Multiplication and Addition

It is well known that in "basic" math, multiplication is addition. It is known that in higher tiers of mathematics, such as rings, that this is not the case. Yet in "basic" math multiplication is addition. So then consider...

a + 0 = a

yet...

a * 0 = 0

This case of multiplication is not addition, if anything it is subtraction. For it to be addition it should be...

a * 0 = a

So then we may say that multiplication is NOT addition in "basic" mathematics. Yet this is problematic because in every other case not involving (0). Multiplication IS addition.

I stand by to offer another solution...for those who wish to hear it.

 

Multiplying a by a positive integer, n, is the same as adding a to itself n times. But 0 is NOT a positive integer so that does not apply,

 

Ok, well stated. We must then still change it to multiplication is addition with only positive integers, and not zero. So then in general multiplication is NOT addition. Only in special cases is it so....So then a rewrite of many school books at the least. Yet there is a way to make the "definition" fit....with negative integers as well.

Assign all numbers a dimensional unit quantity as well as a numerical quantity. In this way, all cases of multiplication will be addition.

Excellent reply, thanks for your time!

 

Allow numbers to be defined as...

"A numerical quantity, inside a dimensional quantity. "

Allow the dimensional quantity to be notated with (_).

1 = the numerical quantity
(1_)=(_)= the dimensional quantity
(1) = the "number one"

classical 2 = novel (2) = (2,2_) = (2,_,_)
classical 3 = novel (3) = (3,3_) = (3,_,_,_)

Allow multiplication to be defined as...
"A given numerical quantity, placed into a given dimensional quantity. Then all numerical quantities are added"

Allow the definition of addition to remain unchanged.
Allow the definition for the negative symbol to remain unchanged: "The opposite of"
Allow the definition for a negative dimensional unit to exist as...
"Place the opposite of a given numerical quantity into the dimensional unit quantity."

Classical
(2*3)=6
Novel
(2*3_)=(3_*2)=(2_*3)=(3*2_)=6

2= the numerical unit
3_ = (_,_,_) = the dimensional unit

(2,2,2) = the numerical quantity given, placed into the dimensional unit quantity given.
(2 + 2 + 2) = the numerical quantity given, placed into the dimensional unit quantity given. Then all numerical quantities in all dimensional unit quantities are added.

Classical
(-2*3)
Novel
(-2 * 3_)

-2 = the numerical quantity given
3_ = the dimensional quantity given

(_,_,_) = the dimensional quantity
(-2,-2,-2) = The numerical quantity given placed into the dimensional quantity.
(-2 + -2 + -2) = (-6)

Therefore multiplication of negative integers are addition by definition.

I wish to keep this short, but I can show expressions for -(_,_,_). Or "negative" dimensional units. Upon request.

 

Consider the case where a+0=a, a fundamental property of addition. Yet, a×0=0, which seems contradictory to the notion that multiplication is just repeated addition. In fact, it appears more like subtraction if we aim to reconcile it with addition.

The proposed solution, a×0=a, would indeed align multiplication with addition, but it fails to address the reality where a×0=0. Thus, while in basic mathematics multiplication often resembles addition, it's clear that this simplistic view does not hold universally, especially when 00 is involved.

Mathematics, particularly at higher levels, demands a more nuanced understanding where concepts like multiplication may deviate from the simple rules observed in basic arithmetic.

Many students find it difficult to do such types of question. Many students also miss their assignment due to not knowing the basic concepts. If you are one of them facing such difficulties, I would suggest you to visit MathsAssignmentHelp website once. You can also contact them on +1 (315) 557-6473.

 

Please see the thread "The Unified Number"

I have addressed (a * 0 = 0). This mathematical construct is relative to 0 used as "space" or as "value", in the binary expression.

BOTH (a*0=a) and (a*0=0) are true. Again it is relative to zero used as a dimensional quantity, or a numerical quantity in the expression.