# Proving Equation: m + (n + (p + q)) = ((m+n) + p) + q

• mossfan563
In summary, the equation m + (n + (p + q)) = (m + n) + (p + q) = ((m + n) + p) + q can be proven using the commutative and associative properties of addition and multiplication, as well as the distributive property.
mossfan563

## Homework Statement

I have to prove this equation:
m + (n + (p + q)) = (m + n) + (p + q) = ((m + n) + p) + q

## Homework Equations

Commutative property of addition and multiplication
(m+n) = (n+m), (mn) = (nm)
Associative property of addition and multplication
(m+n)+p = m+(n+p), (mn)p = m(np)
Distributive property
m * (n+p) = mn + mp

## The Attempt at a Solution

Only being allowed to use certain axioms, I ruled out all except the ones in the relevant equations.
So far I'm pretty much stumped as to how to prove the equation. Going from the first part to the second is difficult. I tried to manipulate the first part of the equation to use commutative and I got nowhere. How do I get myself in the right direction?

You should use associativity, not commutativity.

## 1. How can I prove the equation m + (n + (p + q)) = ((m+n) + p) + q?

This equation can be proven using the associative property of addition, which states that the grouping of numbers in an addition equation does not affect the final result. In other words, you can add the numbers in any order and still get the same answer. In this case, we can group the numbers in any way we want without changing the final sum.

## 2. What is the purpose of proving equations?

The purpose of proving equations is to demonstrate that a mathematical statement is always true, regardless of the values of the variables involved. This helps to establish the validity and accuracy of mathematical concepts and theories.

## 3. Can the equation be proven using any other properties?

Yes, this equation can also be proven using the commutative property of addition, which states that the order of numbers in an addition equation does not affect the final result. In other words, you can add the numbers in any order and still get the same answer.

## 4. Is it possible to prove this equation using a visual representation?

Yes, this equation can be proven using a visual representation such as a number line or a set model. These visual representations can help to illustrate the concept of grouping and rearranging numbers in an addition equation.

## 5. Are there any other equations that follow the same pattern as this equation?

Yes, there are many other equations that follow the same pattern as this equation, such as the distributive property of multiplication and the associative property of multiplication. These properties also involve grouping and rearranging numbers to show that the final result remains the same.

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