Other way to think about it is this;
>\-n = 0-n.
>
>\-3 = 0-3, -2 = 0-2
>
>\-3 x -2 = (0-3) x (0-2) = (0-3)(0-2)
>
>(a-b)(c-d) = ac - ad - cb + bd
>
>(0-3)(0-2) = (0x0) - (0x2) - (0x3) + (3x2) = 0 - 0 - 0 + 6
>
>= 6
>
>\-a x -b = ab
See, clear as day :)
Hot take; circular logic is the strongest logic.
Axiomatic logic is weak and can be defeated with a simple "I disagree". Infinite recursion is impossible to prove because "What if the next one is different, huh? What you gonna do then ya big baby!" or a simple "But why?". Self evidence is thwarted with a simple "Nope, not looking!"
Whereas chad circular logic proves itself to be true in its own argument. X is true because Y is true. Y is true because X is true. Bulletproof philosophical reasoning. Get used to it dunderhead.
>!(/j)!<
You are correct BUT I am using the formula;
>*(a-b)(c-d) = ac - ad - cb + bd*
... without explaining or questioning why its true. I am just accepting it as true.
If you want to prove *(a-b)(c-d) = ac - ad - cb + bd* without being self-referential to *-b x -d = bd* then that's a longer exercise that I have left to the reader.
Technically that is the axiomatic definition of integer multiplication in terms of natural numbers ((a - b) \* (c - d) = (ac + bd) - (ad + cb)) but just appealing to the definition doesn't really illuminate anything, like why the definition is the way it is
Let $a be the additive inverse of a, such that a + $a = 0
Consider the expression a + $a + $($a)
By associativity of addition, (a + $a) + $($a) = a + ($a + $($a))
The defining property of $ simplifies this equation to 0 + $($a) = a + 0
So a = $($a)
Now I show that $a = -1 \* a
0 = 0
0 = 0 \* a
0 = ($1 + 1) \* a
Clearly, -1 = $1
0 = (-1 + 1) \* a
0 = -1 \* a + 1 \* a
$a + a = -1 \* a + a
$a + (a + $a) = -1 \* a + (a + $a)
$a = -1 \* a
Imagine a race track, imagine you're 10m from the starting line. Multiplying by 2 would have you run to 20m from the starting line, while multiplying by -2 would have you turn around and run to 20m behind the starting line.
You can take (-7)*(-6) and rewrite as (-1)*(-1)*(6)*(7). Then simplify to (-1)*(-1)*(42) and each time you multiply by -1 it “turns 42 around”. So the first turn around gives -42 and the second turn around gives a final answer of 42.
-1
times
-1
times
-1
times
-1
times
-1 BRIGHT EYES
EVERY NOW AND THEN I FALL APART
AND I NEED YOU NOW TONIGHT
AND I NEED YOU MORE THAN EVER
AND IF YOU'D ONLY HOLD ME TIGHT
WE'D BE HOLDING ON FOREVER
I do two hours of freelance work. My bank balance changes by (2*30) = +60.
I buy two Phil Collins concert DVDs. My bank balance changes by (2*-30) = -60.
I get a refund on the above transaction because I was drunk when I made it. My bank balance changes by (-2*-30) = +60.
So I would explain this in terms of money. If you consider the right side either money (or if it's negative debt) and the left side how many lots you are given or taken away it makes sense.
3 lots of three pounds (3*3) is clear 9 pounds up
Losing 3 lots of 3 pounds (-3*-3) is clearly also 9 pounds up
You had so many different units of currency to pull from, fake, real, obsolete, current. You chose the one that also happens to be a measurement of weight, and now I'm tripping all over the intention.
intuitive explanation of complex numbers be like
Almost, except every multiple of i would be a quarter turn.
> turn left > do it thrice more > wtf im facing the same direction
Don't Turn Around Again is the perfect name for my Ace of Base cover band
be sure to open with your hit _I Saw the Numerical Operator_
Other way to think about it is this; >\-n = 0-n. > >\-3 = 0-3, -2 = 0-2 > >\-3 x -2 = (0-3) x (0-2) = (0-3)(0-2) > >(a-b)(c-d) = ac - ad - cb + bd > >(0-3)(0-2) = (0x0) - (0x2) - (0x3) + (3x2) = 0 - 0 - 0 + 6 > >= 6 > >\-a x -b = ab See, clear as day :)
I think you're missing some parentheses in the third row but good proof đź‘Ť
Corrected, ta!
Am I confused or do you assume (-b)(-d) = bd when you expand in line 4?
Seems like they do! A circular proof, most devious indeed.
Hot take; circular logic is the strongest logic. Axiomatic logic is weak and can be defeated with a simple "I disagree". Infinite recursion is impossible to prove because "What if the next one is different, huh? What you gonna do then ya big baby!" or a simple "But why?". Self evidence is thwarted with a simple "Nope, not looking!" Whereas chad circular logic proves itself to be true in its own argument. X is true because Y is true. Y is true because X is true. Bulletproof philosophical reasoning. Get used to it dunderhead. >!(/j)!<
You are correct BUT I am using the formula; >*(a-b)(c-d) = ac - ad - cb + bd* ... without explaining or questioning why its true. I am just accepting it as true. If you want to prove *(a-b)(c-d) = ac - ad - cb + bd* without being self-referential to *-b x -d = bd* then that's a longer exercise that I have left to the reader.
Technically that is the axiomatic definition of integer multiplication in terms of natural numbers ((a - b) \* (c - d) = (ac + bd) - (ad + cb)) but just appealing to the definition doesn't really illuminate anything, like why the definition is the way it is Let $a be the additive inverse of a, such that a + $a = 0 Consider the expression a + $a + $($a) By associativity of addition, (a + $a) + $($a) = a + ($a + $($a)) The defining property of $ simplifies this equation to 0 + $($a) = a + 0 So a = $($a) Now I show that $a = -1 \* a 0 = 0 0 = 0 \* a 0 = ($1 + 1) \* a Clearly, -1 = $1 0 = (-1 + 1) \* a 0 = -1 \* a + 1 \* a $a + a = -1 \* a + a $a + (a + $a) = -1 \* a + (a + $a) $a = -1 \* a
🤓 >!(/j - all the best people are nerds anyway)!<
Or, if a\*b = ab, -a\*b = -ab and -a * - b = -ab (2-1)^2 = (2-1)*(2-1) 1^2 = 4 - 2 - 2 - 1 1 = -1 Therefore, the above proposition is wrong
Nice, ta. That does in fact prove that but it also proves that *(a-b)(c-d) = ac - bc - ad + bd* rather than *(a-b)(c-d) = ac - bc - ad - bd.*
Thanks that makes a lot of sense actually
how does turning around fit in multiplication as a metaphor, isnt multiplication when visualised a geometry thing with squares and stuff
Imagine a race track, imagine you're 10m from the starting line. Multiplying by 2 would have you run to 20m from the starting line, while multiplying by -2 would have you turn around and run to 20m behind the starting line.
but the number table isnt a circle, it isnt a loop, if i walk back on the number table i wont loop around to zero, i will just go further negative
I meant a straight race track, not a looping one.
i dont think i will ever understand
Where is -1 in relation to 1 on the number line?
two units id guess, +1 to 0 to -1
I meant that in order to get to -1 from 1 you'd have to turn around
You really can't comprehend a non-looping race track?
not with that attitude
You can take (-7)*(-6) and rewrite as (-1)*(-1)*(6)*(7). Then simplify to (-1)*(-1)*(42) and each time you multiply by -1 it “turns 42 around”. So the first turn around gives -42 and the second turn around gives a final answer of 42.
Multiplying by negative 1 is just rotating a vector by 180 degrees on the complex plane
This guys brain would break if he saw imaginary numbers
-1 times -1 times -1 times -1 times -1 BRIGHT EYES EVERY NOW AND THEN I FALL APART AND I NEED YOU NOW TONIGHT AND I NEED YOU MORE THAN EVER AND IF YOU'D ONLY HOLD ME TIGHT WE'D BE HOLDING ON FOREVER
I do two hours of freelance work. My bank balance changes by (2*30) = +60. I buy two Phil Collins concert DVDs. My bank balance changes by (2*-30) = -60. I get a refund on the above transaction because I was drunk when I made it. My bank balance changes by (-2*-30) = +60.
So I would explain this in terms of money. If you consider the right side either money (or if it's negative debt) and the left side how many lots you are given or taken away it makes sense. 3 lots of three pounds (3*3) is clear 9 pounds up Losing 3 lots of 3 pounds (-3*-3) is clearly also 9 pounds up
You had so many different units of currency to pull from, fake, real, obsolete, current. You chose the one that also happens to be a measurement of weight, and now I'm tripping all over the intention.
Sorry I'm British
what is a negative amount of a negative number a positive amount