Number of concepts of Multiplication and Addition

[thedy]

Hi,can someone exaplain me,why do we have a few concepts of multiplication,like repeated addition,scaling,grouping etc...but only one concept for addition that means to put something together and make it more or bigger
I was solving in past,why multipliplication produce new unit and so on...but now,I am trying to look at this problem in different way,
By the way how was multiplication as mathematical operation invented?And specially unit concept of multiplication...In history I mean of course

Thanks for answers.I appreciate each of your answers...

 

[Simon Bridge]

Multiplication was invented too long ago to be sure how...
There are many more ways to arrive at the same multiplication than there are ways to arrive at the same addition. But they are not "different concepts". The concept is the same, the different approaches are just that, approaches. Many paths can lead to the same place.
Multiplication builds on addition. The multiplication sign is just a shorthand notation to avoid having to write down lots of plus signs. The plus sign is also shorthand for writing down lots of 1s in groups and numerals are shorthand for lots of 1s in one group.
So multiplication and addition are the same thing... the different approaches just show younthat these apparently different things are actually just repeated addition, so we don't need a special tule to handle them.

 

[symbolipoint]

Simple question for very simple idea - very difficult answer

Try to think simply. Addition comes from counting. We connect a number with another number and if we want to know the count of both these numbers together, then this is ADDITION, and we show this with the ADDITION symbol between them. That is the plus-sign.

People must have became frustrated when they wanted to show the addition of a repeated number, so they learned to make this addition more efficient, and so repeated addition of the same number defines MULTIPLICATION.

 

[Mark44]

So multiplication and addition are the same thing... the different approaches just show younthat these apparently different things are actually just repeated addition, so we don't need a special tule to handle them.

People must have became frustrated when they wanted to show the addition of a repeated number, so they learned to make this addition more efficient, and so repeated addition of the same number defines MULTIPLICATION.

I'll accept that multiplication of an integer by an integer represents repeated addition, but explaining that multiplication is repeated addition falls short with, say ##\pi \times \sqrt{3}##.

 

[symbolipoint]

I'll accept that multiplication of an integer by an integer represents repeated addition, but explaining that multiplication is repeated addition falls short with, say ##\pi \times \sqrt{3}##.

Maybe you would like to explain that one, for the non-rational products.

 

[Mark44]

I'll accept that multiplication of an integer by an integer represents repeated addition, but explaining that multiplication is repeated addition falls short with, say ##\pi \times \sqrt{3}##.

Maybe you would like to explain that one, for the non-rational products.

Maybe you missed my point. The claim was made twice in this thread, that multiplication is repeated addition. What I'm saying is that in some cases, such as the example I gave, this is an oversimplification.

 

[symbolipoint]

Maybe you missed my point. The claim was made twice in this thread, that multiplication is repeated addition. What I'm saying is that in some cases, such as the example I gave, this is an oversimplification.

The reason I ask for the explanation is that once a person learns to understand and perform multiplication for rational numbers, the person is not likely to feel uncomfortable about what it means to multiply irrational numbers. The feeling in ones mind is to just make an approximation for the result. The person is not likely to feel the need to over-analyze the meaning or the process.

 

[Mark44]

My comment was not about how multiplication is performed, but rather, the claim that multiplication is repeated addition. If the operation is to multiply 5/6 and 3/8, you'll have a difficult time convincing someone that there is repeated addition going on here.

 

[Simon Bridge]

I think you missed the point of the prev replies... but fair enough. Perhaps you'd like to give a better, less of an oversimplification, reply?
Personally I have seldom had trouble convincing someone that multiplication of rational numbers involves repeated addition in the manner roughly as I outlined above.

 

[symbolipoint]

My comment was not about how multiplication is performed, but rather, the claim that multiplication is repeated addition. If the operation is to multiply 5/6 and 3/8, you'll have a difficult time convincing someone that there is repeated addition going on here.

I easily overlooked that idea. At some point in my learning, I began to understand multiplying non-whole numbers as two lengths crossing at right angles to form an area. No effort was further spent on trying to understand through the repeated addition idea.

 

[Mark44]

I think you missed the point of the prev replies... but fair enough. Perhaps you'd like to give a better, less of an oversimplification, reply?

Not particularly. I see addition and multiplication as basically different operations. Of course, for multiplication involving whole numbers, a X b can be rewritten repeated addition -- b + b + b + ... + b, with a addends.

Personally I have seldom had trouble convincing someone that multiplication of rational numbers involves repeated addition in the manner roughly as I outlined above.

How would you flesh out what you wrote in post #2 to write 5/6 X 3/8 as repeated addition?

Or better yet, what about my other example, ##\pi \times \sqrt{3}##?

 

[pwsnafu]

How would you flesh out what you wrote in post #2 to write 5/6 X 3/8 as repeated addition?

Not Simon.
I learned ratios before fractions. And of course they are equivalent to each other.
##(5: 6) \times (3:8) = 5 \times 3 : 6 \times 8 = 5+5+5 : 6+6+6+6+6+6+6+6##

 

[Mark44]

I learned ratios before fractions. And of course they are equivalent to each other.
##(5: 6) \times (3:8) = 5 \times 3 : 6 \times 8 = 5+5+5 : 6+6+6+6+6+6+6+6##

For "repeated addition" I would expect to see only addition.

 

[Simon Bridge]

The usual pedagogical starting point for multiplication is repeated addition of integers.
You'll also find this as a definition of "multiplication" in dictionaries like Mirriam Webster.

Historically, numbers were understood, in many cultures, in terms of correspondences between lengths ... to multiply fractions you can change the unit length, this is equivalent to the modern process of finding a common denominator ... then notice the relationship with finding an area.
You find an area by counting the unit areas that fit inside ... which can be organized into repeated addition of a fixed row, or groups of unit areas that are systematically arranged... in a rectangle, this amounts to multiplying the lengths of the sides... You get to the multiplication of irrationals by applying the rules for the others, like considering how you can get an area of pi-squared: where does it come from?

Although this is what most people mean - hence the M-W definition - it is incomplete (or, at least, inefficient) to describe the totality of the modern mathematical concept, and there is some contention that teaching "multiplication is repeated addition" can be harmful in the long run. For instance, see 4 articles: https://www.maa.org/external_archive/devlin/devlin_01_11.html (other three linked from this page.) I'll let the author speak - and the reader can relate that text to what each has said here.

That should cover things... what I would like to see is a better reply to post #1, that does not use MIRA, does justice to the whole concepts, and is within OP's education level.

 

[thedy]

But why does addition have far less concepts of what addition is?And why does multiplication have so many concepts or examples better said in real life? I do not see symetry in this. By symetry I mean identical quantity of examples in addition and multiplication.If people invented multiplication like repeated addition first,what was motivation to get other examples what multiplication is.Scaling,ratios...

With multiplication is my fantasy unlimited how to see and when to use multiplication. It will almost always have meaning,but with addition have not.

And addition is still the same over years...

thanks