Question

I always get confused with multiplication/addition and division/subtraction

Answer 1

For positive integers, you're correct that multiplication can be thought of as repeated addition.   3\*4 is the same as 3 4's added together or 4+4+4.     So you could write 4+4+4 instead of 3\*4.

But what if you wanted to represent the idea of multiplying two unknown quantities x and y?   Then x\*y is x y's added together.   And that means you'd have to write something like:   **y+y+y...+y   (x times)**.      That's a lot messier than just writing x\*y.   

And it gets further complicated when you're dealing with fractions.   (1/2)\*4 doesn't represent adding 4 to itself (1/2) times at all.    For that it can help to think of the multiplication sign \* as representing the word "of".   (1/2)\*4  becomes one half of 4, or 2.     And this sort of corresponds to multiplication with integers if you think of 3\*4 as meaning 3 "of" 4's.   It's poor english, but it roughly makes sense.

Note also that multiplication doesn't always increase.   (1/2)\*4 = 2 which is less than 4.   

Finally, division should definitely not be though of as subtraction.   It should instead be thought of as the opposite of multiplication -- which means you should really understand multiplication well before even trying to understand division.

x/y  means  "the number of y's I'd need to produce x".    12/3 means the number of 3's you'd need to produce 12.    Notice the word "of" again.   In other words,  12/3 means the number you'd need to multiply by 3 to get 12.    Since 4\*3 = 12,   12/3 must equal 4.

And that works even with fractions.    4 / (1/2) means the number you'd need to multiply by 1/2 to get 4.     Since 8 \* (1/2) = 4,  that means that 4 / (1/2) = 8.     You need 8 halves to make 4.

Answer 2

> I think there's something wrong that I'm thinking about in math.

Exactly.

Learn multiplication and division correctly.

Answer 3

Ultimately all four operations on the integers can just be represented by repeatedly taking the subsequent number of the previous number. Addition of x+y is just taking the subsequent number from x over and over again y times. so 5+3 is 5+1+1+1.  


So why do these things? Three reasons. First is that these operations model things we want to study or reason about. If you want to how many squares there are on a checker board you could count them all individually or you could count that there are 8 rows and 8 columns and multiply them to get 64 squares. If you need 5 ounces of something to make a a recipe that serves 2 people and you want to make enough for 6 people multiplication models solving that problem really well.

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The second reason is that just taking the subsequent number over and over and over again is a long process. You can do it but it's harder, even for a computer.

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The third reason is that these abstractions make it easier for us to reason about problems we are solving. Computation is only part of the story. Sometimes you'll have to think about some problem in a way that gets complicated if we rely on primitive notions like addition. This is a big part of mathematics, to find an idea that takes a complicated concept and reduces it to something easy to understand and reason about, so when we need to think through a problem we aren't tracking all the details of just adding things over and over again.

Answer 4

Indeed, multiplication can be understood as an abbreviation of a repetitive addition. But it can be understood in other ways, too.  

For example, multiplication can be understood as a machine controlled by one factor to make the other factor bigger or smaller. From this point of view, it is no longer a matter of adding or subtracting units.  

Let’s use 12 as “the other factor.” If we want to make it grow until it triples, we put three in our dial (the second factor):  

12 • 3 = 36  

To double its “size,” we put a 2 in our dial:  

12 • 2 = 24  

If we want the other factor just as it is, that is, no bigger nor smaller, we put a 1 in our dial:  

12 • 1 = 12  

If we want to shrink the other factor so that it is half of what it was, we put 1/2 in our dial:  

12 • (1/2) = 6  

Also: 12 • 0.5 = 6  

If we want to shrink our other factor to a third of what it was, we put 1/3 in our dial:  

12 • (1/3) = 4  

So why do people express a formula as multiplications instead of additions? One answer could be: to scale things up or down proportionally.  

For example, in the formula F = m•a, we express the relation between force, mass and acceleration. We’re talking about, say, a rocket in space being pushed by the constant force of the burning of it’s fuel.  

Let’s solve for acceleration:  

a = F / m  

Here, it’s clear that the formula states that, in order to double the acceleration without altering the mass, you will need to double the constant force that is pushing the object.  

Also, it’s also clear that, to double the acceleration without altering the force, you will need to be pushing half of the original mass.  

As you can see, we’re talking about making the values bigger or smaller. We are not talking about repeated addition or repeated subtraction.

Answer 5

Here's an analogy. I hope it's of some help.

All positive whole numbers can ultimately be formed by just repeatedly adding 1. For example, the number 3 is the same as 1+1+1, and the number 5 is the same as 1+1+1+1+1.

Therefore, in a certain sense, we don't "really need" a number like 5, because we could always rephrase things using a sum of 1's. For example, instead of saying "I had five cookies", you could instead say "I had 1+1+1+1+1 cookies" or even "I had a cookie and a cookie and a cookie and a cookie and a cookie."

So yes, you \*could\* avoid mentioning the number 5, and just use 1's instead. However, if you thought that collections of five things were something you were likely to encounter in your life, you might find it convenient to have a single word or symbol like "five" or "5" to refer to that situation. In other words, we might find it helpful to think of 5 as a single "thing".

Similarly, the more you encounter multiplication, the more useful it is to think of multiplication as its own "thing". If someone mentions 10 times 30, it can be convenient to be able to say that that's 300 without needing to write it as 30+30+30+30+30+30+30+30+30+30.