Algebraic explanation of - times - = +

In summary, the conversation is about understanding the rule of negative numbers in algebra and how to solve for the product of two negative numbers. The conversation includes a discussion of additive inverses and the use of algebraic identities to prove that a negative times a negative equals a positive. The speaker also provides a link to a resource for further explanation.
  • #1
G037H3
280
2
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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  • #2
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
 
  • #3
I understand that completely. What I'm asking for is an explanation of what I posted.
 
  • #4
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?
 
  • #5
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
 

FAQ: Algebraic explanation of - times - = +

What is the algebraic explanation for multiplying two numbers and getting a positive result?

The algebraic explanation for multiplying two numbers and getting a positive result is based on the properties of multiplication. In algebra, a positive number multiplied by another positive number will always result in a positive number. This is because when multiplying two positive values, the product will always be larger than either of the original values.

What is the algebraic explanation for multiplying two numbers and getting a negative result?

The algebraic explanation for multiplying two numbers and getting a negative result is based on the properties of multiplication. In algebra, a negative number multiplied by another negative number will always result in a positive number. This is because when multiplying two negative values, the product will always be larger than either of the original values, but it will also have a negative sign.

What is the algebraic explanation for multiplying a positive and a negative number?

The algebraic explanation for multiplying a positive and a negative number involves understanding the concept of opposite values. In algebra, a positive number multiplied by a negative number will always result in a negative number. This is because when multiplying two values with opposite signs, the product will always be smaller than either of the original values, but it will also have a negative sign.

What is the algebraic explanation for multiplying zero by any number?

The algebraic explanation for multiplying zero by any number is based on the property of zero. In algebra, any number multiplied by zero will always result in zero. This is because zero is considered to be a neutral number, meaning it has no effect on the value of the other number. So, no matter what number you multiply by zero, the product will always be zero.

What is the algebraic explanation for multiplying a number by itself?

The algebraic explanation for multiplying a number by itself is based on the concept of exponents. In algebra, when a number is multiplied by itself, it is the same as raising that number to the power of 2. This is known as the square of a number. The result of multiplying a number by itself is always a positive number, no matter if the original number was positive or negative.

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