# The Distributive Property of Multiplication <over> Addition

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#### VicNicSor

##### Banned
There's this law in math, which is called:
“The Distributive Property of Multiplication over Addition”.

1) 3x(4+6) = 3x10 = 30

2) 3x(4+6) = 3x4+3x6 = 30

It means the second way of solving the problem, that is, the '3' is "distributed" between '4' and '6'.

The question is, what does "over" mean here? In Russian, we use "regarding" here.

Thank you.

#### VicNicSor

##### Banned
I edited that typo two minutes later after posting, how did you manage to catch it?

#### JulianStuart

##### Senior Member
In this case, one rule takes precedence over another.

#### VicNicSor

##### Banned
In this case, one rule takes precedence over another.
One question: does, grammatically, the word "over" have anything to do with the word "property"/"law"?

“The Distributive Property of Multiplication over Addition”.
“The Distributive Law of Multiplication over Addition”.

Or is the phrase "Multiplication over Addition" independent of whether it is "property", "law", or any other noun?...

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#### bennymix

##### Senior Member
From Macmillan dictionary, these meanings are relevant.

1b moving across the space above someone or something

Government troops fired a few shots over their heads.

Flocks of geese flew over, calling loudly.

==

3a from one side of something to the other

going or looking across from one side of something to the other

Several bridges over the River Danube were destroyed.

---

[Me} Yes, there is a distributive law, axiom or rule: multiplication and division distribute over addition and subtraction.

The king rules over all his subjects. It's like covering.

#### VicNicSor

##### Banned
So, Benny, unlike Julian, you mean that "over" here is to be understood literally?:
click

I.e., those yellow arrows show us the way of the "distribution", that is, it goes "above" (over) the addends?

#### bennymix

##### Senior Member
I don't think Julian and I disagree. If you diagram 'precedence,' you may put that which takes precedence at a higher, hence more encompassing level.

Further, I don't think you can say that only a physical sense of 'over' is literal. Look at the examples in the dictionary.

over (adverb, preposition) definition and synonyms | Macmillan Dictionary

Yes, I think the diagram explains why we say 'distributes over' in this case.

#### VicNicSor

##### Banned
I don't think Julian and I disagree. If you diagram 'precedence,' you may put that which takes precedence at a higher, hence more encompassing level.

But what Julian meant is, as I understand, if we have "3x4+3x6", we must do the multiplication first, and only then -- the addition. Multiplication takes precedence over addition...

#### VicNicSor

##### Banned
I've just watched the video to the end, and the man says:
That’s why the 'technical' name is the Distributive Property of Multiplication over Addition. You’re distributing the multiplication over all the members of a group that are being ADDED.

So, "over" has this meaning:
in or to many different parts of an area
The drought has spread over much of the southern US.
(Macmillan dictionary)

#### JulianStuart

##### Senior Member
And you believed that If you were being picky ( ) you would say "He should have said "...over the group of numbers (not over an operation like addition or subtraction)". I think he provided a mnemonic rather than a grammatical or semantic comment. But, whatever helps you get though this

#### VicNicSor

##### Banned
And you believed that If you were being picky ( ) you would say "He should have said "...over the group of numbers (not over an operation like addition or subtraction)". I think he provided a mnemonic rather than a grammatical or semantic comment. But, whatever helps you get though this
I.e., you mean the meaning is indeed rather this, right?
But what Julian meant is, as I understand, if we have "3x4+3x6", we must do the multiplication first, and only then -- the addition. Multiplication takes precedence over addition...

#### JulianStuart

##### Senior Member
Yes - in the list describing the order in which we perform opeations, multiplication comes above addition - this is the sense of over in takes precedence over -> comes first/above the other.

#### VicNicSor

##### Banned
Yes - in the list describing the order in which we perform opeations, multiplication comes above addition - this is the sense of over in takes precedence over -> comes first/above the other.
But what does it have then to do with "distribution"?...

“The distributive property of multiplication over addition”.

= Multiplication has a property of distribution of the factor [of 3, in our example] over "addition" (the members of the group).

To me, it works semantically

#### JulianStuart

##### Senior Member
The distributive property of multiplication (takes precedence) over (the additive property of) addition.

#### bennymix

##### Senior Member
Have we not exhausted this topic?

Can we move to advanced issues, like why we say, "I've going over to my friend's apt," when his apartment is on a lower floor than mine?

#### srk

##### Senior Member
6. across;
.....throughout:
.....They hitchhiked all over Europe.
“The Distributive Property of Multiplication over Addition”.
The way multiplication is applied across summed terms.

Distribute the fertilizer evenly over the lawn. (The way-it-ought-to-be-done law for application of fertilizer to lawns.)

What something in mathematics actually means is often poorly described by the words used to identify it. It may take a paragraph or two to define the term, along with a prototype in terms of symbols: x(y+z)=xy+xz.

Try understanding topological compactness by looking "compact" up in our dictionary. Take "imaginary" seriously in understanding complex numbers, and see how far it gets you.

I liken it to choosing a file name when saving a Word document I've spent a lot of effort preparing. I'm lucky if I can locate the file a month later.

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