of digits by a European. Both are valid. The fundamental question is what the basis for measurement is. The perfect digit in decimal system will turn to a recurring fraction in binary system.
Assume a two dimensional Cartesian coordinate system. If you assume, for example, each division corresponds to PI or PI/2, or PI/3… When you are multiplying 1 x 1, you are actually multiplying PI x PI, or PI/2 x PI/2 or PI/3 x PI/3. The concept of explaining the origin of + number from multiplication of two negative numbers is valid what ever may be the value of each division in the coordinate system. Let it be 1 meter or PI inch or e mile, or SQRT(2) feet, or anything.
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18Feb2012, 11:57 AM
(This post was last modified: 18Feb2012, 11:57 AM by Kanad Kanhere.)
(15Feb2012, 11:35 PM)mihraj Wrote: I have read discussions going on here until now since 10052010. A few could not digest the approach presented by the initiator of this topic. However a detailed analysis of their new approaches shows that they are not fundamentally differing at all. The concept of negative can be best understood by the analogy such as opposite direction in space, movement in reverse direction, going below a reference (positive is going above), etc. However, if you do not want to deviate much from generic/abstract approach while explaining the concept simple, the initiators approach was the best. The only challenge to the concept was the question raised for multiplication of ‘PI’. An answer was given later, but not with clarity (emphasized unnecessarily on approximation).
I don't think the main objection was with the definition of multiplication in terms of addition. It was pointed out that the definition can be easily refined for fractional numbers. The main objection was that the main comment just describes a convention in which it so happens that ve times ve is +ve. That is no mathematical explanation. I can as well create a convention in a system (e.g. sarcasm in language) that will result in ve times ve as +ve.
(15Feb2012, 11:35 PM)mihraj Wrote: The existence irrational number does not challenge the validity of concept, as it appears. It will be clear, if we correlate numbers with physical quantities. Consider two different units for length – inch, meter. While USA measures the length in terms of inch, they are comparing the length w.r.t a standard length which we defined as 1 inch. Similarly, while Europeans measure the length in terms of meter, they are comparing the length w.r.t a standard length which we defined as 1 meter. Since 1meter is not an integral multiple of 1 inch, the length measured by US person and expressed as a perfect integer inches will be measured and expressed as fractional number with uncountable number of digits by a European. Both are valid. The fundamental question is what the basis for measurement is. The perfect digit in decimal system will turn to a recurring fraction in binary system.
Assume a two dimensional Cartesian coordinate system. If you assume, for example, each division corresponds to PI or PI/2, or PI/3… When you are multiplying 1 x 1, you are actually multiplying PI x PI, or PI/2 x PI/2 or PI/3 x PI/3. The concept of explaining the origin of + number from multiplication of two negative numbers is valid what ever may be the value of each division in the coordinate system. Let it be 1 meter or PI inch or e mile, or SQRT(2) feet, or anything.
The approach of scaling or usage of units doesn't again explain anything in terms of Number Theory. For e.g. suppose I want to calculate PI*2. What is suggested that scale the first number by PI, so basically it will result in 1*2, which is 1 + 1 = 2 and then again scale back to get the actual result that is 2*PI. This is useless result as it is just coming back to the basic question.
Multiplication for fractional numbers is NOT defined just by addition, but by the commutative and associative law and division. This can be extended for a series and thus we can calculate the multiplication of a series such as (a + b/10 + c/100 ....) * (x + y/10 + z/100 ...).
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It is true that Mathematics is the most abstract form of knowledge. But if trace the history of mathematics, whatever available to us, we can understand that the knowledge originated as need of human being from among human being and has some direct correlation with the physical reality in and around us. It is not knowledge came from heaven or one that can be understood only in abstract terms devoid of all physical reality. It is true that after some stage of development, the mathematical concepts originated as a practical need of man, will develop to further stages in an abstract world. However, when we teach our children basics of mathematics; it is strongly advisable to teach them the concept linking very much with the daytoday life around. Many children are loosing interest in the subject.... something difficult to crack for many children. It is getting treated as an area of privileged students (smart in mathematics by birth... may because of some specific combination of active regions in left and right brain). Even for adults, it is compulsory to understand the practical/historical background in which a specific mathematical concept originated. I will cite an example. The concept of vector which has now developed into various directions such as vector calculus,... But how this concept originated? Why they defined a peculiar rule of addition for vectors [R = SQRT{A^2 + B^2 + 2*A*B*Cos(AB angle)}]. Not because some mathematician meditated and felt so. It came from physical reality as a need to quantify some observations in physics  the real world. Not all physical quantities which have magnitude and direction are vectors. Many times it is debatable a physical quantity is having direction or not. (One of my professors from Delhi University had a pet question. After introducing a physical quantity, he used give us a home work to find out, it is a vector or scalar). Only the physical quantities which obey the law of vector addition are considered as vectors. It may be surprising to many that wavelength is a vector, which seems to be scalar. In the physical world wavelength obeys vector addition law. So vector concept originated as a tool to understand this strange physical world, not that sacrosanct mathematical concepts are imposed on physical world. If some mathematical concepts or equation or analysis goes against the physical observations, mathematics to be refuted not the physical world. (It is true that some times some clues comes from mathematical world for physicist. That is again shows correlation of physical world and mathematics, even though in abstract level.
You may wonder why I stress for the physical world interpretation for math. Simple, if you think mathematics is sacrosanct like Quran/Bible, you may deny even physical world sometimes, in favour of abstract concepts in mathematics. Two year back, I had a shocking experience while observing the electromagnetic noise emitting from a Piezo injector of an automobile. The frequency spectrum captured by high end spectrum analyzer provided an unbelievable realization of limitations of highly celebrated Fourier series in mathematics, of course in some specific physical phenomenon It lead to debates. Consulted many experts in world. It took many months for me to accept the reality I saw with my eyes because I blindly believed Fourier series in very abstract sense.
Now coming specific to the discussion over here. I appreciate the attempt of initiator of this discussion  to understand the concepts in basic mathematics (number theory) in simple terms. The realization that negative of negative is originated from the practical realization of early human being. You can make similar observation in language too. You can have more examples and interpretations, let it be from language, or logic, or physics..... It only strengthens the initiators ideas, not denies. There is nothing sacrosanct about mathematical explanation. Remember, there is no mathematical proof for the most fundamental concepts in mathematics – axioms. It is just believed to be true, as commonsense. Now, from where this commonsense came – from observations and generalizations of the physical world. When you have different more advanced observations, you will have different generalization and commonsense. So you have to go back and challenge the earlier concepts. This is what happened when modern math differ with Euclidean math. Let us welcome all attempts to understand the mathematics in layman’s words. However, concrete reality and abstract concepts are different, though the later originated from the first. So some gaps and gray areas exist, obviously. Because both are not one and the same. The major gap, I could understand here is how to interpret the irrational numbers. No issues with fractions, recurring decimals, etc.
No need of commutative or associative law or any such stuff to understand fractional multiplication. Assume the multiplication of any number. First express it in p/q form. If you need to multiply p/q x r/s, multiply p by r (i.e r times p) then multiply q by s (i.e q times s). Then check how many times qs is present in pr. i.e if pr is cut into equal qs pieces, what is the size of each piece? Example, 0.25 x 5.2 means ¼ x 26/5 means 26/20. If we equally divide 26 Kg sugar for 20 people, each one gets 1Kg and 300g sugar.
However, as mentioned earlier, irrational number (which cannot be expressed in p/q form) seems to be a major challenge to generalize the debate initiator idea for number system. The best way as known to me is explaining with scaling approach. (Irrational quantity in one number system becomes rational in another number system. Irrational number in one scale becomes rational in another scaling). A way to link the concept to physical reality… If anyone knows any better approach, please share. Also complex numbers seems to be another issue.
The formula for calculating the area of circle (PI x r^2) is derived from basic integral calculus. But this formula was known to earlier civilizations too. They arrived at it with an approach of assuming the circle is divided into small pieces forming the shape of a rectangle. The irrational number ‘e’ is a converging infinite series. ……
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18Feb2012, 10:46 PM
(This post was last modified: 18Feb2012, 10:48 PM by karatalaamalaka.)
(15Feb2012, 11:35 PM)mihraj Wrote: Assume a two dimensional Cartesian coordinate system. If you assume, for example, each division corresponds to PI or PI/2, or PI/3… When you are multiplying 1 x 1, you are actually multiplying PI x PI, or PI/2 x PI/2 or PI/3 x PI/3. The concept of explaining the origin of + number from multiplication of two negative numbers is valid what ever may be the value of each division in the coordinate system. Let it be 1 meter or PI inch or e mile, or SQRT(2) feet, or anything.
The definition of irrationality has nothing to do with scaling. The point is this, one way of classifying real numbers is to identify them as rational and irrational. Some, countably infinite numbers are rational, almost all, i.e., an uncountably infinite many, are irrational.
Nothing stops you from starting your own number system where 'pi' is rational. But the only thing you are doing then is redefining 'rational'. That is essentially what the 'degreeminutesecond' unit for measuring angle does. But to define a whole number system like that is unreasonable and extremely cumbersome, simply because a system in which you scale all numbers by '1/pi' (so that 'pi' is rational), has useful integers being irrational. One consequence then will be that you'd have to go to a shop and ask the clerk to give you Rs. 25/pi = Rs. 7.95774715.... worth of candy. Remember, 'pi' has an intrinsic property it is the ratio of the circumference of a circle to its diameter. Similarly, sqrt(2) has the property that sqrt(2)^2 = 2. Most representable irrational numbers have intrinsic utility.
No matter what rescaling and redefining you do, you cannot escape the fact that there are two types of numbers rational and irrational.
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The example given is not a good one. I never meant so. In most practical transactions, we do not use irrational number, even fractions rarely. I do not disagree with the fact that irrational numbers, in fact infinite number of them, exists. But I am just trying to make the explanantion applicable to irrational numbers too for the idea that negative x negative = positive. I do not favour switching of scales too, but believe in standardisation.
Between any two number points in a cartesian coordinate infinite irrational numbers exists, also infinite rational numbers. (A function defines as Y = 1 if x is rational and 1 if x is irrational will be two strait lines in cartesian coordinate with discontinuity in each points in both strait lines) So it will not matter, if we assume any rational number as basis or irrational number as basis of scaling and start life. If we mix up the scales all issues will pop up. More serious than grocery shop issues. (We are unable to standardise many engineering tool, and control precision issues in the world of six sigma because US and European world are following different scales). Take one quantity as scale and move along, let it be irrational number in another scale. World will be comfortable to live.
My point is that existence of irrational number is a specific characterisitcs of number theory (mathematical abstraction). It no way affect practical life of man. Its existence cannot challenge the basic understanding that negative x negative is positive.
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18Feb2012, 11:13 PM
(This post was last modified: 18Feb2012, 11:13 PM by karatalaamalaka.)
(18Feb2012, 10:45 PM)mihraj Wrote: It is true that Mathematics is the most abstract form of knowledge... It is not knowledge came from heaven or one that can be understood only in abstract terms devoid of all physical reality. It is true that after some stage of development, the mathematical concepts originated as a practical need of man, will develop to further stages in an abstract world.
There are both mathematics of utility and totally abstract mathematics. A simple concept like 'a sphere (or any geometrical object) in ndimensions' can be argued as being unrelated to the real world. Now, you can argue that ndimensional geometric objects do have uses in modeling real data and systems, but when mathematicians studied it first, there really weren't any practical applications. Or take the example of number theory. In history, whether during Fermat's, Gauss', or GH Hardy's, days there really was no 'practical' use for most of the number theory they were doing. As much was acknowledged by the mathematicians themselves. Fast forward to 1980s and they form the basis of modern ecommerce. These are just two of several such examples in the development of mathematics. If we set these aside, and go deeper into the epistemology, we encounter philosophical problems such as whether mathematics is Platonic or utilitarian, etc.
Quote:However, when we teach our children basics of mathematics; it is strongly advisable to teach them the concept linking very much with the daytoday life around.
Agreed. But we are not all kids, in physical or mathematical terms, here. But into our adulthood, as we learn more about the sciences and mathematics, foundational questions such as the opening post invariably arise.
Quote:The frequency spectrum captured by high end spectrum analyzer provided an unbelievable realization of limitations of highly celebrated Fourier series in mathematics, of course in some specific physical phenomenon It lead to debates. Consulted many experts in world. It took many months for me to accept the reality I saw with my eyes because I blindly believed Fourier series in very abstract sense.
What do you mean by limitations of Fourier series? The existence of frequency is a very real thing. Fourier transformations provide us with a mathematical tool to understand this. If you are a mathematician or mechanical engineer, you probably first encounter Fourier series in solving partial differential equations. In such a case, you can start to believe that it is a very abstract concept. However, folks in optics and some fields of electrical engineering first encounter it as a practical tool to understanding frequency content. For example, the Fourier representation of white light will be significantly different from light that is seen through red glass because of differing frequency content (spectrum).
Quote: Now coming specific to the discussion over here. I appreciate the attempt of initiator of this discussion  to understand the concepts in basic mathematics (number theory) in simple terms. The realization that negative of negative is originated from the practical realization of early human being. You can make similar observation in language too. You can have more examples and interpretations, let it be from language, or logic, or physics..... It only strengthens the initiators ideas, not denies. There is nothing sacrosanct about mathematical explanation. Remember, there is no mathematical proof for the most fundamental concepts in mathematics – axioms.
You are raising important questions here. But intuitive reasoning for negative numbers is not necessarily always consistent. For example, as Strogatz points out, two wrongs don't make a right. As to the commonsense of axioms, the most famous failure of commonsensical axioms is Euclid's assertion that given two points in space, there can only be one line that passes through both the points. Till the 1800s, commonsense was used to accept this assertion as an axiom. However, it is only true on a plane. On a sphere for example, there are infinitely many 'lines' passing through two points. This, obviously to someone between 300 BC1800 AD, the falsity of the assertion had no practical consequence. However, since the 1800s, geodesics have had plenty of uses. Simplest being the planning of airline routes (sphere) and more complicated uses include Einstein's general theory of relativity and the nature of space and time (Reimannian surface).
