Multiplication and Addition

[conway]

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.

 

[HallsofIvy]

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,

 

[conway]

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!

 

[conway]

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.

 

[Theodore28]

You raise an interesting point about the relationship between addition and multiplication in mathematics. While it's true that in basic arithmetic, multiplication is often thought of as repeated addition, this analogy breaks down when considering the case of multiplying by zero.

Indeed, the property \(a \times 0 = 0\) may seem more akin to subtraction, as it effectively "removes" \(a\) from the total. However, this interpretation does not align with the broader understanding of multiplication in mathematics.

In mathematical structures like rings and fields, multiplication often exhibits properties beyond mere addition or subtraction. The distributive property, for instance, shows how multiplication interacts with addition, but it's clear that multiplication is not simply repeated addition in these contexts.

Your observation underscores the complexity of mathematical operations and the need for a nuanced understanding beyond simple analogies. I'm eager to hear your alternative solution and explore further insights into this fascinating topic .

 

[RobertSmart]

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.

 

[conway]

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.

 

[conway]

A grade calculator is a useful tool designed to help students and educators determine academic performance by calculating final grades based on individual assignment scores, weights, and grading criteria. This tool simplifies the often complex process of aggregating grades, ensuring accurate and transparent results. By inputting scores for quizzes, tests, homework, and other assessments, users can easily see their overall grade and identify areas needing improvement. Grade calculators are essential for effective academic planning and goal setting, providing a clear and immediate understanding of current academic standing.

?

 

[conway]

I want to do a quiz competition

No you don't. And if you did, I feel sorry for you. Bragging about how fast and accurate you are in math. Is like bragging about being a good calculator. A meaningless thing, considering how cheap and plenty they are.

You should be more concerned with creativity, imagination and above all the capacity to love.

 

[conway]

You raise an interesting point about the relationship between addition and multiplication in mathematics. While it's true that in basic arithmetic, multiplication is often thought of as repeated addition, this analogy breaks down when considering the case of multiplying by zero.

Indeed, the property \(a \times 0 = 0\) may seem more akin to subtraction, as it effectively "removes" \(a\) from the total. However, this interpretation does not align with the broader understanding of multiplication in mathematics.

In mathematical structures like rings and fields, multiplication often exhibits properties beyond mere addition or subtraction. The distributive property, for instance, shows how multiplication interacts with addition, but it's clear that multiplication is not simply repeated addition in these contexts.

Your observation underscores the complexity of mathematical operations and the need for a nuanced understanding beyond simple analogies. I'm eager to hear your alternative solution and explore further insights into this fascinating topic .

You may find it in the "other" section titled "The Unified Number"