# Justification of addition in Spivak, Ch.1

• B
• Von Neumann
In summary, the conversation discusses the properties provided by Spivak which allow one to justify a specific argument, specifically the equation x + 3 = 5. The properties P1, P2, and P3 are mentioned, and it is questioned whether the step in question is justifiable from any of these properties. The conversation then delves into the concept of addition as a function and the transitive property of equality, ultimately providing justification for the argument.
Von Neumann
TL;DR Summary
Based on the properties provided, why does x + 3 = 5 lead to (x + 3) + (-3) = 5 + (-3)?
I am 100% reading too much into this, but I am curious which of the properties provided by Spivak allow one to justify a specific argument. For reference/context, the properties are:

P1: If a, b, and c are any numbers, then
$$a +(b + c) = (a + b) +c$$

P2: If a is any number, then
$$a + 0 = 0 + a = a$$

P3: For every number a, there is a number -a such that
$$a + (-a) = (-a) + a = 0$$

Specifically, I am curious about the following:

$$x + 3 = 5$$

$$x + 3 + (-3) = 5 + (-3)$$

From this point, I understand the argument as the following:

By Property P3,

$$x + 0 = 2$$

By Property P2 ,then

$$x = 2$$

Is the step in question justifiable from any of the listed properties? Or, rather, is this basic property of addition intended to be assumed? Most properties are introduced from first principles, so I don't know if this should be any different. However, directly before the proof Spivak says "It is then possible to find the solution of certain simple equations by a series of steps (each justified by P1, P2, or P3) ...".

Here is the best that I can come up with:

$$x + 3 = 5$$

By P2,

$$x + 3 + 0 = 5$$

By P3,

$$x + 3 + (-3 + 3) = 5$$

By P1,

$$x + (3 + (-3)) + 3 = 5$$

Then, (by basic algebra?)

$$x + (3 + (-3)) = 5 + (-3)$$

By P3,

$$x + 0 = 2$$

By P2,

$$x = 2$$

Stephen Tashi
If I understand your question you are trying to prove:

##a=b \implies a+c = b+c##

Is this correct?

It can be justified using something very elementary (more elementary than the properties you listed) that you probably read over:

Addition is a function ##+: \mathbb{R} \times \mathbb{R} \to \mathbb{R}##.

This means, for every ##(x,y) \in \mathbb{R}\times \mathbb{R}##, there is a unique number ##x+y\in \mathbb{R}##.

So, if ##a=b##, then ##(a,c)=(b,c)## and thus by uniqueness it follows that ##a+c=b+c##.

Ps: Spivak didn't formally introduce the function notion at this point, so don't worry about this too much. But it is a very good question and shows that you don't take things for granted!

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jim mcnamara and Von Neumann
Von Neumann said:
Here is the best that I can come up with:

$$x + 3 = 5$$

By P2,

$$x + 3 + 0 = 5$$

That seems to assume "A quantity may be substituted for it's equal in any algebraic expression". I seem to remember that phrase given as an axiom in a secondary school textbook. But to get to ##x+3+0 = 5## correctly, you'd have to use the transitive property of ##=##.

##(x+3)+0 = (x+3)## by P2
##(x+3) = 5 ## given as the initial step
##(x+3)+0 = 5 ## transitive property of ##=##

jim mcnamara and Von Neumann
Math_QED said:
If I understand your question you are trying to prove:

##a=b \implies a+c = b+c##

Is this correct?

It can be justified using something very elementary (more elementary than the properties you listed) that you probably read over:

Addition is a function ##+: \mathbb{R} \times \mathbb{R} \to \mathbb{R}##.

This means, for every ##(x,y) \in \mathbb{R}\times \mathbb{R}##, there is a unique number ##x+y\in \mathbb{R}##.

So, if ##a=b##, then ##(a,c)=(b,c)## and thus by uniqueness it follows that ##a+c=b+c##.

Ps: Spivak didn't formally introduce the function notion at this point, so don't worry about this too much. But it is a very good question and shows that you don't take things for granted!

This is exactly my question! I think I need to get better at formulating exactly what I have a problem understanding. Your response makes sense. I was worried about overthinking this aspect.

jim mcnamara
Stephen Tashi said:
That seems to assume "A quantity may be substituted for it's equal in any algebraic expression". I seem to remember that phrase given as an axiom in a secondary school textbook. But to get to ##x+3+0 = 5## correctly, you'd have to use the transitive property of ##=##.

##(x+3)+0 = (x+3)## by P2
##(x+3) = 5 ## given as the initial step
##(x+3)+0 = 5 ## transitive property of ##=##

Yes! Those seem to be the steps I was performing implicitly in going directly from

$$x + 3 = 5$$

to

$$x + 3 + 0 = 5$$

I think the prologue of the text will make a lot more sense with this information. Thanks for responding!

## 1. What is the justification for addition in Spivak, Ch.1?

The justification for addition in Spivak, Ch.1 is based on the concept of "closure." This means that when adding two numbers, the result will always be a number that is also in the set of real numbers. In other words, the result of adding two real numbers will always be a real number.

## 2. How does Spivak justify the commutative property of addition?

Spivak justifies the commutative property of addition by stating that the order in which numbers are added does not affect the result. In other words, when adding two numbers, it does not matter which number is added first, the result will be the same.

## 3. What is the basis for the associative property of addition in Spivak, Ch.1?

The basis for the associative property of addition in Spivak, Ch.1 is the idea that grouping numbers differently when adding them will not change the result. In other words, when adding three or more numbers, it does not matter how they are grouped together, the result will be the same.

## 4. How does Spivak justify the existence of an additive identity in Ch.1?

In Spivak, Ch.1, the existence of an additive identity is justified by stating that there is a number, 0, which when added to any number, will result in that number. In other words, 0 is the identity element for addition.

## 5. What is the justification for the inverse property of addition in Spivak, Ch.1?

The justification for the inverse property of addition in Spivak, Ch.1 is based on the idea that for every real number, there exists another real number that when added together, will result in the additive identity, 0. In other words, every number has an additive inverse.

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