3 ms·

The teacher mentions juggling units when adding and multiplying, but is 2 cm × 3 cm the same thing as 2 × 3?

cturner 6y ago
Both examples use the same operator, but we resolve them to different functions. Mentally, we use the tuple of argument types to select the appropriate function. Addition does not provide equivalent functions for all multiplication cases. Hence, multiplication is not repeated addition.
- salawat 6y ago
  You need to stop conflating the tyoe mumbo jumbo with multiplication. We don't define it based on what machines do with it, we define it based on what WE do with it. In the context of teaching, dimensional analysis comes far later in the process of education. The pedants answer is that there is more than one multiplication operator taught. It's most basic incarnation is, for all intents and purposes, repeated addition. It's more advanced forms, which include extra auditing for dimensional analysis isn't merely repeated addition. Though I'd challenge that too. Dimensional analysis is an extra aspect stapled onto and seperate from the arithmetic operation. Therefore makes up an additional level of auditing seperately as a whole, and therefore inappropriate to define as an inherent property of the operator. You multiply numbers. People need to stop trying to teach unrelated things bundled by some programmer or language writer to kids/adults. It doesn't make one seem smart. It just confuses people by blurring orthogonal aspects of problem solving, which puts people off of it.
d--b 6y ago
Arguably, what they're saying is that 2 x 3 should be understood as a unit-less "scaler" 2 that multiplies a number (of things) "3".
caddemon 6y ago
Yeah I don't think they are the same. You can of course also have 2 cm * 3, which still fits with the repeated addition idea, as long as you frame it as 2 cm + 2 cm + 2 cm. I'm not an educator by any stretch, but that seems like an intuitive way to introduce units without causing confusion about previous number teachings.
vishnugupta 6y ago
> but is 2 cm × 3 cm the same thing as 2 × 3? This made me think too, in particular the teacher's example. So; here's what I think it should be calculated. 3 cm + 3 cm = 2 X 3 cm (and not 3 cm + 3 cm = 2 cm X 3 cm). What you are doing above is adding two objects each of who are 3 cm long. And the way you compute "2 X 3 cm" is "(2 X 3) cm" which then becomes "6cm". However 3 cm + 3 cm != 2 cm X 3 cm as the author states which is what leads them to wrong conclusion. In other words 2 cm X 3 cm leads to an entirely new thing in geometry which is "area". So while 3 cm + 3 cm is still length (and hence it should be translated to 2 X 3 cm i.e., one scalar unitless number and other number with unit) 2 cm x 3 cm is not length. And the resultant area is correctly written as 6 cm^2; by multiplying units as well as the numbers. Now, how does one multiply numbers (2 X 3)? It's indeed by repeated addition! The author, however is onto something. Which is when you multiply a something with unit (like 3Kg, 2Cm) with a unitless number that resultant product is still something (that weight, length) that you began with. However when you multiply two things both of which have units the resultant product is something entirely new (i.e., area as opposed to length, pressure as opposed to weight etc.,)
- harperlee 6y ago
  Another way to say it is that 2cm * 3cm is shorthand for (2)(1cm)(i) * (3)(1cm)(j). We are complecting 3 semantics: measure and unit (which compose a module, in vector terms) and “meaning” (dimension, in vector terms). This is why in high school physics you are taught to always write down the units of each number: typing them helps avoid a big amount of errors. Note that “typing” and “vector terms” are not essential, just helps me express it with the HN crowd easily.
- insert_coin 6y ago
  The author is confusing math with physics. Math: 3x3 = 3^2 physics: cmxcm = cm^2 3cm x 2cm = (3x2)(cm x cm) = (math)(physics) The context gives the meaning. Or in computers, how many bits would the representation of a matrix of 2b x 2b occupy? 2b x 2b = 4b (not 4b^2 since memory is linear)

Both examples use the same operator, but we resolve them to different functions. Mentally, we use the tuple of argument types to select the appropriate function. Addition does not provide equivalent functions for all multiplication cases. Hence, multiplication is not repeated addition.

salawat 6y ago
You need to stop conflating the tyoe mumbo jumbo with multiplication. We don't define it based on what machines do with it, we define it based on what WE do with it. In the context of teaching, dimensional analysis comes far later in the process of education. The pedants answer is that there is more than one multiplication operator taught. It's most basic incarnation is, for all intents and purposes, repeated addition. It's more advanced forms, which include extra auditing for dimensional analysis isn't merely repeated addition. Though I'd challenge that too. Dimensional analysis is an extra aspect stapled onto and seperate from the arithmetic operation. Therefore makes up an additional level of auditing seperately as a whole, and therefore inappropriate to define as an inherent property of the operator. You multiply numbers. People need to stop trying to teach unrelated things bundled by some programmer or language writer to kids/adults. It doesn't make one seem smart. It just confuses people by blurring orthogonal aspects of problem solving, which puts people off of it.

d--b 6y ago

Arguably, what they're saying is that 2 x 3 should be understood as a unit-less "scaler" 2 that multiplies a number (of things) "3".

caddemon 6y ago

Yeah I don't think they are the same. You can of course also have 2 cm * 3, which still fits with the repeated addition idea, as long as you frame it as 2 cm + 2 cm + 2 cm. I'm not an educator by any stretch, but that seems like an intuitive way to introduce units without causing confusion about previous number teachings.

vishnugupta 6y ago

> but is 2 cm × 3 cm the same thing as 2 × 3? This made me think too, in particular the teacher's example. So; here's what I think it should be calculated. 3 cm + 3 cm = 2 X 3 cm (and not 3 cm + 3 cm = 2 cm X 3 cm). What you are doing above is adding two objects each of who are 3 cm long. And the way you compute "2 X 3 cm" is "(2 X 3) cm" which then becomes "6cm". However 3 cm + 3 cm != 2 cm X 3 cm as the author states which is what leads them to wrong conclusion. In other words 2 cm X 3 cm leads to an entirely new thing in geometry which is "area". So while 3 cm + 3 cm is still length (and hence it should be translated to 2 X 3 cm i.e., one scalar unitless number and other number with unit) 2 cm x 3 cm is not length. And the resultant area is correctly written as 6 cm^2; by multiplying units as well as the numbers. Now, how does one multiply numbers (2 X 3)? It's indeed by repeated addition! The author, however is onto something. Which is when you multiply a something with unit (like 3Kg, 2Cm) with a unitless number that resultant product is still something (that weight, length) that you began with. However when you multiply two things both of which have units the resultant product is something entirely new (i.e., area as opposed to length, pressure as opposed to weight etc.,)

harperlee 6y ago
Another way to say it is that 2cm * 3cm is shorthand for (2)(1cm)(i) * (3)(1cm)(j). We are complecting 3 semantics: measure and unit (which compose a module, in vector terms) and “meaning” (dimension, in vector terms). This is why in high school physics you are taught to always write down the units of each number: typing them helps avoid a big amount of errors. Note that “typing” and “vector terms” are not essential, just helps me express it with the HN crowd easily.
insert_coin 6y ago
The author is confusing math with physics. Math: 3x3 = 3^2 physics: cmxcm = cm^2 3cm x 2cm = (3x2)(cm x cm) = (math)(physics) The context gives the meaning. Or in computers, how many bits would the representation of a matrix of 2b x 2b occupy? 2b x 2b = 4b (not 4b^2 since memory is linear)