Number Theory logic

shahar

shahar

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I want to prove that "Adding 2 negative numbers can't give a positive number:"
For example:
(-a)+(-b)=5
I can I do it?
 
shahar said:
I want to prove that "Adding 2 negative numbers can't give a positive number:"
For example:
(-a)+(-b)=5
I can I do it?
Click to expand...
Please show what you've tried, so we can see what's going wrong.

In particular, we need to know what axioms or assumptions you are willing to take as the basis for a proof. A proof requires some context.
 
Given:
a > 0
b > 0
(-a)+(-b)=5
Hence,
(-1 * a) + (-1 * b) = 5
(-1) * (a+b)=5
a+b=-5 =>a+b!=5 Because -5 not equal 5.
I do all the steps by myself alone.
Can I get comments?
 
Proofs depend on axioms, definitions, and previously proved theorems. If we do not know what those are, we cannot comment.
 
shahar said:
(-a)+(-b)=5
Click to expand...
This suggests that your goal, rather than the general "Adding 2 negative numbers can't give [any] positive number" is "there are not two negative numbers whose sum is 5". Or are you thinking that one example proves the general claim? (It doesn't.)

shahar said:
a+b=-5 =>a+b!=5 Because -5 not equal 5.
Click to expand...
True, if (-a)+(-b)=5, it is not possible for a+b to equal 5; but that doesn't prove anything about your claim, because no one has said that a+b=5; there is no contradiction (yet).

Am I right that you are trying to show a contradiction?

In any case, you haven't shown your context, so it's hard to suggest what to try. That context might include not only what definitions, axioms, and theorems you have, but also why you want to prove this particular fact. Are you taking a course in number theory or in proof, or reading a book on such a topic? If this is based on an exercise there, quoting that exactly (in any language) could help us.
 
shahar said:
Given:
a > 0
b > 0
(-a)+(-b)=5
Hence,
(-1 * a) + (-1 * b) = 5
(-1) * (a+b)=5
a+b=-5 =>a+b!=5 Because -5 not equal 5.
I do all the steps by myself alone.
Can I get comments?
Click to expand...
Have you proven that the sum of two positive numbers is a positive number? (If you define a and b to be Natural numbers it almost doesn't need to be stated. You should easily be able to extend that to positive real numbers.) Then a + b = -5 would be a contradiction.

-Dan
 
shahar said:
I want to prove that "Adding 2 negative numbers can't give a positive number:"
For example:
(-a)+(-b)=5
I can I do it?
Click to expand...
If [imath]a=-2~\&~b=-3[/imath] then [imath](-a)+(-b)=5>0[/imath].
 
Dr.Peterson said:
This suggests that your goal, rather than the general "Adding 2 negative numbers can't give [any] positive number" is "there are not two negative numbers whose sum is 5". Or are you thinking that one example proves the general claim? (It doesn't.)


True, if (-a)+(-b)=5, it is not possible for a+b to equal 5; but that doesn't prove anything about your claim, because no one has said that a+b=5; there is no contradiction (yet).

Am I right that you are trying to show a contradiction?

In any case, you haven't shown your context, so it's hard to suggest what to try. That context might include not only what definitions, axioms, and theorems you have, but also why you want to prove this particular fact. Are you taking a course in number theory or in proof, or reading a book on such a topic? If this is based on an exercise there, quoting that exactly (in any language) could help us.
Click to expand...
Here the source:
It ask as a hidden remark in the text, so I don't thinking the writer of the text want to prove it, but she ask it in page - 2- in the sentence:
* האם יתכנו שני מספרים שליליים שסכומם
(5+)?​
 
shahar said:
Here the source:
It ask as a hidden remark in the text, so I don't thinking the writer of the text want to prove it, but she ask it in page - 2- in the sentence:
* האם יתכנו שני מספרים שליליים שסכומם
(5+)?​
Click to expand...

Yes, from the context, it appears that they are asking not for a proof, but for more informal reasoning, based on having tried to find such pairs of numbers.

Here is what Google gives for a translation of that exercise:

Version C - specific to the result (+5)​
1. Please give an example of two numbers whose sum is +5, so that one of them is positive and one of them is negative.​
2. Please give another example of two numbers whose sum is +5, so that one of them is positive and one of them is negative.​
3. How many pairs of numbers whose sum is (+5) exist, assuming that one of the numbers is positive and one of the numbers is negative?​
​
* Can there be two negative numbers whose sum is (+5)?​

So, what informal reasoning can you give for your answer to this question?
 
Dr.Peterson said:
Yes, from the context, it appears that they are asking not for a proof, but for more informal reasoning, based on having tried to find such pairs of numbers.

Here is what Google gives for a translation of that exercise:

Version C - specific to the result (+5)​
1. Please give an example of two numbers whose sum is +5, so that one of them is positive and one of them is negative.​
2. Please give another example of two numbers whose sum is +5, so that one of them is positive and one of them is negative.​
3. How many pairs of numbers whose sum is (+5) exist, assuming that one of the numbers is positive and one of the numbers is negative?​
​
* Can there be two negative numbers whose sum is (+5)?​

So, what informal reasoning can you give for your answer to this question?
Click to expand...
In clause 1:
I can give example:
(-7)+(2)
In 2:
(3)+(-8)
In 3:
Infinitive numbers of solution.
How can I continue?
 
shahar said:
How can I continue?
Click to expand...
You need to do what the paper is trying to get students to do: Think beyond just answering direct questions.

Here is what they said at the top:

The teaching of mathematics is based on practice of the subjects studied. Such practice is usually carried out through the solution of a large number of exercises, according to a predetermined algorithm, which may lead to the acquisition of skill in algorithmic solution of the exercises. The question arises, do students who have acquired this skill necessarily understand the subject being studied? In order to answer this question, it is advisable to include in the teaching of mathematics questions that may indicate an understanding of the mathematical subject being studied. One of the ways to construct such questions is by "inverting" the question. The reversal is done as follows: the final result answer of the question is taken, and the students are asked to determine what data of the question led to this result answer. Sometimes, it is necessary to slightly change the wording of the final result, in order to create a question that also requires understanding.​

So, rather than just answer the questions they first ask, tell us what you were thinking as you tried to solved them. How did you choose numbers to use? What convinces you that there are infinitely many answers?

Now think about the extra question, about adding two negatives. How would you try to find such a pair? What goes wrong when you do that? What about that might suggest, or even convince you, that there is no such pair?
 
I know
Dr.Peterson said:
You need to do what the paper is trying to get students to do: Think beyond just answering direct questions.

Here is what they said at the top:

The teaching of mathematics is based on practice of the subjects studied. Such practice is usually carried out through the solution of a large number of exercises, according to a predetermined algorithm, which may lead to the acquisition of skill in algorithmic solution of the exercises. The question arises, do students who have acquired this skill necessarily understand the subject being studied? In order to answer this question, it is advisable to include in the teaching of mathematics questions that may indicate an understanding of the mathematical subject being studied. One of the ways to construct such questions is by "inverting" the question. The reversal is done as follows: the final result answer of the question is taken, and the students are asked to determine what data of the question led to this result answer. Sometimes, it is necessary to slightly change the wording of the final result, in order to create a question that also requires understanding.​

So, rather than just answer the questions they first ask, tell us what you were thinking as you tried to solved them. How did you choose numbers to use? What convinces you that there are infinitely many answers?

Now think about the extra question, about adding two negatives. How would you try to find such a pair? What goes wrong when you do that? What about that might suggest, or even convince you, that there is no such pair?
Click to expand...
I know that in adding opposite numbers one number is positive than the negative number that in its' absolute value to get result of positive number ("5").
That is what I notice. Did what I find can help to answer? If the answer is yes so Why?
 
shahar said:
I know

I know that in adding opposite numbers one number is positive than the negative number that in its' absolute value to get result of positive number ("5").
That is what I notice. Did what I find can help to answer? If the answer is yes so Why?
Click to expand...
What you say here is not very clear, but if you mean that when you add two numbers with opposite signs, the sum is the difference of the absolute values, with the sign of the number with the larger absolute value, then, yes, that helps to answer the three questions. It is not a full explanation of your thinking.

But that doesn't touch the question you are asking about, which involves two numbers that are both negative. What is the rule in that case, and how would you use it to answer that question?

A set of rules like these can provide a substitute for formal axioms as the basis for a "proof". But a proof of any sort must say enough to convince the listener of the conclusion. If you want to learn to write proofs, writing carefully will be essential, perhaps even before you learn about axioms.
 
It occurs to me that you do not understand why the rule that the sum of two negative numbers is negative is true.

This may not be a proof that meets all the standards of modern rigor, but it may help

[math]a + (-a) = 0 \text { by definition.}\\ a + b = a \iff b = 0.\\ a + b > a \iff b > 0.\\ c > 0 \text { and } c + d = 0 \implies d \ne 0 \text { and } d \not > 0 \implies d < 0.\\ \text {But } d = - c.\\ \text {THUS, } a > 0 \implies - a < c. [/math]
Got that?

[math] p > 0 \implies - p < 0.\\ q > 0 \implies - q < < 0.\\ (p + q) > 0.\\ \{p + (-p)\} = 0 = \{q + (-q)\} \implies \\ \{p + (-p)\} + \{q + (-q)\} = 0 + 0 = 0.\\ \therefore \ (p + q) + \{(-p) + (-q) \} = 0 \implies \\ \{(-p) + (-q)\} < 0. [/math]
 
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