# Justify each step with a property, definition or an operation

#### mathdaughter

##### New member
Please justify each step with a property, operation, or definition:

A: Original: 3 + (-2)
step 1: 1

B: Original: 3 - (-2)
step 1: 5

C: Original: 3 - (-2)
step 1: 3 + 2
step 2: 5

B: subtraction operation
C: step 1: commutative property; step 2: addition operation

Am I right?

Thanks,

EDIT:

C: step 1: distributive property

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#### j-astron

##### Junior Member
I don't think it is the commutative property for C.

Addition being commutative means the order of the operands doesn't matter. 2+3 = 3+2

What you need is the property that subtracting a number is the same thing as adding its negative, whatever that property is called.

Without knowing more about what you've been taught specifically, and the instructions given in your class, it's hard to say whether you are saying enough for parts A and B, or whether you are meant to say more. For instance, B is the same as C, but you do it in only one step, which implies some property is being used or assumed that is more explicitly shown in C...

#### Dr.Peterson

##### Elite Member
Please justify each step with a property, operation, or definition:

A: Original: 3 + (-2)
step 1: 1

B: Original: 3 - (-2)
step 1: 5

C: Original: 3 - (-2)
step 1: 3 + 2
step 2: 5

B: subtraction operation
C: step 1: commutative property; step 2: addition operation

EDIT:

C: step 1: distributive property
The distributive property would require an addition inside parentheses.

What is being applied here is a property that says that -(-a) = a. I don't know that this has a universal name; it might be called the "negative of a negative property", or something like that. Or, you might just have been given a property that says subtraction is addition of the negative (perhaps presented as a definition of subtraction: a - b = a + (-b). Possibly you are expected to mention both of these together.

Can you list the properties you have been taught?

#### JeffM

##### Elite Member
The distributive property would require an addition inside parentheses.

What is being applied here is a property that says that -(-a) = a. I don't know that this has a universal name; it might be called the "negative of a negative property", or something like that. Or, you might just have been given a property that says subtraction is addition of the negative (perhaps presented as a definition of subtraction: a - b = a + (-b). Possibly you are expected to mention both of these together.

Can you list the properties you have been taught?
I am not sure that it is technically a fundamental property. It is certainly a theorem derivable from the definition of additive inverse, a convention of notation, and the following two axioms (or theorems: it has been almost 50 years since I studied this).

$$\displaystyle \text {AXIOM 1: } p = q \text { and } q = r \implies p = r.$$

$$\displaystyle \text {AXIOM 2: } u + v = w + v \iff u = w.$$

Now for the definition of additive inverse.

$$\displaystyle a + b = 0 \iff a \text { and } b \text { are additive inverses of each other.}$$

Now for a convention of notation.

$$\displaystyle b = -\ a \iff b \text { is the additive inverse of } a.$$

Now we can deduce the needed conclusion.

$$\displaystyle \text {By notational convention, the additive inverse of } -\ a \text { is } -\ (-\ a).$$

$$\displaystyle \therefore \text {, by the definition of additive inverse, } -\ (-\ a) + (-\ a) = 0.$$

$$\displaystyle \text {Again by the definition of additive inverse, } a + (-\ a) = 0.$$

$$\displaystyle \therefore \text {, by AXIOM 1, } -\ (-\ a) + (-\ a) = a + (-\ a).$$

$$\displaystyle \therefore \text {, by AXIOM 2, } -\ (-\ a) = a.$$

I must admit that I see no reason whatsoever for this kind of formalism during first year algebra. Bourbaki for Adolescents.

EDIT: What makes it worse is that we already have enough confusion by using the minus sign for three distinct purposes: as an operator sign for subtraction, as a negative value indicator for numerals, and as an additive inverse indicator for pronumerals. We would be much better off making these distinctions clear rather than mucking around with foundations of mathematics as an introductory topic. It is logically prior, but people who are just past arithmetic cannot possibly see why it is germane to anything. Notice, by the way, that the deduction shown above has nothing formally to do with negative numbers although it can easily be extended to them.

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#### Dr.Peterson

##### Elite Member
The main point here is that mathdaughter has to answer this question within the context of what she has been taught, which presumably includes a short list of "properties, operations, and definitions". These would not be presented as an axiomatic system, just as facts that tell us what manipulations are valid. But we need to see that list in order to help; I can't find any single list online that includes everything that is needed.

#### JeffM

##### Elite Member
But we need to see that list in order to help; I can't find any single list online that includes everything that is needed.
And right there is part of the problem. If we see the list, I bet it will not include all that is needed to justify the student's reasoning. The student will be expected to supplement the list with "common sense." So the "rigor" being taught is an outright fraud. The student will have to rely in part on her intuition about numbers. Furthermore, the difficulty in understanding created by the historical accident of using the same symbol for three completely different concepts is neither acknowledged nor addressed by providing a list of rules and terminology rather than a comprehensible explanation.

I make no pretense of being a mathematician; I do not know any advanced mathematics. My academic training was in history. But I have been tutoring high school kids in math since I retired from business. And I have seen many bright kids driven from math by a superficial formalism that does not have any practical purpose for 99% of the kids who could benefit from a solid grasp of elementary algebra. I am angry at how badly served are our kids and their parents by our very expensive schools.

#### Dr.Peterson

##### Elite Member
And right there is part of the problem. If we see the list, I bet it will not include all that is needed to justify the student's reasoning. The student will be expected to supplement the list with "common sense." So the "rigor" being taught is an outright fraud. The student will have to rely in part on her intuition about numbers. Furthermore, the difficulty in understanding created by the historical accident of using the same symbol for three completely different concepts is neither acknowledged nor addressed by providing a list of rules and terminology rather than a comprehensible explanation.

I make no pretense of being a mathematician; I do not know any advanced mathematics. My academic training was in history. But I have been tutoring high school kids in math since I retired from business. And I have seen many bright kids driven from math by a superficial formalism that does not have any practical purpose for 99% of the kids who could benefit from a solid grasp of elementary algebra. I am angry at how badly served are our kids and their parents by our very expensive schools.
Yes, but let's try to help the student rather than just bash the system.

#### mathdaughter

##### New member
Can you list the properties you have been taught?

#### JeffM

##### Elite Member
Yes, but let's try to help the student rather than just bash the system.
And just WHY do you think I spend each Wednesday afternoon helping students from the local high school individually? Or come here every day? Nice try at deflection, Doctor. (By the way, are you a doctor of education?)

Trying to help individual students does not mean that I must forget about a system that gives a virtual monopoly of public education to the graduates of the weakest schools and weakest departments in what is laughably called the American university system (cf Evergreen State), about the monetary receipts of university education departments as they are paid to teach the latest, invariably unsuccessful, fads self-servingly dreamed up by their very own faculties, or about the profiteering by text book publishers as they put out new "best practices" texts like clockwork after the previous "best practices" have been proven not merely ineffective, but positively deleterious. Is there any metric that clearly demonstrates American education to be more effective now than it was fifty years ago?

This unbelievably corrupt "system" is an implicit but conscious conspiracy to enrich those who have a direct monetary incentive to fail in educating children. Teachers are bloodsuckers battening off the denial of an education to helpless children. The teachers and their accomplices legitimately have contempt for the taxpayers that they annually fleece. The worse the education, the more money can be extorted.

I shall not tolerate being maligned in my personal efforts to help individuals by the creatures of such a fundamentally evil "system." I know how much it must sting to have the callous greed of the education establishment mentioned, but I invest too much personal time and effort to individual children to find an attempt to silence me by implying that criticism is not enough.

#### Dr.Peterson

##### Elite Member
Thanks.

I imagine the "definition of subtraction" (which can be stated in several very different ways) is something like I suggested before,

a - b means a + (-b)

If so, then applying this to 3 - (-2) yields 3 + -(-2).

None of the listed properties seems likely to be

-(-a) = a

which is needed to get from that to 3 + 2, so I would suppose that this is considered to be just an unnamed fact -- that the negative of -2 is 2. That is, at this point we are just carrying out the negation operation. That is not listed as an operation, but perhaps you are expected to think of this as the "inverse property of addition" (which says simply that every number has an additive inverse, namely its negative). On the other hand, since going from 3 - (-2) to 3 + 2 is listed as just one step, "definition of subtraction" is probably what was intended. That's all I would say here.

Yes, this whole thing is a little too arbitrary and subjective; as was pointed out, there is something of a false rigor in this that is not always helpful to students. So don't let it worry you; the main point you are supposed to take away from this is not that you have to be really careful to name everything you do (it will not be required going forward), but that numbers follow certain patterns that form the basis of algebra, allowing you to know what changes you can make to an expression or equation without messing things up.