6 ms·

Mathematics is a uniquely beautiful field to me. The commutative property has always struck me as special in its own way. 2 x 3 = 3 x 2 feels so obvious, but mu

Mathematics is a uniquely beautiful field to me. The commutative property has always struck me as special in its own way. 2 x 3 = 3 x 2 feels so obvious, but multiplication is really just addition, and 2 + 2 + 2 = 3 + 3 is far less intuitive, yet states the very same claim. Most fascinating to me is that many theories are effectively 1-way functions. Entire branches of mathematics have been developed to prove otherwise trivially stated claims. It is something to marvel at, if from a safe distance.

rat9988 7y ago
>2 x 3 = 3 x 2 feels so obvious You just got used to it because you learned it very young. I'm pretty sure it wasn't that obvious when you learned it.
- red75prime 7y ago
  If I remember correctly, in my school multiplication was explained in terms of rectangles made of unit squares. Commutativity was literally visible to the naked eye then.
  - quickthrower2 7y ago
    I also think you define integers as successors. Z, S(Z), S(S(Z)) etc. you can prove that a * b = b * a, given that you have defined the + function and * functions.
  - lonelappde 7y ago
    It's both intuitively obvious geometrically, and algebraically is conceptually tricky to grasp. Or, more precisely it's trivial to decide that arithmetic operations are commutative or not, but tricky to decide correctly for all operations a child knows.
- r_c_a_d 7y ago
  When my kids started school I was surprised to learn that at age 4 it is not obvious that the number of items on a whiteboard remains constant if all the items are moved to a different part of the board. So what seems obvious, probably is something we just don't remember learning.
ncmncm 7y ago
The notion that multiplication is ("just") repeated addition is the hardest misconception to overcome blocking progress in mathematics. Repeated addition is an algorithm that works when you have a non-negative integer argument. Imagining that the algorithm defines the operation limits conception.
- apalmer 7y ago
  I would like to understand what multiplication is in your (and i guess advanced mathematics)
  - sls 7y ago
    The comment to which you replied says that thinking about multiplication as "just" repeated addition is problematic, so let's look at that. Consider 3 x 2. If we take that approach, it seems ok - we understand it to mean "add together 3 2's" - 2 + 2 + 2, which gives the correct answer of 6. What about -3 * -2? What does it mean to add a negative number of times? What about pi * pi? What does it mean to add something pi times? What about the matrix M * the matrix N? etc. The parent's point is that this repeated addition thing is just an algorithm one can use to calculate a multiplication for some operands, specifically whole numbers, not a general definition of the operation of multiplication.
    - jacobolus 7y ago
      If we want to define π×π, the best we can do is some kind of algorithm for generating an approximation to π (e.g. as a continued fraction or as a positional decimal fraction) to any desired degree, and then an algorithm for multiplying such approximations. We can prove some bounds on the error introduced by our multiplication algorithm, and that gives us a way of approximating the product to any desired precision. For example if we want to know π^2 to about 3 decimal digits (~9.87), we can start by using an approximation of π to about 4 decimal places (~3.142) and then multiplying the two decimals. For rational numbers, multiplication algorithms are usually built on breaking a number down into constituent pieces, multiplying every pair of pieces from the two multiplicands, and then adding up all of the partial products. Matrix multiplication has the additional complication that the elementary terms involved (entries in different places in the matrix) cannot be added to each-other. But the basic procedure is still the same: break the two multiplicands down into basic units which we already know a multiplication table for, compute all of the partial products, then sum them up.
jacobolus 7y ago
More interesting, in more general contexts multiplication is not commutative. For example, if we apply the same pair of 3-dimensional rotations in opposite orders, we generally get different results. Scaling and planar rotation (in combination, a.k.a. “complex numbers”) are conveniently among the types of commutative multiplication. > multiplication is really just addition This is a misleading summary. That multiplication of integers per se can be re-expressed as addition (or if you like, as counting) depends on the basic parts involved being very simple and uniform. The basic multiplication table for integers is just × | -1 0 1 ––––––––––––– -1 | 1 0 -1 0 | 0 0 0 1 | -1 0 1 Since every other integer is just some sum of these basic parts, and multiplication distributes over addition, that covers it. To multiply two integers, first break each one down into some sum of a collection of –1, 0, and 1, then look up each partial product in the basic multiplication table above, and finally sum up all of the results. This process boils down to counting. Since the basic integer multiplication table is commutative, so is integer multiplication in general. But other kinds of numbers have richer structure based on a richer multiplication table of basic elements.
why-el 7y ago
Well, I am not sure if you've come across some of Bertrand Russell's work, but he qualified Mathematics as having what he called "supreme beauty". The older I get, and I am not a mathematician by any stretch of the imagination, the more I get what he means. This stuff is surely a gift for all of us.
emilgouliev 7y ago
2 + 2 + 2 = 3 + 3 becomes much more obvious when you think of a 3x2 rectangle and rotate it by 90 degrees to a 2x3 rectangle.
- chii 7y ago
  but then isn't this just a tautological argument? why does multiplying the sides of a rectangle give you the area?
  - dboreham 7y ago
    Because of little squares being added.
  - red75prime 7y ago
    Multiplication in the form we know it arose from practical needs of area measurement, not as some abstract operation in semirings and rings. Multiplying the sides gives you the area because multiplication was invented/discovered to do that.
  - emilgouliev 7y ago
    you don't need to invoke a concept of area to justify this. you can just think of a grid of dots. since multiplication is repeated addition, each row represents one step of the repeated addition. 2x3 = 3 + 3 ... ... 3x2 = 2 + 2 + 2 .. .. .. since rotation doesn't change the number of dots, we expect 2x3 and 3x2 grid of dots to contain the same number of dots, which proves that multiplication is commutative.

rat9988 7y ago

>2 x 3 = 3 x 2 feels so obvious You just got used to it because you learned it very young. I'm pretty sure it wasn't that obvious when you learned it.

red75prime 7y ago
If I remember correctly, in my school multiplication was explained in terms of rectangles made of unit squares. Commutativity was literally visible to the naked eye then.
- quickthrower2 7y ago
  I also think you define integers as successors. Z, S(Z), S(S(Z)) etc. you can prove that a * b = b * a, given that you have defined the + function and * functions.
- lonelappde 7y ago
  It's both intuitively obvious geometrically, and algebraically is conceptually tricky to grasp. Or, more precisely it's trivial to decide that arithmetic operations are commutative or not, but tricky to decide correctly for all operations a child knows.
r_c_a_d 7y ago
When my kids started school I was surprised to learn that at age 4 it is not obvious that the number of items on a whiteboard remains constant if all the items are moved to a different part of the board. So what seems obvious, probably is something we just don't remember learning.

ncmncm 7y ago

The notion that multiplication is ("just") repeated addition is the hardest misconception to overcome blocking progress in mathematics. Repeated addition is an algorithm that works when you have a non-negative integer argument. Imagining that the algorithm defines the operation limits conception.

apalmer 7y ago
I would like to understand what multiplication is in your (and i guess advanced mathematics)
- sls 7y ago
  The comment to which you replied says that thinking about multiplication as "just" repeated addition is problematic, so let's look at that. Consider 3 x 2. If we take that approach, it seems ok - we understand it to mean "add together 3 2's" - 2 + 2 + 2, which gives the correct answer of 6. What about -3 * -2? What does it mean to add a negative number of times? What about pi * pi? What does it mean to add something pi times? What about the matrix M * the matrix N? etc. The parent's point is that this repeated addition thing is just an algorithm one can use to calculate a multiplication for some operands, specifically whole numbers, not a general definition of the operation of multiplication.
  - jacobolus 7y ago
    If we want to define π×π, the best we can do is some kind of algorithm for generating an approximation to π (e.g. as a continued fraction or as a positional decimal fraction) to any desired degree, and then an algorithm for multiplying such approximations. We can prove some bounds on the error introduced by our multiplication algorithm, and that gives us a way of approximating the product to any desired precision. For example if we want to know π^2 to about 3 decimal digits (~9.87), we can start by using an approximation of π to about 4 decimal places (~3.142) and then multiplying the two decimals. For rational numbers, multiplication algorithms are usually built on breaking a number down into constituent pieces, multiplying every pair of pieces from the two multiplicands, and then adding up all of the partial products. Matrix multiplication has the additional complication that the elementary terms involved (entries in different places in the matrix) cannot be added to each-other. But the basic procedure is still the same: break the two multiplicands down into basic units which we already know a multiplication table for, compute all of the partial products, then sum them up.

jacobolus 7y ago

More interesting, in more general contexts multiplication is not commutative. For example, if we apply the same pair of 3-dimensional rotations in opposite orders, we generally get different results. Scaling and planar rotation (in combination, a.k.a. “complex numbers”) are conveniently among the types of commutative multiplication. > multiplication is really just addition This is a misleading summary. That multiplication of integers per se can be re-expressed as addition (or if you like, as counting) depends on the basic parts involved being very simple and uniform. The basic multiplication table for integers is just × | -1 0 1 ––––––––––––– -1 | 1 0 -1 0 | 0 0 0 1 | -1 0 1 Since every other integer is just some sum of these basic parts, and multiplication distributes over addition, that covers it. To multiply two integers, first break each one down into some sum of a collection of –1, 0, and 1, then look up each partial product in the basic multiplication table above, and finally sum up all of the results. This process boils down to counting. Since the basic integer multiplication table is commutative, so is integer multiplication in general. But other kinds of numbers have richer structure based on a richer multiplication table of basic elements.

why-el 7y ago

Well, I am not sure if you've come across some of Bertrand Russell's work, but he qualified Mathematics as having what he called "supreme beauty". The older I get, and I am not a mathematician by any stretch of the imagination, the more I get what he means. This stuff is surely a gift for all of us.

emilgouliev 7y ago

2 + 2 + 2 = 3 + 3 becomes much more obvious when you think of a 3x2 rectangle and rotate it by 90 degrees to a 2x3 rectangle.

chii 7y ago
but then isn't this just a tautological argument? why does multiplying the sides of a rectangle give you the area?
- dboreham 7y ago
  Because of little squares being added.
- red75prime 7y ago
  Multiplication in the form we know it arose from practical needs of area measurement, not as some abstract operation in semirings and rings. Multiplying the sides gives you the area because multiplication was invented/discovered to do that.
- emilgouliev 7y ago
  you don't need to invoke a concept of area to justify this. you can just think of a grid of dots. since multiplication is repeated addition, each row represents one step of the repeated addition. 2x3 = 3 + 3 ... ... 3x2 = 2 + 2 + 2 .. .. .. since rotation doesn't change the number of dots, we expect 2x3 and 3x2 grid of dots to contain the same number of dots, which proves that multiplication is commutative.