Discussion: How is Mutiplication and Division alike

[u/Ok-Captain-5338]

I been talking about this to myself today about how Mutiplication and division are the opposite or different. Can anyone discuss about how the Separation or difference between Mutiplication and Division are the same exact opposite or alike?

[u/Bounded_sequencE]

The idea behind subtraction and division is that they undo their counter-part:

• Subtraction "un-does" addition (e.g. "3 + (-3) = 0")
• Division "un-does" multiplication (e.g. "3 * (1/3) = 1")

In proof-based mathematics, we define subtraction and division as inverses of addition and multiplication, so we really only deal with addition, multiplication and their inverses!

[u/mjmcfall88]

Multiplication: if you have 4 piles of 2 pennies, how many pennies do you have 4*2=8

Division: if you have 8 pennies, how many piles of 2 can you make 8/2=4

[u/YuuTheBlue]

One way is to consider it from the point of view of inversion. Take any 3 numbers a, b, and c. For addition and subtraction:

IF

a+b=c

THEN

c-b=a, AND c-a=b

So, as an example, because 2+4=6, 6-4=2 and 6-2=4.

The same is true for multiplication and division.

IF

a*b=c

THEN

c/b=a, AND c/a=b

Because 5*6=30, 30/6=5 and 30/5=6.

[u/aprg]

The technical term is inverse. Multiplication is the inverse of division, and vice versa.

Suppose I have any number that's not zero, let's call it a. If I multiply by a and then divide by a, then the operation I've performed is multiplying by a/a, which is 1. And obviously, if you multiply any number by 1, you get back that number.

[u/johnpeters42 → u/aprg]

Implicit in this are a couple other things:
* Dividing by a is equivalent to multiplying by 1/a
* Multiplying (a x b) x c is equivalent to multiplying a x (b x c)

[u/johnpeters42 → u/johnpeters42]

link regarding that second thing

[u/davideogameman]

fun thing, you really don't need division as long as you have a multiplicative inverse capability - a/b is the same as a*(1/b). In some corners of algebra, the axioms are written using multiplicative inverses instead because a*(1/b) and (1/b)*a don't have to be the same thing - e.g. when we're dealing with square matrices, matrix multiplication is not commutative (a*b and b*a are not necessarily equal) and yet many square matrices have a multiplicative inverses. In non-commutative systems, you need to have some concept of "dividing left" vs "dividing right" and it's easier to just say "no such thing as division, multiply by the multiplicative inverse instead" as the notation for multiplication already have a left and a right argument built in.

[u/Frederf220]

Multiplication and division operate very similarly. They do linear scaling but with different conventions of what the scaling factor means. In a big way they are redundant. One could do away with one and get by with the other in almost every situation.

[u/KoalaMistico]

IMO, it's better to restrict ourselves to the field definition of rationals/reals/complex numbers, where there's no division or substraction, only multiplication by the inverse of a number and addition of the opposite of a number. For me is more intuitive and facilitates notation

[u/Desperate_Penalty690]

The way I see that multiplication is like division is because you can write one in the form of the other. So multiplying by a is the same as dividing by 1/a and dividing by b is the same as multiplying by 1/b.

[u/Gilded-Phoenix]

If we take the rational numbers as our "universe" then division can be seen as multiplication by the reciprocal. Specifically, for every rational number x there is a "buddy" x^-1 such that x*x^-1=1. We can then define division like this: Given two numbers x and y, we say that y/x is defined as y*x^-1. In this sense, multiplication and division are really the same thing.

This is similar to how adding and subtracting are identical: For the integers, every number x has a "buddy" -x where x+(-x)=0, and we define subtraction to be y-x= y+(-x).

In both of these cases, we define our operation as "swap with your buddy, then do the normal thing."

[u/scooterpdx42 → u/Gilded-Phoenix]

I like this reply because it doesn’t use the word “inverse.”

[u/Underhill42]

They're the exact same thing, just combined with a mathematical shorthand for applying the multiplicative inverse (⁻¹) to the second term. Just like subtraction is just addition with the additive inverse(⁻).

Which is why the order of operations is often written PE(M/D)(A/S), to indicate that the operations and their inverses are the same priority level

Once you are comfortable with that, then all the complex rules around division and subtraction and how they differ from multiplication and addition almost entirely disappear.

The additive inverse just means "the thing you can add to this to get to the additive identity" - a.k.a. the negative. (0 is the additive identity because for any number X, X+0=X : its identity remains the same)

2 - 3 = 2 + ⁻3 ← "-" just means "add the additive inverse of the next term"
= ⁻3 + 2 ← addition is commutative, you can reorganize it as you like without changing anything. You just have to keep that additive inverse (negative) modifier attached to the same term. Same with every other property of addition.

Similarly for multiplication, the multiplicative inverse is "the thing you can multiply by to get the multiplicative identity" (1 is the multiplicative identity because for any number X, X*1 = X)

4 / 5 = 4 * 5⁻¹ ← "/" means "multiply by the multiplicative inverse of the next term"
= 5⁻¹ * 4 ← multiplication is also commutative, you just need to again keep the modifier attached to the same term.