Understanding Positive and Negative Numbers: Proving Properties and Applications

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In summary, the speaker defines positive and negative numbers as equivalence classes of pairs of natural numbers, with the natural numbers being associated with the positive integers and the negative integers being associated with the negative of the corresponding positive integer. The properties mentioned ((-a)(-b)=ab and a<b then ac<bc for c>0 and ac>bc for c<0) can be proven, but are more tedious than anything else. The speaker provides a link to a paper they wrote on basic number systems for further clarification.
  • #1
C0nfused
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Hi everybody,
How do we define positive and negative numbers? Also, how do we prove that (-a)(-b)=ab and also that if a<b then ac<bc for c>0 and ac>bc for c<0 ?
Thanks
 
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What level explanation do you want?

One way to define the integers is to define an equivalence relation on the set of pairs of natural numbers: (a,b)~ (c,d) if and only if a+ d= b+ c. It's easy to show that that is an equivalence relation and so separates all such pairs in "equivalence classes". The set of integers IS the set of equivalence classes (with appropriately defined operations). There is exactly one equivalence class that consists of all pairs (a,a): that is, (a,b) with a= b. That turns out to be the additive identity and we call it "0". You can show that there is a one-to-one correspondence between equivalence classes [(a,b)] such that a> b and we associate that with the natural number n where a= b+n (all pairs in that class having the same n). We can show that [(a,b)]+ [(b,a)]= [(a,a)]= 0. If a> b so that [(a,b)] is associated with n, we call [(b,a)] "-n". The set of negative numbers.

The proof of the properties you mention are more tedious than anything else.

Here's a link to a paper I wrote on basic number systems:
http://academic.gallaudet.edu/courses/MAT/MAT000Ivew.nsf/ID/918f9bc4dda7eb1c8525688700561c74/$file/NUMBERS.pdf [Broken]
 
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  • #3
Thanks for your answer. I am not very familiar yet with these things, but I will check your paper and your answer and if i have any problems I will post them. Impressive work by the way!
 

1. What are positive and negative numbers?

Positive and negative numbers are values used to represent quantities that are greater than or less than zero, respectively. They are essential in mathematics and science for measuring and comparing values.

2. How do you identify a positive or negative number?

Positive numbers are always greater than zero and are often written with a plus sign (+) in front. Negative numbers are always less than zero and are often written with a minus sign (-) in front. Examples of positive numbers include 5, 10, and 3.5, while examples of negative numbers include -2, -15, and -0.5.

3. What is the relationship between positive and negative numbers?

Positive and negative numbers are opposite values on a number line. Positive numbers appear to the right of zero, while negative numbers appear to the left. The distance from zero to a positive number is the same as the distance from zero to its negative equivalent. For example, the distance from zero to 5 is the same as the distance from zero to -5.

4. How do you perform operations with positive and negative numbers?

Adding a positive number to another positive number results in a larger positive number. Subtracting a positive number from a positive number results in a smaller positive number. Adding a negative number to a positive number results in a smaller number (or subtracting a positive number from a negative number results in a larger negative number). Subtracting a negative number from a positive number results in a larger number (or adding a positive number to a negative number results in a smaller negative number).

5. Why are positive and negative numbers important?

Positive and negative numbers are important because they allow us to represent and manipulate values that are greater than or less than zero. This is crucial in fields such as science, economics, and engineering, where measurements and calculations often involve values that can be either positive or negative.

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