Algebraic explanation of - times - = +

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Discussion Overview

The discussion revolves around the algebraic explanation of the rule that states a negative times a negative equals a positive. Participants explore various interpretations and explanations of this concept, referencing historical texts and algebraic identities.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant presents a complex algebraic argument from Euler's "Elements of Algebra," attempting to illustrate the reasoning behind the rule that - times - = +.
  • Another participant offers a simpler explanation based on the concept of additive inverses, demonstrating that -1 times -1 equals 1 through a series of algebraic manipulations.
  • Some participants express confusion regarding the clarity and coherence of the argument presented in Euler's text, describing it as "rambling algebra identities."
  • A participant seeks clarification specifically on the argument from Euler's work, indicating a desire for a deeper understanding of the reasoning rather than the conclusion itself.
  • A link to the original text is provided to facilitate further discussion and examination of the argument in question.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the clarity of Euler's argument, with some finding it confusing while others express understanding of the underlying principle that - times - = +. The discussion remains unresolved regarding the interpretation of Euler's explanation.

Contextual Notes

The discussion highlights limitations in the clarity of the original text and the varying levels of understanding among participants. There are unresolved questions about the specific algebraic identities referenced and their implications for the rule in question.

G037H3
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I'm pretty sure this isn't a homework style question. o_O

could someone please explain this, I may be overly tired, but it's confusing me:

"A further illustration of this rule is generally given by algebraists as follows: First, we know that +a multiplied by +b gives the product +ab; and if +a be multiplied by a quantity less than b, as in b-c, the product must necessarily be less than ab; in short, from ab we must subtract the product of a, multiplied by c; hence a times (b-c) must be expressed by ab-ac; therefore it follows that a times -c gives the product -ac. If now we consider the product arising from the multiplication of the two quantities (a-b), and (c-d), we know that it is less than that of (a-b) times c, or of ac-bc; in short, from this product we must subtract that of (a-b) times d; but the product (a-b) times (c-d) becomes ac-bc-ad, together with the product of -b times -d annexed; and the question is only what sign we must employ for this purpose, whether + or -. Now we have seen from that the product ac-bc we must subtract a quantity less than ad, we have therefore subtracted already too much by the quantity bd; this product must therefore be added; that is, it must have the + sign prefixed, hence we see that -b times -d gives +bd for a product; or -minus multiplied by -minus gives +plus."
 
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A negative times a negative equaling a positive is best understood from what it means to be negative. If a + b = 0 we call b “-a” or we say that b is a’s additive inverse. Given that:
1 + -1 = 0
-1(1 + -1)=-1*0
-1(1 + -1) = 0
-1*1 + (-1)(-1) =0
So -1 + (-1)(-1) = 0
Thus -1’s additive inverse is (-1)(-1), but -1’s additive inverse is 1. So (-1)(-1) = 1.
From here it gets simple, consider
(-a)(-b) = (-1)a(-1)b = (-1)(-1)ab = 1ab = ab
 
I understand that completely. What I'm asking for is an explanation of what I posted.
 
The context part you have quoted isn't very clear. It's all just a bunch of rambling algebra identities.

What is it you are having problems understanding?
 
Tac-Tics said:
The context part you have quoted isn't very clear. It's all just a bunch of rambling algebra identities.

What is it you are having problems understanding?

I understand why - x - = +, i simply want to understand the argument given

it's in Euler's Elements of Algebra x.x
 

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